Optical hybrid approaches to quantum information
Abstract
This article reviews recent hybrid approaches to optical quantum information processing, in which both discrete and continuous degrees of freedom are exploited. There are wellknown limitations to optical singlephotonbased qubit and multiphotonbased qumode implementations of quantum communication and quantum computation, when the toolbox is restricted to the most practical set of linear operations and resources such as linear optics and Gaussian operations and states. The recent hybrid approaches aim at pushing the feasibility, the efficiencies, and the fidelities of the linear schemes to the limits, potentially adding weak or measurementinduced nonlinearities to the toolbox.
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Schematics for the optical implementation of entanglement distribution between two stations in a hybrid quantum repeater.
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Contents
 1 Introduction
 2 Optical quantum information
 3 Hybrid approaches
 4 Summary and outlook
1 Introduction
Quantum information is a relatively young area of interdisciplinary research. One of its main goals is, from a more conceptual point of view, to combine the principles of quantum physics with those of information theory. Information is physical is one of the key messages, and, on a deeper level, it is quantum physical. Apart from its conceptual importance, however, quantum information may also lead to realworld applications for communication (quantum communication) and computation (quantum computation) by exploiting quantum properties such as the superposition principle and entanglement. In recent years, especially entanglement turned out to play the most prominent role, representing a universal resource for both quantum computation and quantum communication. More precisely, multipartite entangled, socalled cluster states are a sufficient resource for universal, measurementbased quantum computation [1]. Further, the sequential distribution of many copies of entangled states in a quantum repeater allow for extending quantum communication to large distances, even when the physical quantum channel is imperfect such as a lossy, optical fiber [2, 3].
Many if not most experiments related to quantum information are conducted with quantum optical systems. This includes the preparation, manipulation, and measurement of interesting and useful quantum optical states, in particular, entangled states; possibly supplemented by additional atomic systems for storing and processing quantum states.
Why is quantum optics the preferred field for quantum information demonstrations? Based on mature techniques from nonlinear optics for state preparation such as parametric down conversion, together with the most accessible means for manipulating optical states with linear elements such as beam splitters, there is a long list of optical proofofprinciple demonstrations of various quantum information processing tasks. Some of these experiments are performed with singlephoton states, leading to a discretevariable (DV) encoding of quantum information, where, for instance, a qubit space is spanned by two orthogonal polarizations (‘photonic qubits’) [4]. In other experiments, continuousvariable (CV) states, defined in an infinitedimensional Hilbert space, are utilized, for example, expressed in terms of the quadrature amplitudes of an optical, bosonic mode (‘photonic qumodes’) [5]. Typically, the DV experiments involve some heralding mechanism, rendering them conditional, and hence less efficient; nonetheless, fidelities in the DV schemes are fairly high [6]. Conversely, in the CV regime, unconditional operations and high efficiencies are at the expense of lower fidelities [7].
Beyond experimental smallscale demonstrations, towards potential realworld applications, light is clearly the optimal choice for communication. Moreover, in order to mediate interactions between distant matter qubits on a fixed array (for instance, in a solidstate system), individual photons or intense light pulses may be utilized. In such a scenario, the light field acts as a kind of quantum bus (‘qubus’) for applying entangling gates to the matter qubits. This kind of optical, qubusmediated quantum logic could become part of a full quantum computer. In addition, it is exactly this qubus approach which can be exploited for quantum communication in a quantum repeater, where the optical qubus propagates through a fiber (or even through free space) between neighboring repeater stations and locally interacts with each matter qubit for nonlocal, entangledstate preparation of the two distant qubits. In analogy to classical, optical/electronic hybrid computers, the above schemes may be referred to as ‘hybrid’ approaches, as they combine the useful features of both light and matter; the former as an ideal medium for communicating, the latter well suited for storing quantum information.
In addition to the hybrid notion mentioned in the preceding paragraph, there is a related, but somewhat different definition of (optical) hybrid quantum information protocols. These are inspired by practical as well as fundamental limitations of those optical quantum information schemes, which are solely based upon either discrete or continuous degrees of freedom. A hybrid scheme, similar to a classical, digital/analog hybrid computer, would then exploit at the same time both DV and CV states, encodings, gates, measurements, and techniques, in order to circumvent those limitations.
1.1 CV versus DV
It is well known that a very strong version of universal quantum computation (‘CV universality’), namely the ability to simulate any Hamiltonian, expressed as an arbitrary polynomial of the bosonic mode operators, to arbitrary precision, is not achievable with only linear transformations, i.e., Gaussian transformations [8]. Gaussian transformations are rotations and translations in phase space, as well as beam splitting and squeezing unitaries, all transforming Gaussian states into Gaussian states. Similarly, a fully Gaussian qumode quantum computer can always be efficiently simulated by a classical computer [9]. A single nonGaussian element such as a cubic Hamiltonian would be sufficient to both achieve universality and prevent classical simulability. In general, however, nonGaussian transformations are difficult to realize on optical Gaussian states; nonetheless deterministic protocols for approximating such gates exist [10] (the ‘GKP’ scheme).
Although the physical states in quantum optical approaches, representing quantized harmonic oscillators, would always live in infinitedimensional Fock space, there is a weaker, but possibly more useful notion of universality (‘DV universality’). It refers to the ability to approximately simulate any DV multiqubit unitary with a finite set of gates, logically acting on a finite subspace of the infinitedimensional optical Fock space, spanned by states with only a few photons (photonic qubits). In this case, a universal set must also contain a nonlinear interaction, unless we accept probabilistic operations [11] (the ‘KLM’ scheme). Similarly, exactly fulfilling finite tasks supposedly simpler than universal quantum computation, such as a complete, photonic twoqubit Bell measurement, would be impossible with only linear elements (including squeezers) [12, 13, 14]. The problem is that nonlinear interactions on the level of single photons are hard to obtain. It is very difficult to make two photons “talk” to each other.
1.2 Going hybrid
In a hybrid scheme, where DV and CV degrees of freedom are exploited at the same time and the goal is to circumvent the limitations of the linear approaches, it can be useful to consider two special notions of nonlinearity: one is that of weak nonlinearities, the other one is that of measurementinduced nonlinearities. The former one would then be effectively enhanced through the use of sufficiently intense light fields; an approach first utilized for quantum nondemolition measurements with CV states [15]. More recently, weak nonlinearities were applied to combined DVCV, i.e., hybrid systems, where the weak nonlinear interaction is not only enhanced, but also mediated between the DV components through a bright CV qubus state. This enables one to perform various tasks from projecting onto the complete, photonic DV Bell basis to implementing universal, photonic twoqubit entangling gates, using either DV (threshold) photon detectors [16], CV homodyne detectors [17, 18, 19, 20], or no detectors at all [21, 22].
The concept of a measurementinduced nonlinearity was initiated by the seminal works of KLM [11] and GKP [10]. The KLM protocol is a fully DV scheme, using complicated, entangled, multiphoton ancilla states, measurements of photon number, and feedforward, whereas the GKP scheme may be considered one of the first hybrid protocols. It relies upon DV photon number measurements, to create nonGaussian states from CV Gaussian resources, and to eventually realize nonGaussian (cubic) CV gates. The GKP proposal also contains the hybrid concept of encoding logical DV states into physical CV states.
At this point, before going into more detail in the following sections, let us summarize the key elements of the optical hybrid approaches to quantum information processing reviewed in this article. The goal of circumventing the limitations of the most practical, linear optical schemes, maintaining to some extent their feasibility and efficiency, and still achieving a true quantum advantage over classical schemes may be reached through all or some of the following ingredients:

hybrid states and operations, i.e., a combination of DV and CV elements,

qubus systems for mediating entangling gates,

weak nonlinearities,

measurementinduced nonlinearities.
The paper is organized as follows. In Section 2, we give a brief introduction to optical quantum information. This includes a description of how to encode quantum information into photonic qubits and qumodes, how to process such quantum information using linear and nonlinear transformations, and how these tools may be exploited to achieve (theoretically and experimentally) efficient (scalable?) quantum computation and communication. Section 3 then presents the concept of optical hybrid protocols, discussing various hybrid schemes for both quantum computation and communication. Finally, we summarize and conclude in Section 4.
2 Optical quantum information
How can we encode quantum information, for instance, a qubit, into optical states? Do quantum optical states naturally allow for different types of encoding? These questions, together with the issue of processing quantum information encoded in optical states in an efficient way, will be addressed in this section. In particular, we discuss that there are qualitatively different levels of ‘efficient’ optical quantum information processing. These depend on the type of encoding, and on the scalability and the feasibility of the optical resources necessary for their implementation.
2.1 Linear versus nonlinear operations
In Fig. 2.1.1, a table is shown summarizing possible optical interactions and transformations for state preparation and manipulation.
The most accessible and practical interactions are those described by linear and quadratic Hamiltonians. Here, quadratic refers to the order of a polynomial of the mode operators for the modes that participate in the interaction. Each () corresponds to the annihilation/lowering (creation/raising) operator of a quantized harmonic oscillator, each representing a single mode from a discrete set of (spatial, frequency, polarization, etc.) modes into which the electromagnetic field is most conveniently expanded [23].
2.1.1 Optical interactions and transformations in terms of the annihilation and creation operators representing a discrete set of modes of the optical field.
The unitary transformations generated from quadratic Hamiltonians are linear. An arbitrary quadratic Hamiltonian transforms the mode operators as
(1) 
Here, the matrices and satisfy the conditions and according to the bosonic commutation relations . This transformation is also referred to as linear unitary Bogoliubov transformation (LUBO) [24]. It combines passive and active elements, i.e., beam splitters and squeezers, respectively; the ’s in Eq. (1) describe phasespace displacements and come from the linear terms in the Hamiltonian. As an example, in a singlemode squeezer, , the quadrature would be ‘squeezed’, , and the quadrature correspondingly ‘antisqueezed’, , with and . The quadratures here are dimensionless variables playing the roles of position and momentum with .
Comparing the LUBO to a purely passive (photon number preserving), linear transformation,
(2) 
with an arbitrary unitary matrix , we observe that there is no mixing between the annihilation and creation operators in the passive transformation. Despite this difference, also the active, more general LUBO is only linear in the mode operators. Therefore, general linear optical transformations are here referred to as LUBOs, including squeezers. As squeezing, however, typically involves a nonlinear optical interaction (such as ) ^{1}^{1}1for which the actual, fully quantum mechanical Hamiltonian is cubic; with the usual parametric approximation, considering the socalled pump field classical, the pump mode operator becomes a complex number which is then absorbed into the squeezing parameter of the resulting quadratic Hamiltonian., it may as well be excluded from the ‘linearoptics’ toolbox (see Fig. 2.1.1).
Initially, squeezing was not really considered a useful tool for DV quantum information processing. Moreover, it is hard to apply squeezing to an arbitrary state such as a photonic qubit (see next section) ‘online’ in a controlled and efficient way. Hence, usually, squeezing will be explicitly excluded from the linearoptics toolbox for DV quantum information [6, 11]. In the CV approaches, however, optical squeezed states are the essential resource for creating Gaussian CV entangled states [7]. In this case, squeezed states are first created ‘offline’ and then linearly transformed, according to the passive transformation in Eq. (2). In more recent experiments, it was demonstrated that even online squeezing may be shifted offline using squeezed ancilla states [25, 26, 27]. Only these very new approaches would allow for efficient online squeezing of photonic DV states, as potentially needed in hybrid schemes.
In the hybrid context, and in the view of the recent offlinesqueezing experiments [25, 26, 27], it is sensible to define the complete set of linear resources and operations as all offline prepared, optical Gaussian states and all general LUBOs which are equivalent to Gaussian transformations mapping a Gaussian state back to a Gaussian state, see Fig. 2.1.1. Particularly practical resources here are coherent states, as these are readily and directly available from a laser source. To these coherentstate sources, we then add deterministic, online squeezing, and as a Gaussian, linear measurement, homodyne detection.
The link between the elementary quantum optical devices such as phase shifters, beam splitters, and singlemode squeezers on one side and an arbitrary LUBO as in Eq. (1) on the other side is provided through two important results:
The former result, the socalled ‘BlochMessiah reduction’, can be derived through singular value decomposition, with and , a pair of unitary matrices and , and nonnegative diagonal matrices and , [28]. The two results together imply that any multimode LUBO, i.e., any linear multimode transformation as in Eq. (1), can be implemented with singlemode phase shifters, singlemode squeezers, and twomode beam splitters. The displacements in Eq. (1) (the ’s) can be also realized using highly reflective beam splitters.
As shown in Fig. 2.1.1, going beyond the regime of linear resources and operations means to include cubic or higherorder interactions leading to nonlinear transformations. Such a nonlinear interaction would normally map a Gaussian state onto a nonGaussian state, described by nonGaussian Q and Wigner functions, see Fig. 2.1.2. These interactions are typically very weak; an example would be the extremely weak Kerr effect in an optical fiber. Therefore, for sufficiently long interaction times, unwanted photon losses will normally dominate over the desired nonlinear transformation.
Rather than performing nonlinear transformations online, we may first create offline nonlinear resources [10, 11] such as photon number (Fock) states as well as other nonGaussian states such as ‘cat states’ (i.e., superposition states of coherent states, see Fig. 2.1.2 on the right). Typically, this offline preparation would be probabilistic, i.e., conditional, depending on, for instance, certain photon number measurement outcomes for a subset of modes [30, 31, 32].
2.1.2 Examples of singlemode states generated through highly nonlinear interactions. Shown is the Q function for a Gaussian coherent state evolving into various nonGaussian states subject to a quartic selfKerr interaction with .
Let us now explicitly consider the encoding and processing of quantum information using optical resources and linear/nonlinear optical transformations.
2.2 Qubits versus qumodes
The information being processed through a quantum computer is most commonly represented by a set of twolevel systems (qubits), in analogy to classical, digital bit encoding. These qubit states could be superpositions of two different, electronic spin projections, or, in the optical context, superpositions of two orthogonal polarizations. As the bosonic Fock space is infinitedimensional, however, there is much more room for encoding quantum information into optical states. Allowing for states with more than a single photon per mode is a possible way to represent multilevel systems. Alternatively, a multilevel state can be expressed through sufficiently many modes with at most one photon in total.
Apart from discrete photon numbers, an optical state may be described by its amplitude and phase. The corresponding quantum phasespace variables could be considered the quantum analogues of classical, analog encoding. Such phasespace representations would completely determine the state of a quantized optical mode (qumode).
2.2.1 Photonic qubits
Consider the free electromagnetic field with a Hamiltonian with photon number operator for mode . Note that the sum includes the zeropoint energy ‘1/2’ for every mode. Then the number (Fock) states , eigenstates of , form a complete, orthogonal basis for each mode. Dropping the mode index, we have the wellknown relations for annihilating and creating photons, and , respectively. The vacuum state, containing no photons, is defined as .
Using this number basis, there are now (at least) two different ways to encode an optical qubit. The first encoding is called ‘singlerail’ (or ‘occupation number’), as it relies upon just a single optical mode,
(3) 
This encoding, however, is rather inconvenient, because even simple singlequbit rotations would require nonlinear interactions. For example, the Hadamard gate, acting as , transforms a Gaussian state (the vacuum) into a nonGaussian state (a superposition of vacuum and onephoton Fock state).
In contrast, for the socalled ‘dualrail’ encoding,
(4) 
singlequbit rotations become an easy task (see Fig. 2.2.1). A 50:50 beam splitter, for instance, would turn into , and similarly for the other basis state. The linear transformation here is a simple, special case of the general passive transformation in Eq. (2), and the two modes are spatial.
2.2.1 Using a beam splitter to switch between the computational and the Hadamard transformed, conjugate basis. Here, measuring the photons at the output of the beam splitter would project the input state onto the conjugate basis.
The most common dualrail encoded, photonic qubit is a polarization encoded qubit,
(5) 
for two polarization modes, where one is horizontally, the other one vertically polarized. So polarization encoding is by no means different from dualrail encoding; it is rather a specific manifestation of dualrail encoding. Singlequbit rotations are then particularly simple, corresponding to polarization rotations.
The drawback of the dualrail encoding is that for realizing twoqubit entangling gates, it is necessary to make two photons (each representing a dualrail qubit) ‘talk’ to each other. This kind of interaction between two photons would require some form of nonlinearity. Later we will discuss various possibilities for such twophoton entangling gates. We will also discuss an extension of dualrail to multiplerail encoding, where every logical basis state is represented by a single photon that can occupy any one of sufficiently many, different modes (not just two as for dualrail encoding).
Towards photonic qubit processing, the most accessible, optical resources and their optical manipulation can be summarized as follows.
Resources: singlephoton states (i.e., conditionally prepared Fock states, superpositions of Fock states), producible through nonlinear interactions (i.e., parametric down conversion); polarizationencoded states; singlephoton states approximated by weak coherent states.
Processing: passive linear optics (beam splitters, phase shifters, polarization rotators).
Measurements: photon counting; on/off detectors.
However, creating a Fock state with many photons is hard, as well as counting large photon numbers. For this purpose, the on/off detector is more realistic, as it does not discriminate between different photon numbers, but only between the vacuum state (‘no click’) and the nonvacuum state (‘click’).
For efficiently generating and measuring states with many photons, it is more practical to enter the regime of Gaussian states with CV homodyne detections (see next section). We may further add squeezing to the above toolbox. The truly nonlinear regime for processing is attainable by including, for instance, measurementinduced nonlinearities plus feedforward operations [11].
Despite the difficulty for realizing a twophoton entangling gate, there is a clear advantage of the singlephoton encoding. Single photons are fairly robust against noise. Therefore, typically, processing singlephoton states can be achieved with high fidelity, though, in most cases, only conditional operations are possible, at potentially very low success probabilities. As extra resources for processing DV quantum information, the atomic counterpart of the photonic polarization (spin) states are the electronic spin states whose qubit representations we shall introduce and utilize later.
We will now introduce a kind of complementary way to encode quantum information, in terms of qumodes. This type of encoding leads to states which are rather sensitive to noise, but which can be processed in an unconditional fashion; even entangling gates can be achieved through deterministic, linear optics.
2.2.2 Photonic qumodes
Consider again the free electromagnetic field with a Hamiltonian with photon number operator for mode . This time we shall rewrite the Hamiltonian as , with the position and momentum operators and for each oscillator (mode) . The zeropoint energy ‘1/2’ is still present and becomes manifest in the vacuum fluctuations of the position and momentum of every mode.
Further, we have the commutators and . After rescaling and into dimensionless variables, we arrive at with , corresponding to , as before. The position and momentum (quadrature) eigenstates may serve as a CV basis to represent the infinitedimensional state of an optical qumode.
The vacuum state can now be written, for example, in the position basis, , with the wave function . It must not be confused with the (unphysical, unnormalized) zeroposition eigenstate . The position probability distribution of the vacuum state is a normalized Gaussian, . The first and second moments of the vacuum state are easily calculated as (dropping the mode index) and , and similarly for the momentum. Thus, the quadrature vacuum variances are . The Wigner function of the vacuum state is , with . For finite and , with , we obtain the Wigner function for a displaced vacuum, i.e., a coherent state .
2.2.2 Encoding of qumodes into CV Gaussian squeezed and coherent states. Largely squeezed states would approximately resemble position or momentum eigenstates.
The coherent state is also the eigenstate of the nonHermitian annihilation operator, , with, correspondingly, complex eigenvalues . Its mean photon number is therefore . As a displaced vacuum, it may be written as , with the wellknown displacement operator ^{2}^{2}2whenever there is no ambiguity we may drop operator hats.
(6) 
Further, with regard to photon number, the coherent state obeys Poissonian statistics, and can be expanded as
(7) 
As opposed to the linear displacement operator, the squeezing operator, , with , is quadratic in and . Together with the quadratic singlemode phase rotations (by an angle ),
(8) 
and the quadratic twomode beam splitter interactions, this completes the linear regime of Gaussian transformations. Note that
(9) 
as can be easily understood from Eq. (7).
2.2.3 Optimal unambiguous state discrimination of equally likely, binary coherent states using a beam splitter, an ancilla coherent state, and on/off detectors. The inconclusive event here corresponds to the detection of the twomode vacuum for the two output modes with probability .
In the linear, CV Gaussian regime, the optical encoding into qumodes (see Fig. 2.2.2) is achieved either through approximate eigenstates (largely squeezed states), for which projection measurements are well approximated by homodyne detections; alternatively, the overcomplete and nonorthogonal set of coherent states may serve as a basis for qumodes. Perfectly projecting onto this basis is only possible for sufficiently large amplitudes , for which the coherent states become nearorthogonal. Nonetheless, two coherent states can also be unambiguously discriminated in the regime of small amplitudes using a beam splitter, an ancilla coherent state, and on/off detectors (see Fig. 2.2.3, real). This unambiguous state discrimination (USD) is probabilistic, but errorfree.
Remarkably, the linear optical scheme for the USD of two arbitrary coherent states [33] such as achieves the quantum mechanically optimal USD for two pure nonorthogonal states , where the success probability for a conclusive result equals (assuming real) [34, 35, 36]. For discriminating a larger set of coherent states, optimal USD becomes more subtle. The quantum mechanical optimum for more than two symmetrically distributed coherent states can be approached using linear optics and feedforward [37].
Let us summarize the most common optical resources and potential optical manipulations of qumodes.
Resources: Gaussian states; squeezing by means of nonlinear interactions (optical parametric amplification); nonGaussian states (e.g., ‘cat states’), producible, in principle, directly from nonlinear interactions.
Processing: Gaussian transformations (LUBOs: phasespace displacements, passive linear optics, active squeezers).
Measurements: Gaussian measurements (e.g., homodyne detection); nonGaussian measurements (e.g., on/off detectors).
The creation of nonGaussian resource states such as ‘cat states’ becomes more feasible when conditional state preparation is allowed. In this case, a hybrid approach is useful, as discussed later. A drawback of the CV qumode encoding is that these states are fairly sensitive to losses and noise. The quality of CV Gaussian entangled states, unconditionally and efficiently producible from squeezed light through beam splitters, is fundamentally limited by the constraint of finite squeezing (energy). Fidelities drop quickly in the presence of excess thermal noise, or simply when photons leak into the environment. Nonetheless, most operations are unconditional and homodyne detection is possible with nearunit efficiency.
Finally, once again we note that additional atomic systems and their degrees of freedom may be utilized. In the case of sufficiently large atomic ensembles, the collective spin variables can play the analogous roles of the CV qumode phasespace variables.
2.3 Implementing efficient quantum computation efficiently?
A necessry criterion for a quantum computer to give a true advantage over classical computers is that its realization does not require exponential resources. In other words, the exponential ‘speedup’ quantum computation is usually associated with must not be at the expense of an exponential increase of physical resources. The exponentially large dimension of the Hilbert space of logical qubits, , should be exploited with a number of physical resources scaling as (or a polynomial of ) rather than . If this criterion is satisfied, the quantum computation is considered to be ‘efficient’.
Besides this theoretical, inprinciple ‘efficiency’, an actual physical implementation of a quantum computation should be experimentally ‘efficient’ as well. While the former type of efficiency is fundamental, the latter one depends on current technology, and an inprinciple efficient quantum computation protocol may be infeasible and impractical today, but implementable in the future. Before we shall assess some of the existing proposals for optical quantum computation with regard to these criteria, let us first recall the most commonly used quantum gate sets for DV as well as for CV universal quantum computation.
2.3.1 Universal sets
For several qubits, a combination of arbitrary singlequbit rotations with one fixed twoqubit entangling gate is known to be sufficient for universality such that any unitary gate can be exactly realized on any given multiqubit state [38]. This universal set, however, is too large (in fact, it is infinitely large) to be implemented in an errorresistant fashion. Therefore, a discrete, finite set of elementary gates must be chosen which no longer achieves exact multiqubit gates (as the set of unitary gates is continuous), but rather an approximate realization to arbitrary precision. To be efficient, a sufficiently good approximation must not require an exponential number of elementary gate applications. A convenient universal set of gates is
(10) 
where we omitted the operator hats. Here, is the Hadamard gate, , and acts as an entangling gate, with
(11) 
The gates describe singlequbit rotations about the axis by an angle with being one of the usual Pauli operators , and , . Note that removing the gate from the elementary gate set means that only socalled Clifford unitaries can be realized which are known to be insufficient for a quantum computational speedup over classical computation. For both universality and speedup when computing with computational basis states, the nonClifford phase gate must be included here.
Universality for qumodes can be defined and achieved similarly to the qubit case. Arbitrary singlequmode transformations, together with beam splitters for two qumodes, are sufficient to simulate any Hamiltonian expressed as an arbitrary polynomial of the qumode position and momentum operators [8]. As a discrete, elementary gate set for approximate simulations to any precision, one may choose [9]
(12) 
with , , and real. ^{3}^{3}3commonly, both in the DV and the CV case, the CNOT or SUM gates are used for the canonical entangling gate instead of ; however, these are equivalent up to local DVHadamard or CVFourier transformations. In this case, represents the Fourier transform operator to switch between the position and momentum basis states, . The entangling gate is an controlled displacement, , with
(13) 
while the roles of the Pauli gates are now played by the WeylHeisenberg (WH) momentum and position shift operators, and , respectively, and . Finally, the phase gates are included in order to simulate any Gaussian (CV Clifford) transformation () and to achieve full CV universality including nonGaussian (CV nonClifford) gates (). Recall that the Gaussian transformations map Gaussian states onto Gaussian states; they correspond to quadratic Hamiltonians with linear inputoutput relations for the qumode operators as in Eq. (1). Gaussian operations on Gaussian states can be efficiently simulated classically [9].
Both for the DV and the CV case, for those encodings discussed so far, there is always at least one universal gate that is not realizable through linear transformations alone. In singlephoton singlerail encoding, even a singlequbit Hadamard gate, transforming a Gaussian vacuum state into a nonGaussian superposition of vacuum and onephoton Fock state would be highly nonlinear. The hardest part of universally processing dualrail encoded qubits would be the entangling gate which has to act upon at least two photons. Ultimately, the universal processing of even a single qumode requires some form of nonlinearity.
2.3.2 Nonlinear versus linear optics
The most obvious approach now to optically implement an entire set of universal quantum gates would be directly through nonlinear interactions. The twoqubit gate is accomplished by applying a quartic crossKerr interaction on two photonic occupation number qubits,
(14) 
The same interaction leads to a gate for two photonic dualrail qubits, with the crossKerr interaction acting on the second rail (mode) of each qubit such that only the term acquires a sign flip. This is the conceptually simplest and theoretically most efficient (only one optical device needed!) method to complete the set of universal gates in dualrail encoding. In the lowphoton number subspace here, we may even decompose the crossKerr twomode unitary into a beam splitter, two selfKerr onemode unitaries,
and another beam splitter (see Fig. 2.3.1).
Thus, a sufficiently strong onemode selfKerr interaction would be enough to fulfill the criteria for DV universality [38] on the finitedimensional multiqubit subspace of the infinitedimensional, multimode optical Fock space. At the same time, the quartic onemode selfKerr interaction, together with Gaussian, linear transformations (LUBOs), would be also sufficient for the strong notion of full (asymptotically arbitrarily precise) CV universality [8].
2.3.1 Implementing a controlled sign gate () on two singlerail qubits using crossKerr (CK) or selfKerr (SK) nonlinearities. The first beam splitter (BS) transforms the term into , while the other terms stay in the vacuum and onephoton space. As the SK interactions affect sign flips only for the twophoton components, only the term acquires a sign flip.
The problem of this approach, however, is that an effective coupling strength of for the self/crossKerr interactions is totally infeasible on the level of single photons. Therefore, it is worth examining carefully if there is a way to implement universal quantum gates through linear optical elements, ideally just using beam splitters and phase shifters. A very early proposal for linearopticsbased quantum computation indeed does work with only linear elements [39]. It is based upon socalled multiplerail encoding, where a level system is encoded into a single photon and optical modes, with the basis states , . Any unitary operator can be realized in the space spanned by this basis, as we only need , . This linear transformation, as in Eq. (2), is easily achieved through a sequence of beam splitters and phase shifters [29].
As a result, universal quantum computation is, in principle, possible using a single photon and linear optics. This kind of realization would be clearly efficient from an experimental point of view. In fact, implementing a universal twoqubit gate in a dimensional Hilbert space, would only require the modest set of resources of an optical ‘ququart’, in ‘quadrail’ encoding corresponding to a single photon and four optical modes. In fact, for small quantum applications, by adding to the polarization of the photons (their spin angular momenta) extra degrees of freedom such as orbital angular momenta, this kind of approach can be useful [40, 41, 42, 43].
Nonetheless, the drawback of the multiplerailbased linearoptics quantum computer [39] is its bad scaling. In theory even, this type of quantum computer is inefficient. Scaling it up to computations involving qubits, we need basis states, and hence optical modes. All these modes have to be controlled and processed in a linear optical circuit with an exponentially increasing number of optical elements. For example, a 10qubit circuit would only require 10 photons and 20 modes in dualrail encoding, while it consumes modes (for just a single photon) and at least as many optical elements in multiplerail encoding.
2.3.3 Teleportationbased approaches
There are now indeed a few proposals that aim at circumventing the inprinciple (scaling) inefficiency of the linear multiplerail protocol and the experimental infeasibility of the direct nonlinear optical approach. Certainly, the two most important and conceptually unique proposals are the KLM [11] and the GKP [10] schemes. In both schemes, a new concept of measurementinduced nonlinearities is exploited. The KLM scheme is a fully DVbased protocol, demonstrating that, in principle, passive linear optics and DV photonic auxiliary states are sufficient for (theoretically) efficient, universal DV quantum computation. Inducing nonlinearity through photon counting measurements renders the KLM scheme nondeterministic. However, the probabilistic quantum gates can be made asymptotically neardeterministic by adding to the toolbox feedforward and complicated, multiphoton entangled auxiliary states with sufficiently high photon numbers, and by employing quantum teleportation [44]. KLM is ‘inprinciple efficient’, as the number of the ancillary photons grows only polynomially with the success rate. Fidelities are always, in principle, perfect in the KLM approach.
We shall briefly describe the basic elements of KLM. The GKP scheme combines linear CV resources with linear operations and nonlinear measurements; therefore, a discussion of GKP, which achieves both faulttolerant DV universality and (nonfaulttolerant) CV universality, in the spirit of Ref. [8], is postponed until the section on hybrid schemes. Similarly, the alternate concept of weak nonlinearities relies on hybrid systems, and will also be discussed later.
2.3.2 Implementing a probabilistic controlled sign gate (CSIGN ) on two singlerail qubits using two nondeterministic NSS gates. The resulting twoqubit gate works in a similar way to the deterministic implementation described in Fig. 2.3.1 using Kerr nonlinearities.
The essential ingredient for the nondeterministic realization of a twophoton twoqubit entangling gate (in dualrail encoding) is the onemode nonlinear sign shift (NSS) gate [11]. It acts on the qutrit subspace of the optical Fock space as . Placing two such NSS gates in the middle between two beam splitters will then act as a controlled sign gate, , on two singlerail as well as two dualrail qubits. In fact, we may replace the deterministic Kerrbased circuit of Fig. 2.3.1 by the equivalent circuit depicted in Fig. 2.3.2. The latter, however, becomes nondeterministic with NSS gates operating only probabilistically.
In the original KLM proposal, the NSS gate can be realized with success probability, corresponding to a success probability of for the full controlled sign gate as shown in Fig. 2.3.2. In subsequent works, this efficiency was slightly improved [45]. There are also various, more general treatments of these nondeterministic linearoptics gates deriving bounds on their efficiencies [46, 47, 48]. Experimental demonstrations were reported as well [49], even entirely in an optical fiber [50].
Probabilistic quantum gates cannot be used directly for quantum computation. The essence of KLM (see Fig. 2.3.3) is that nearunit success probabilities are attainable by combining nondeterministic gates on offline entangled states with the concept of quantum gate teleportation [44]. As the necessary Bell measurements for quantum teleportation succeed at most with probability, if only fixed arrays of beam splitters are used [51], entangled ancilla states and feedforward must be added to boost efficiencies beyond to near .
2.3.3 Making nondeterministic gates neardeterministic through singlerail quantum teleportation. The Bell measurement is performed by means of the linearoptics circuit plus photon counting. For an entangled twomode state with one ancilla photon, teleportation succeeds only in one half of the cases. For larger ancillae with sufficiently many photons, teleportation can be made almost perfect. In order to teleport a gate neardeterministically onto an input state, the corresponding gate must be first applied offline and probabilistically to the multiphoton entangled ancilla state.
Even though KLM is ‘inprinciple efficient’, it is still highly impractical, as neardeterministic operations would require ancilla states too complicated to engineer with current experimental capabilities. It is therefore extremely important to further enhance the efficiencies of linearoptics quantum computation with regard to the resource scaling. Steps into this direction have been made already by merging the teleportationbased KLM approach with the fairly recent concept of oneway (cluster) computation [1].
2.3.4 Clusterbased approaches
In the preceding section, we introduced the notion of measurementinduced nonlinearities, which, combined with the more general (and implementationindependent) concept of measurementbased quantum computation, enables one to obtain the necessary nonlinear element in (linear) optical approaches to quantum information processing. As opposed to the standard, circuit model of quantum computing, where any computation is given by a sequence of reversible, unitary gates, in measurementbased quantum computing, universal quantum gates are encoded ‘offline’ into an entangledstate resource; suitable measurements, performed ‘online’ on this resources state, and, typically, some form of feedforward will then lead to the desired unitary evolution. Feedforward may sometimes be postponed until the very end of the computation, or even totally omitted through ‘reinterpretation’ of the ‘Pauliframe’; nonetheless, in some form it will be needed in order to render measurementbased quantum computation deterministic despite the randomness induced by the measurements.
There are now further subcategories of measurementbased quantum computing. First, that based on full quantum teleportation [44] involving online nonlocal measurements such as Bell measurements, as described in the preceding section. Secondly, there is an ultimate realization of measurementbased quantum computing that requires all entangling operations be done offline and allows only for local measurements applied on the offline resource state – the cluster state [1]. In such a cluster computation, a multiparty entangled cluster state is first prepared offline. The actual quantum computation is then performed solely through singleparty projection measurements on the individual nodes of the cluster state. By choosing appropriate measurement bases in each step, possibly depending on earlier measurement outcomes, any unitary gate can be applied to an input state which typically becomes part of the cluster at the beginning of the computation, see Fig. 2.3.4.
2.3.4 Oneway (cluster) computation for qubits. Certain singlequbit basis states become pairwise entangled to form a multiqubit cluster state. Local projection measurements on the individual qubits (potentially including feedforward with a measurement order going from left to right) are then enough to realize universal quantum computation. A multiqubit input state attached to the left end of the cluster could, in principle, be universally processed with the output state occurring at the right end of the cluster. The vertical edges allow for twoqubit gates.
2.3.5 Elementary ‘onebit’ teleportation circuit for qubit cluster computation. The gate represents a horizontal edge connecting two nodes of the cluster state. An input state is teleported into the left node, and a subsequent, local singlequbit measurement in the binary basis leaves the second node, up to a Hadamard gate and a Pauli correction depending on result , in the unitarily evolved state , with the rotation angle for a rotation controlled by the actual choice for the measurement basis.
The essence of cluster computation as described by the oneway model of quantum computation [1] can be summarized as follows: the cluster state is independent of the computation; universality is achieved through choice of measurement bases. This is illustrated in Fig. 2.3.5 for an elementary ‘onebit’ teleportation circuit between just two nodes of a qubit cluster state; once an input state is part of the cluster (after it got teleported into the cluster), a local singlequbit measurement is then sufficient to apply an arbitrary rotation. At the very beginning of a cluster computation, the cluster computer may be initiated in a product of ‘blank’ basis states such that no extra encoding teleportation step is needed; an arbitrary state can be anyway prepared within the cluster through local measurements. For example, an arbitrary singlequbit rotation requires just three elementary steps as shown in Fig. 2.3.5. In such a concatenation of elementary steps, for a given desired evolution, the later choice of the measurement bases depends on earlier measurement outcomes whenever nonClifford gates are involved.
Apart from the conceptual innovation, it turned out that, in particular, with regard towards linear optical quantum computation proposals, the cluster approach helps to reduce the resource costs significantly [52, 53]. As there are no more nonlocal Bell measurements needed in clusterbased computation, but only local projections, the problem of the nondeterministic linearoptics Bell measurements can be circumvented. Such Bell measurements may only be used for preparing a DV optical cluster state offline [53], whereas the online computation is perfectly deterministic. Eventually, scalability and resource costs in linearoptics quantum computation are determined by the efficiency with which cluster states can be grown using probabilistic entangling gates [54, 55].
In a very recent extension of the DV cluster model, the analogous CV cluster computation approach is considered [56]. In the CV model, full CV universality can be approached by applying linear Gaussian and nonlinear nonGaussian measurements to a Gaussian, approximate CV cluster state. A discussion of universal CV cluster computation combining CV and DV measurements will be presented in the section on hybrid schemes.
We conclude this section by noting that typically there is a tradeoff between the DV and CV optical approaches. The DV schemes are necessarily probabilistic and only at the expense of special extra resources can they be made neardeterministic; fidelities are usually quite high, nearunit fidelities. Conversely, in the CV schemes, fidelities tend to be modest and are necessarily below unity; nonetheless, CV operations are typically deterministic. These characteristic features were already present in the earliest experiments of DV and CV quantum information processing, namely those demonstrating quantum teleportation of an unknown quantum state between two parties [4, 57, 5].
Let us now consider the possibility of optically realizing efficient quantum communication. Even for this supposedly simpler task than universal quantum computation, similar constraints and nogo results exist, when the toolbox is restricted to only linear operations. Especially when efficient and reliable quantum communication is to be extended over large, potentially intercontinental distances, it turns out that this is, in principle, possible, but would require not much less resources than needed for doing optical quantum computation.
2.4 Implementing efficient quantum communication efficiently?
The goal of quantum communication is the reliable transfer of arbitrary quantum states, possibly drawn from a certain alphabet of states. Quantum communication is “the art to transfer quantum states” [58]. This may then lead to various applications such as the secure distribution of a classical key (quantum key distribution [59, 60, 61]) or the connection of spatially separated quantum computers for distributed quantum computing or a kind of quantum internet [62]. As light is an optimal information carrier for communication, one may send quantum states encoded into a stream of single photons or a multiphoton pulse through an optical channel. However, quantum information encoded in fragile superposition states, for example, using photonic qubits or qumodes, is vulnerable against losses and other sources of excess noise along the channel such that the fidelity of the state transfer will exponentially decay with the length of the channel.
For instance, the term of a singlerail qubit would partially leak into the vacuum modes of the channel, , such that tracing over the channel mode leads to the final signal state , with the transmission parameter and the attenuation length . If a photon that did make it through the channel is to be detected and, in particular, resolved against detector dark counts, this will become exponentially harder for longer channels. Similarly, a qumode in a coherent state would be transformed as , corresponding to the signal map . Although the coherent state remains pure, its transmitted amplitude would be decreased by an exponential factor. Any nonclassical qumode state would exponentially decohere into a mixed state.
2.4.1 Concept of a quantum repeater. The state is teleported to the remotest station after a longdistance entangled pair is created over the total channel. For this purpose, a supply of shortdistance pairs is distributed over sufficiently short channel segments such that highfidelity entangled pairs can be distilled and connected through entanglement swapping. Only this combination of entanglement distillation and quantum teleportation in a fully nested protocol enables one to suppress the exponential decay of the transfer efficiencies or fidelities as obtained in direct state transmission. As the full protocol typically contains probabilistic elements, sufficient local memories are required.
In longdistance, classical communication networks, signals that are gradually distorted during their propagation in a channel are repeatedly recreated through a chain of intermediate stations along the transmission line. For instance, optical pulses traveling through a glass fiber and being subject to photon loss can be reamplified at each repeater station. Such an amplification is impossible, when the signal carries quantum information. If a quantum bit is encoded into a single photon, its unknown quantum state cannot be copied along the line [63, 64]; the photon must travel the entire distance with an exponentially decreasing probability to reach the end of the channel.
The solution to the problem of longdistance quantum communication is provided by the socalled quantum repeater [2, 3], see Fig. 2.4.1. In this case, prior to the actual quantumstate communication, a supply of standard entangled states is generated and distributed among not too distant nodes of the channel. If sufficiently many of these imperfect entangled states are shared between the repeater stations, a combination of entanglement purification and swapping extends this shared entanglement over the entire channel. Through entanglement swapping [65], the entanglement of neighboring pairs is connected, gradually increasing the distance of the shared entanglement. The entanglement purification [66, 67] enables one to distill (through local operations) a highfidelity entangled pair from a larger number of lowfidelity entangled pairs, as they would emerge after a few rounds of entanglement swapping with imperfect entangled states and at the very beginning after the initial, imperfect entanglement generation and distribution between two neighboring repeater stations.
The essence of longdistance quantum communication as realized through the quantum repeater model [2, 3] can be summarized as follows: provided sufficient local quantum memories are available and some form of quantum error detection is applied, quantum communication over arbitrary distances is possible with an increase of (spatial or temporal) resources scaling only subexponentially with distance. Similar to what we concluded for efficient quantum computation, an inprinciple efficient realization depends on a nonexponential resource scaling. Otherwise, without fulfilling this criterion, we could as well choose to directly transmit quantum states, similar to performing exponentially many gate operations in an inefficient quantum computer. ^{4}^{4}4the naive approach of dividing the total channel into several segments that are connected through quantum teleportation without incorporating any form of quantum error detection and without using quantum memories is not enough to render quantum communication efficient with regard to resource scaling; however, it may still help to enhance practicality of a scheme, for instance, in order to resolve singlephoton signals against detector dark counts (“quantum relay” [59, 68, 69, 70]).
The main distinction between the communication and the computation scenario is that in the former case we may now use probabilistic operations; in particular, quantum error correction may be replaced by quantum error detection. However, this supposed advantage becomes a real advantage only provided that quantum states can be reliably stored during the waiting times for classical signals communicating successful events. So eventually, there is a tradeoff between the requirements on memory times and local quantum gates, with the rule of thumb that only less efficient (in terms of fidelity) and more complex quantum error detection/correction schemes would lead to a reduced need for efficient memories [71, 72].
When it comes to turning the inprinciple solution to scalable quantum communication over arbitrary distances in form of a quantum repeater into a realistic implementation, what are the currently available resources regarding quantum memories and gates? For a repeater segment of the order of km, with an optical fiber at minimal absorption of 0.17 dB/km corresponding to km, we have a classical communication time of at least ms (ideally considering the vacuum speed of light ), in order to verify successful entanglement creation, swapping, or distillation events between two neighboring stations. In order to extend this to a total distance of km, even if no intermediate stations are involved such that only a single heralded entanglement creation event is to be confirmed over the distance , a classical communication time of about 6.6 ms would be needed. Although memory times approaching 60 ms are achievable in electron spin systems [73], and single excitations may be stored and retrieved over a time scale of 110 ms [74, 75], longer memory times would have to rely upon nuclear spins. However, even in this case, currently available memories ( s [76, 77]) do not match those needed in a fully nested repeater protocol. Moreover, the local quantum logic for error detection (requiring, effectively, a small quantum computer at each station), would further increase the actual memory requirements, for instance, when using probabilistic linearoptics gates [49, 50], as discussed in the preceding section.
The maximum distance for experimental quantum communication is currently about 250 km [78, 59]. Although extensions to slightly larger distances may be possible with present experimental approaches [79], there are also various proposals for actually implementing a quantum repeater. The most recent proposals are based on the nonlocal generation of atomic (spin) entangled states, conditioned upon the detection of photons distributed between two neighboring repeater stations. The light, before traveling through the communication channel and being detected, is scattered from either individual atoms, for example, in form of solidstate single photon emitters [80, 81], or from an atomic ensemble, i.e., a cloud of atoms in a gas [82] (the ‘DLCZ’ scheme).
In these heralded schemes, typically, the fidelities of the initial entanglement generation are quite high, but the heralding mechanism leads to rather small pair generation rates. Other complications include interferometric phase stabilization over large distances [83, 84, 85, 86] and the purification of atomic ensembles. Yet some elements towards a realization of the DLCZ protocol have been demonstrated already [87, 88, 89, 90].
The DLCZ scheme is initially based upon “hybrid” CV Gaussian entangled states, where hybrid here means that the entanglement is effectively described by a twomode squeezed state with the two modes being a (symmetric) collective atomic mode for a large ensemble of atoms,
(16) 
and a Stokeslight mode, [82, 91]; here, , are the ground and metastable states of the th atom before and after a Raman transition in a configuration, respectively; is induced by a classical pump (with being the excited state), while produces the forwardscattered Stokes light. In the regime of only low excitations, corresponding to short interaction times and Stokes mean photon numbers less than unity, the terms of the twomode squeezed state,
(17) 
with and “squeezing parameter” , are dominating such that the resulting atomlight state becomes approximately . To this order, dropping terms and higher, by creating the same state at the nearest repeater station equipped with atomic memories and combining the Stokes fields coming from both stations, labeled by and , at a central beam splitter in order to erase whichpath information,
(18) 
singlephoton detector events would trigger an entangled state of the form between the ensembles denoted by and . The initial atomlight entanglement is completely swapped onto the atomic memories. Additional atomic “vacuum” contributions originating from detector dark counts would be (to some extent) automatically removed from the final states; a kind of purification “built into” the entanglement swapping process. The effect of atomic spontaneous emissions is suppressed, because a single atomic spin mode gets collectively enhanced for sufficiently large ensembles .
Though, in principle, being “fully CV” at the beginning, the DLCZ protocol uses DV measurements with photon detectors in order to conditionally prepare DV entangled states. Ideally only the single excitations from the initial Gaussian states would contribute to the final DV state. Only in this limit (corresponding to small “squeezing” ), we will obtain nearunit fidelity pairs, , at the expense of longer average creation times, , with . In some sense, we could as well add DLCZ to the list of hybrid schemes according to our definition, as it combines (very weakly excited) CV resources with DV measurements, resulting in DV output states.
This is in contrast to the complementary approaches of the Polzik group who use similar physical systems and interactions, i.e., atomic ensembles with Stokes light, but remain fully CV throughout for lightmatter teleportations [92] and interfaces [93], as their operations consist of CV homodyne detections and feedforward. In their atomlight CV approach, two orthogonal components of the optical Stokes operators and the atomic collective spin operators are each well approximated by (rescaled) qumode phasespace variables. To sum up, we may say now that in DLCZ effectively “flying qumodes” become “static qubits”, while in Polzik’s schemes, “flying qumodes” become “static qumodes”. The advantage of the latter approach is clearly that it is entirely unconditional with no need for any heralding element; this, as typical in fully CV schemes, is at the expense of only imperfect nonunit fidelities.
In either case, however, the interactions take place in free space, with many atoms effectively enhancing the coupling between the collective spins and the light field. Yet other approaches to quantum communication would turn “flying qubits” (e.g. polarizationencoded photonic qubits) into “static qubits” [59, 80, 81, 94]. Later, in the section on hybrid schemes, we shall describe a scenario in which a single DV spin system (an atomic qubit) is to be entangled with a CV qumode; so then the lightmatter coupling is qualitatively different, for instance, taking place and being enhanced in a cavity, and a “flying qumode” will mediate the entangling interaction between two “static qubits”; as a “genuine” hybrid scheme – the CV qumode component will have to have high excitation numbers as opposed to the single excitations of DLCZ – such a scheme, though not being fully unconditional, will contain an only moderate conditional element.
In summary, all those approaches discussed in this section are still fairly demanding with current technology. Especially, in the DV setting, with its highly conditional entangledpair creation and connection protocols, the minimal requirements on the necessary quantum memories [95, 96, 97] are far from being met using stateoftheart resources. Even though a fully CV approach, as implemented in the Polzik experiments, is tempting, because of its unconditionalness for entanglement creation and swapping, entanglement distillation has been shown to be impossible with only Gaussian states and Gaussian operations [98, 99, 100]. So it seems there is always a price to pay when certain types of resources are replaced by supposedly cheaper ones. In the next section, we will now discuss the very recent concept of hybrid quantum information processing, which could be useful for both efficient quantum computation and communication.
3 Hybrid approaches
Let us recall our definition of a hybrid protocol. We refer to a quantum information scheme as hybrid, whenever it is based upon both discrete and continuous degrees of freedom for manipulating and measuring the participating quantum subsystems. In the quantum optical setting, this includes in particular those approaches that utilize light for communication and employ matter systems for storage (and processing) of quantum information [101], as the optical qumodes are most naturally represented by their quantized position and momentum (amplitude and phase quadrature) variables, whereas the atomic spins or any twolevel structures in a solidstate system provide the natural realization of qubits. An important ingredient of such hybrid schemes may then be a particularly intriguing form of entanglement – hybrid entanglement, i.e., an inseparable state of two systems of different dimensionality, for example, between a qubit and a qumode.
We shall also remind the reader of the motivation for combining CV and DV approaches, besides the “natural” motivation of representing and using hybrid lightmatter systems, as stated in the preceding paragraph. CV Gaussian resources can be unconditionally prepared and Gaussian operations are deterministic and (experimentally) efficient. Nonetheless, there are various, highly advanced tasks which would require a nonGaussian element:
The nonGaussian element may be provided in form of a DV measurement such as photon counting. There are also a few simpler tasks which can be performed better with some nonGaussian element compared to a fully Gaussian approach, for instance, quantum teleportation [103] or optimal cloning [104, 105] of coherent states.
Similarly, (efficient) universal quantum computation on photonic qubits would depend on some nonlinear element, either directly implemented through nonlinear optics or induced by photon measurements, as discussed in detail in Sec. 2.3. In addition, there are even supposedly simpler tasks which are impossible using only quadratic interactions (linear transformations) and standard DV measurements such as photon counting. The prime example for this is a complete photonic Bell measurement [12, 13, 14].
It is worth pointing out that the above restrictions and nogo results apply even when linear elements and photon detectors are available that operate with 100% efficiency and reliability (i.e., fidelity). In other words, the imposed constraints are of fundamental nature and cannot be resolved by improving the experimental performance of the linear elements, for example, by further increasing squeezing levels.
Incorporating both nonlinear resources and nonlinear operations into an optical quantum information protocol would enable one, in principle, to circumvent any of the above constraints. Such an approach, unless weak nonlinearities are employed, will most likely be more expensive than schemes that stick to either linear resources or linear operations. There are now many proposals for optically implementing quantum information protocols through a kind of hybrid approach. Among other classifications, two possible categories for such hybrid schemes are:

those based upon nonlinear resources using linear operations,

those based on linear resources using nonlinear operations.
In the latter case, for instance, DV photon number measurements may be applied to CV Gaussian resources. The former type of implementations would utilize, for example, CV homodyne measurements and apply them to DV photonic qubit or other nonGaussian states. Some of these approaches will be presented, with regard to hybrid quantum computing (Sec. 3.3), in Secs. 3.3.3 and 3.3.4, after a discussion on qubitintoqumode encodings in Sec. 3.3.1 and hybrid Hamiltonians (Sec. 3.3.2). As an additional approach for incorporating a nonlinear element into a quantum information protocol, in Sec. 3.3.5, we shall give a brief description of schemes using weakly nonlinear operations. Finally, we will turn to hybrid quantum communication in Sec. 3.4.
The notion of hybrid entanglement and some of its applications as well as qubitqumode entanglement transfer will be discussed in Sec. 3.2. As a start, however, let us give a short summary of some earlier hybrid proposals and implementations and some more recent schemes.
3.1 Overview
The hybrid approaches may either aim at “simpler” tasks such as quantum state engineering and characterization, or at the ultimate applications of universal quantum computing and longdistance quantum communication. Here is a short overview on theory and experiments.
Realizing POVMs:
theory: optimal unambiguous state discrimination of binary coherent
qumode states using a 50:50 beam splitter, a coherentstate ancilla,
and a DV measurement discriminating between zero and nonzero photons (on/off detector)
[33] (see
Fig. 2.2.3);
theory and experiment:
nearminimum error discrimination of using displaced
on/off detectors beating the optimal Gaussian homodynebased discrimination scheme [106].
Quantum state engineering:
generation of
Schrödingercat (coherentstate superposition, CSS) states with a “size” of ;
from Gaussian squeezed vacuum through DV photon subtraction using beam splitters and photon detectors,
theory [108] and experiment [30, 109, 31, 32];
from squeezed vacuum and
onephoton (twophoton) Fock states through beam splitters,
CV homodyne detection, and
postselection yielding squeezed Fock states and approximate
odd (even) CSS states (see Fig. 3.1.1),
theory [110] and experiment [111];
experiments: singlemode photonadded/subtracted coherent state [112]
and thermal state [113].
3.1.1 Conditional preparation of an odd (even) CSS state using DV onephoton (twophoton) Fock states, and CV squeezed vacuum resources, , together with CV homodyne detection and postselection [110].
Quantum state characterization:
theory:
measurement of entanglement and squeezing
of Gaussian states through beam splitters and photon counting [114];
measurement of Bell nonlocality of Gaussian
twomode squeezed states via photon number
parity detection [115];
experiment:
homodyne tomography of onephoton state [107],
homodyne tomography of twophoton state [116].
Quantum communication subroutines:
theory:
entanglement concentration (EC) of pure Gaussian twomode
squeezed states (TMSSs) through DV photon subtraction [117];
entanglement distillation (ED) of noisy Gaussian TMSSs using beam splitters and
on/off detectors [118] or
photon number QNDmeasurements [119, 120];
experiment: EC of pure TMSSs through nonlocal [121]
and local [122] DV photon subtraction;
ED of Gaussian TMSSs
subject to nonGaussian noise such as phase diffusion [123] and
random attenuation [124].
Quantum computational resources:
theory:
conditional creation of nonGaussian cubic
phase states from TMSSs
through DV photon counting, as a resource to implement
CV cubic phase gates (GKP, [10]);
generalization of such cubicgate schemes [125];
universal DV gates through weak Kerrtype
nonlinearities and strong CV Gaussian probes [126].
More details on the GKP scheme will be presented in Sec. 3.2. The pioneering “hybrid” work is the experiment of Lvovsky et al. in which CV, homodynebased quantum tomography is performed for the discrete onephoton Fock state [107]. The reconstructed Wigner function in this experiment has a strongly nonGaussian shape including negative values around the origin in phase space.
According to an even earlier, theoretical proposal, CV quantum teleportation [127] is applied to DV entangled states of polarizationencoded photonic qubits [128], transferring DV Belltype nonlocality through CV homodyne detection and optimized displacements in phasespace for feedforward (“gain tuning”). Optical CV quantum teleportation of DV photonic states was further explored by Ide et al. [129, 130], combining CV tools such as gain tuning with postselection, an ingredient inherited from the conditional DV approaches.
It is important to notice that for an experimental implementation of a hybrid scheme in which DV and CV techniques and resources are to be combined (for example, for CV quantum teleportation of DV states), the standard way of applying such methods has to be generalized. In particular, frequencyresolved homodyne detection, as used, for instance, in CV quantum teleportation of coherent states [5], must be extended to timeresolved homodyning [131] in order to synchronize the CV operations with DV photon counting events. ^{5}^{5}5this extension means that, for instance, measurements on quantum states in the weakexcitation regime are no longer restricted to timeresolved photon counting only (like in the DLCZ repeater proposal), but would include timeresolved homodyning; conversely, an extension from frequencyresolved homodyning to frequencyresolved photon counting could be considered as well [132]. CV operations must act on a faster scale: while the standard CV experiments used singlemode cw light sources with narrow sidebands of kHz, the new generation of hybrid experiments relies upon bandwidths of at least MHz, corresponding to time scales of ns [133].
A beautiful example of a typical hybrid scheme according to our definition is the “offline squeezing” protocol from Ref. [110] for quantum state engineering, experimentally demonstrated in Ref. [111], see Fig. 3.1.1. In this scheme, approximate CSS states are built using linear CV measurements with outcomes within a finite postselection window, linear CV squeezedstate, and nonlinear DV Fockstate resources. The protocol works by squeezing the input Fock state, e.g., , which corresponds approximately to an odd CSS state () and would be hard to achieve ‘‘online” using the standard squeezing techniques such as optical parametric amplification. ^{6}^{6}6in the experiment of Ref. [111], a twophoton state was simply split at a beam splitter; so the squeezed vacuum in Fig. 3.1.1 was just a vacuum state. Postselection through timeresolved homodyne detection led to an output CSS state which was squeezed by 3.5 dB. Theoretically, the fidelity of the CSS state would approach unity for input Fock states with [111]. Though postselection renders the protocol probabilistic, it enables one to preserve the nonGaussianity of the input Fock state. This is different from the nonhybrid “offline squeezing” approach of Refs. [134, 135], where postselection is replaced by a continuous feedforward operation such that “offline squeezing” is applied in a fully Gaussian, deterministic fashion. Later we shall discuss how to realize arbitrary squeezing operations on an arbitrary input state through deterministic, homodynebased cluster computation.
3.1.2 Approximate discrimination of binary coherent states . Complementary to the probabilistic, errorfree USD scheme (Fig. 2.2.3), an appropriate singlemode transformation [(b) optimal displacement , (c) optimal squeezing and displacement , (d) optimal nonlinear nonGaussian transformation ] in front of an on/off photon detector achieves closetooptimal (b), even closertooptimal (c), and optimal (minimumerror) discrimination (d), and would always beat the optimal CV receiver solely based upon homodyne detection (a) [137].
In the nonhybrid, Gaussian CV regime, it is known how useful largely (offline) squeezed states are for engineering all kinds of multiparty entangled states [7] including arbitrary CV graph states [136]. On the other hand, typically, squeezing was explicitly excluded from the toolbox for DV (linearoptical) quantum information processing [6]. One important aspect of the more recent hybrid approaches is that squeezing is no longer considered a resource solely for CV protocols. The above quantumstateengineering schemes may serve as examples for this.
Besides quantum state engineering, however, there are other tasks in quantum information that may benefit from the use of squeezing, especially when the squeezing transformation can be applied online to an arbitrary state at any time during a quantum protocol. One example for this is the nearminimum error discrimination of binary coherentstate signals [137], a protocol complementary to the errorfree USD as depicted in Fig. 2.2.3. In this case, squeezing is needed to obtain the optimal Gaussian transformation that, in combination with DV photon detection, leads to a nearoptimal state discrimination, ^{7}^{7}7The actual optimal (i.e., minimumerror) discrimination would correspond to a projection onto a CSS basis. This socalled Helstrom bound [138] is attainable by replacing the Gaussian transformation in front of the photon detectors by a nonGaussian one [139], see Fig. 3.1.2. Remarkably, this highly nonlinear, optimal measurement can also be achieved by only photon detection and realtime quantum feedback [140, 141]. see Fig. 3.1.2.
The bottom line of the discussion here is that squeezing added as an online tool to the standard linearoptics toolbox may be of great benefit, beyond the more conventional quantumstate engineering schemes. Further examples are given later in the section on hybrid quantum computing, where squeezing is a necessary correction operation in order to implement nonlinear quantum gates in a measurementbased fashion. Online squeezing could also be used to “unsqueeze” the squeezed CSS state emerging in the experiment of Ref. [111].
Finally, concluding this overview of hybrid proposals and experiments, let us at least mention the ultimate application of CSS states for faulttolerant, universal quantum computation presented in Ref. [142] and discuss yet another way to create such states [143].
In Section 2.3, we started using the qubit Pauli operator basis , , and as elementary gates, and rotations along their respective axes, , etc., to describe and realize arbitrary singlequbit unitaries. In analogy, we used a similar notation for the qumode WH (displacement) operator basis, and . Here, in the hybrid context, we shall exploit interactions and operations involving combinations of DV qubit and CV qumode operators and therefore we prefer to use unambiguous notations: for qumodes, still and for the WH group elements, and and for the Lie group generators with ; for qubits, now , , and for the Pauli basis. Now look at the effective interaction obtainable from the fundamental JaynesCummings Hamiltonian, , in the dispersive limit [144],
(19) 
Here, () refers to the annihilation (creation) operator of the electromagnetic field qumode in a cavity and is the corresponding Pauli operator for a twolevel atom in the cavity (with ground state and excited state ). The atomic system may as well be an effective twolevel system with an auxiliary level (a system), as described earlier for the DLCZ quantum repeater. However, this time we do not consider the collective spin of many atoms, but rather a single electronic spin for a single atom (in a cavity). The operators and () are the raising (lowering) operators of the qubit. The atomlight coupling strength is determined by the parameter , where is the vacuum Rabi splitting for the dipole transition and is the detuning between the dipole transition and the cavity field. The Hamiltonian in Eq. (19) generates a controlled phaserotation of the field mode depending on the state of the atomic qubit. This can be written as ()
(20) 
which, compared to Eq. (8), now describes a unitary operator that acts in the combined Hilbert space of a single qubit and a single qumode. We may apply this operator upon a qumode in a coherent state, and may formally write
(21) 
Compared to the uncontrolled rotation in Eq. (9), this time the qumode acquires a phase rotation depending on the state of the qubit, see Fig. 3.1.3. As the eigenvalues of are , applying to the initial qubitqumode state results in
(22) 
a hybrid entangled state between the qubit and the qumode.
The observation that this hybrid entangled state can be used for creating a macroscopic superposition state of a qumode, a CSS state, by measuring the microscopic system, the qubit, is about 20 years old [143]. A suitable measurement is a projection onto the conjugate qubit basis, , equivalent to a Hadamard gate applied to the qubit,
(23) 
followed by a qubit computational measurement. Depending on the result, we obtain for the qumode. The size of this CSS state depends on the distance between the rotated states in phase space, see Fig. 3.1.3, scaling as for typically small values. However, sufficiently large initial amplitudes still lead to arbitrarily “large” CSS states (at the same time increasing their vulnerability against photon losses).
3.1.3 Controlled phase rotation of a qumode in a coherent state, real. Depending on the qubit state, , the phase angle of the controlled rotation will be . When the qubit starts in a superposition state, , we obtain a hybrid entangled state between the qubit and the qumode.
This is a manifestation of a weak nonlinearity which is effectively enhanced through a sufficiently intense light field. Although typically the “singleatom dispersion” itself is rather weak with small phase angles of at most , it does not require strong coupling; in a CQED setting, the only requirement is a sufficiently large cooperativity parameter [145]. In the following sections, we shall also consider using weak nonlinearities for certain quantum gate constructions and, more generally, sets of hybrid operations such as controlled displacements and rotations sufficient for universal quantum computing.
Essential part of any such protocol is the generation of hybrid entanglement, as expressed by Eq. (3.1). Then measuring the qubit appropriately yields macroscopic CSS states, as described; measurements on the qumode may be useful for creating entangled qubit memory pairs in quantum communication, and, for the ultimate application, no measurements at all may still give universal quantum gates, as shown later. Let us now look in a little more detail at the notion of hybrid entanglement.
3.2 Hybrid entanglement
In the preceding section, we encountered the example of an entangled state between a qubit and a qumode. This state, though defined in a combined qubitqumode, hence infinitedimensional Hilbert space, can be formally written in a twoqubit Hilbert space, as we shall explain in this section. Thus, the entanglement of this state can be conveniently quantified.
Besides entanglement measures for quantifying entanglement, entanglement qualifiers, socalled entanglement witnesses, have become useful tools for delineating inseparable states. Such witnesses are wellknown for CV Gaussian states [146, 147] as well as for DV density operators [148]. Optically encoded quantum information, however, may comprise DV photonic qubit states in a Fock subspace or general nonGaussian entangled states. Using the partial transposition inseparability criteria [149, 148], Shchukin and Vogel demonstrated that certain inseparability criteria for states living in a physical bipartite space of two qumodes can be unified under one umbrella in terms of a hierarchy of conditions for all moments of the mode operators [150] (their results were further refined by Miranowicz and Piani [151]). These general conditions then include the previously known criteria expressed in terms of second moments [146, 147] as a special case, even those believed to be independent of partial transposition [146]. For nonGaussian entangled states, the secondmoment criteria would typically fail to detect entanglement, and a possible entanglement witness would have to incorporate higherorder moments.
Another interesting aspect when comparing qubit and qumode entangled states is whether the readily available, naturally given amounts of potentially unbounded CV entanglement can be transferred onto DV qubit systems [152, 153]. In the following subsection, we shall first consider qubitqumode hybrid entangled states, including a discussion on how to witness and possibly quantify their entanglement. Further, we will devote another subsection to the topic of transferring entanglement between qubit and qumode systems.
3.2.1 Qubitqumode entangled states
Consider the following bipartite state,
(24) 
with an orthogonal qubit basis and a pair of linearly independent qumode states and . A specific example of such a state and a possible way to build it was presented in Eq. (3.1) and the preceding discussion.
Clearly, the state in Eq. (24) becomes a maximally entangled, effective twoqubit Bell state when . For , the state is nonmaximally entangled, but still can be expressed effectively as a twoqubit state. This can be seen by writing the two pure, nonorthogonal qumode states in an orthogonal, twodimensional basis, ,
(25) 
where with . Then, using this orthogonal basis,
(26) 
the hybrid entangled state of Eq. (24) becomes
(27) 
a nonmaximally entangled twoqubit state with Schmidt coefficients and , where 1 ebit is obtained only for and . Quantifying the entanglement is straightforward, as the entropy of the reduced density matrix is a function of the Schmidt coefficients.
It is interesting to compare the state of Eq. (24) with a bipartite qumodequmode entangled state of the form
(28) 
assuming the overlap is real [154]. First of all, in this case, a normalization constant is needed, depending on . Secondly, and quite remarkably, such a state may always represent a maximally entangled twoqubit state (in the subspaces spanned by and ), independent of , but depending on the relative phase [154, 155, 156], i.e., the sign in Eq. (28).
The prime example for such qumodequmode entangled states are two of the socalled quasiBell states,
(29) 
with . The state is identical to the twoqubit Bell state when , , and , which is maximally entangled with exactly 1 ebit of entanglement for any . In contrast, the state only equals the oneebit Bell state in the limit of orthogonal coherent states, for .^{8}^{8}8similarly, for the other two (quasi)Bell states, we then have for any value of , but only when .
The amount of entanglement in the qubitqumode and qumodequmode states of Eqs. (24) and (28), respectively, is bounded above by one ebit, corresponding to a maximally entangled twoqubit Bell state. This is different from a ‘‘genuine” CV qumodequmode entangled state such as a Gaussian twomode squeezed state, which contains an arbitrary amount of entanglement for sufficiently high levels of squeezing, see Fig. 3.2.1 in the next section.^{9}^{9}9the entropy of the reduced density matrix of a twomode squeezed state, i.e., its entanglement would exceed one ebit at about 4.5 dB squeezing. Thus, the currently available squeezing levels of about 10 dB [157, 158, 159], corresponding to about 3 ebits (see Fig. 3.2.1), would easily suffice to outperform those hybrid entangled states discussed in the present section.
Let us further mention that the quantification of entanglement of the hybrid states in Eqs. (24) and (28) (we may also refer to the latter as hybrid in the sense that the two physical qumodes each effectively live in a twodimensional logical subspace) becomes more subtle, when they are mapped onto mixed states. For example, a qumode could be subject to an imperfect channel transmission, for instance, in a lossy fiber. Then for the special case of and being coherent states do we still obtain qubitlike expressions, as the coherent states themselves remain pure under amplitude damping (and the resulting mixed states would be expressible and hence quantifiable as twoqubit density operators in the orthogonal basis, see, for example, Ref. [160]). In this case, the density operator decoheres faster for larger amplitudes ; an effect which will become important later in the hybrid quantum communication schemes.
Finally, it is worth pointing out that lower bounds on the entanglement of (pure or mixed) nonGaussian entangled states such as those purestate examples in Eq. (29) can be derived from the standard measures for Gaussian entanglement [161, 162] by simply calculating the entanglement for the Gaussian state with the same secondmoment correlation matrix as the nonGaussian state given; in other words, for a given correlation matrix, the corresponding Gaussianstate entanglement provides a conservative and hence safe estimate on the actual entanglement of the state in question [163].^{10}^{10}10note that Gaussian states also provide an upper bound on the entanglement when only the correlation matrix is known: for given energy, Gaussian states maximize the entanglement [164].
Apart from applying entanglement measures to hybrid and nonGaussian entangled states, it is sometimes enough to have a (theoretical and, in particular, experimental) tool in order to decide whether a given state is entangled or not. Such entanglement qualifying criteria are typically related with certain observables (Hermitian operators) for which the expectation value is nonnegative for all separable states , whereas it may take on negative values for some inseparable states . Entanglement witnesses are wellknown for CV Gaussian states [146, 147] as well as for DV density operators [148].
Qumodequmode entangled states like those in Eq. (28) may be identified through the partial transposition criteria [149, 148] adapted to the case of arbitrary CV states [150, 151]. As a result, all known CV inseparability criteria, including those especially intended for Gaussian states and expressed in terms of second moments [146, 147], can be derived from a hierarchy of conditions for all moments of the mode operators and . Moreover, for nonGaussian entangled states, for which the secondmoment criteria typically fail to detect entanglement, the highermoment conditions would work. The concept for these criteria is as follows.
It is known that for any positive operator , we can write such that is nonnegative for any operator and any physical state . Then we may choose the bipartite decomposition , for which
(30)  
for any coefficients . Hence the matrix is positivesemidefinite for any physical state . Now any separable state remains a physical state after partial transposition of either subsystem such that and remain positivesemidefinite matrices, where and denotes partial transposition for subsystem and , respectively. Then, negativity of or , and hence negativity of any subdeterminant of or is a sufficient criterion for entanglement, from which sets of inequalities can be derived with convenient choices for the local operators and [150].
One such choice for a qumodequmode state would be each qumode’s position and momentum operators, eventually reproducing Simon’s criteria in terms of secondmoment correlation matrices [147]. Another choice, adapted to a qubitqumode state of the form in Eq. (24), is given by and , with some generic qubit state [165]. The resulting expectation value matrix then serves again, using partial transposition, as a tool to detect entanglement, this time for hybrid qubitqumode states. This can be particularly useful for verifying the presence of effective entanglement in a binary coherentstatebased quantum key distribution protocol [166, 167] as a necessary security requirement [168].
The choices for the local operators and discussed so far all lead to secondmoment conditions only. These are experimentally most accessible, but may fail to detect some form of nonGaussian entanglement. For qumodequmode states, a more general choice is the normally ordered form for , , with mode operators and for the two qumodes and , respectively. Inserting this into or yields a hierarchy of separability conditions in terms of the moments of the mode operators. For example, the quasiBell state of Eq. (29) leads to a subdeterminant of the matrix of moments with a sufficient order of the moments such as which becomes [150, 151]. This subdeterminant is negative for any nonzero , proving the entanglement of the state for any .^{11}^{11}11note that for and that is maximally negative, , for , even though we know that has constant entanglement of one ebit for any nonzero . , for real
To conclude this section, let us summarize that it is straightforward to quantify the entanglement of hybrid qubitqumode and nonGaussian qumodequmode states, provided these states can be represented in twodimensional subspaces. Otherwise, in order to detect the inseparability of such states, the partial transposition criteria expressed in terms of matrices of moments can be used. In a recent experiment, employing CV quantum information encoded into the spatial wavefunction of single photons, fourthordermoment entanglement was detected [169].
3.2.2 Qubitqumode entanglement transfer
As the entanglement in Gaussian qumode states is unconditionally available and, in principle, unbounded, it is tempting to consider protocols in which the CV entanglement is transferred onto DV systems. Especially the local memory nodes in a quantum repeater are typically represented by atomic spin states (recall the discussion in Sec. 2.4). So the unconditional generation of CV entanglement and its efficient distribution between two repeater stations should then be supplemented by a transfer of the transmitted ebits (in form of flying qumodes) onto the local, electronic or nuclear, storage spins (in form of static qubits).
Consider the distribution of twomode squeezed states, , with squeezing . In principle, an amount of entanglement in ebits [170],
could then be shared between the ends of the channel. Realistically, of course, the entanglement distribution will be subject to photon losses, leading to a highly degraded, shared CV entanglement; though always a nonzero amount of entanglement remains, unless additional thermal noise sources are present along the channel [146]. We shall take into account lossy channels in a later section. In this section, let us neglect the imperfect channel transmission and consider ideal CV entanglement distribution. In this case, Fig. 3.2.1 shows how many ebits could be, in principle, transferred from the CV to the DV system. Figure 3.2.2 illustrates the principle of entanglement distribution and transfer.
3.2.1 Entanglement of a Gaussian twoqumode squeezed state in ebits (the units of maximally entangled twoqubit states) as a function of the squeezing values in dB. The letters indicate experiments with corresponding squeezing levels: (a) recent CV Shortype quantum error correction scheme [171], (b) first CV quantum teleportation [5], and (c) most recent squeezing “world records” [157, 158, 159].
So what mechanism achieves this entanglement transfer? Are there possibly fundamental complications owing to the mismatch of the Hilbert spaces?^{12}^{12}12later, for a hybrid quantum repeater, we can avoid this mismatch from the beginning by distributing effective twoqubit entanglement of the form of hybrid qubitqumode states. These questions were addressed in Refs.[172, 173, 174], where it was shown that an entangled twomode squeezed state may act as a driving field to two remote cavities. When a twolevel system is placed in each cavity, the resonant JaynesCummings Hamiltonian, , can be used, describing the qubitqumode coupling for qumode and qubit , and similarly for mode and spin . As the driving field is assumed to be an external, broadband field, an additional interaction has to be included that describes the desired coupling between the external qumodes and the internal cavity qumode (and similar for the internal and external qumodes and , respectively), [174]. Thus, is the rate of this wanted coupling between each internal qumode at frequency and a corresponding external driving qumode at frequency . In the weakcoupling regime, where is much smaller than the external bandwidth, the nonunitary dynamics of the (internal) qumodequbit systems is governed by a master equation. The steady state of the cavities can then be shown to become a twomode squeezed state (in the badcavity regime using CQED language [175]).
3.2.2 Using a CV entanglement distributor for DV entanglement distribution. Two entangled qumode pulses travel to the opposite ends of a channel and interact with local atomic qubits placed, for instance, in a cavity. Through these interactions, the CV entanglement is dynamically transferred onto the two qubits.
Eventually, the CV entanglement is transferred onto the DV systems in the steady state. More realistically, additional dissipations have to be taken into account such as spontaneous atomic decay. This kind of unwanted inout coupling will occur at a rate . An important parameter then is the socalled cooperativity, . Only for sufficiently large do we obtain a nearly pure atomic steady state [174].^{13}^{13}13however, the assumption of large may collide with the badcavity/weakcoupling assumption [174, 175]. Later, in the section on hybrid quantum communication, the hybrid quantum repeater shall also depend on sufficiently large [145].
The general conditions for transferring general CV entanglement (including nonGaussian entanglement of the form Eq. (29)) from CV driving fields onto two qubits, both for the dynamical and the steadystate cases, were presented in Ref. [174]. Similar DV entanglement generation schemes were proposed in Refs. [176] and [177] using an indirect interaction of two remote qubits through Markovian and nonMarkovian environments, respectively. Finally, let us mention the interesting concept of transferring entanglement from a relativistic quantum field in a vacuum state onto a pair of initially unentangled atoms [178, 179].
Different from those schemes described in this section, which are basically measurementfree and dynamical entanglement transfer protocols utilizing an optical field as entanglement distributor, later we shall describe how to exploit local measurements on hybrid entangled states in order to nonlocally prepare entangled qubit states. The optical field will then act as a kind of quantum bus mediating the interaction between the qubits.
3.3 Hybrid quantum computing
In this section, we shall now discuss various hybrid approaches to quantum computing. This includes optical hybrid protocols for models of universal quantum computation as well as for certain gates from a universal gate set. More specifically, for processing photonic quantum information, we consider either using linear optical resources such as Gaussian entangled cluster states and performing nonlinear operations on them, or first creating offline nonlinear, nonGaussian resource states and applying linear operations such as homodyne detections. Further, we discuss hybrid schemes for quantum computing and universal quantum logic that do not require any highly nonlinear resources or interactions: only a weakly nonlinear element is needed [17].
However, let us start this section by looking at a few proposals in which either the encoding of quantum information [10, 142] or the unitary gate evolutions, i.e., the interaction Hamiltonians [180], are explicitly hybrid.
3.3.1 Encoding qubits into qumodes
There are various ways to encode a photonic qubit into optical modes such as polarization or spatial modes, as we discussed in the introductory part of this article. In particular, the photon occupation number in a singlerail, singlemode Fock state may serve as a qubit or a more general DV basis. In dualrail or, more generally, multiplerail encoding, a single photon encoded into multimode states can be even universally processed through linear optical elements; though in an unscalable fashion, unless complicated ancilla states and feedforward are employed [11].
Another natural way to encode a logical DV state into a physical, optical multimode state would be based upon at least two qumodes and a constant number of photons distributed over the physical qumodes such that, for example, a logical, dimensional spin state, , , can be represented by two physical qumodes in the twomode Fock state , where and denote the photon numbers of the two modes.^{14}^{14}14the choice of constant total number and varying number differences corresponds to a specific basis in the socalled Schwinger representation. For example, in order to faithfully represent the SU(2) algebra by the Lie algebras of two infinitedimensional oscillators, i.e., two qumodes and , one may replace the usual Pauli matrices , , , and by the socalled quantum Stokes operators , , satisfying the SU(2) Lie algebra commutators , while , for