10.3.1 Gaussian States

A general Gaussian state is defined by having a Gaussian Wigner function or, equivalently, a Gaussian characteristic function (which is the Fourier transform of the Wigner function). For our purposes, we may introduce only Gaussian states with vanishing first moments. The nonzero means can always be removed via local phase‐space displacements and hence do not affect the entanglement properties of the state. We define a normalized Gaussian ‐mode Wigner function (with a zero mean) as

10.9

with the ‐dimensional vector containing the quadrature pairs of all modes, . The elements of the correlation matrix are the second moments symmetrized according to the Weyl correspondence in Eq. 10.8,

10.10

Here, we used and for zero mean values. The last equality in Eq. 10.10 defines the correlation matrix for any quantum state. For Gaussian states of the form Eq. 10.9, the Wigner function is completely determined by the second‐moment correlation matrix.

The correlation matrix is real, symmetric, and positive. Moreover, it must satisfy the ‐mode uncertainty relation ( 7, 8),

10.11

based on the commutation relation,

10.12

Here, the “symplectic matrix” is a block diagonal matrix and contains the matrix as diagonal entries for each quadrature pair,

10.13

The matrix equation in Eq. 10.11 means that the matrix sum on the left‐hand side has only nonnegative eigenvalues. In the simplest case of only one mode, , Eq. 10.11 is reduced to the statement , which is a more precise and complete version of the well‐known Heisenberg uncertainty relation

10.14

The correlation matrix, for example, for a pure one‐mode squeezed state can be written as

10.15

where refers to a position‐squeezed state for any and to a momentum‐squeezed state for any . Both become the one‐mode vacuum state for . All these pure states exhibit minimum uncertainty, attaining the bound given by the Heisenberg uncertainty relation in Eq. 10.14. In general, the purity condition for an ‐mode Gaussian state is given by .

10.3.2 Gaussian Operations

As for the manipulation of Gaussian states, an important class is the set of Gaussian operations (19). The Gaussian operations are those quantum operations (completely positive maps) that map all Gaussian states onto Gaussian states. The subset of Gaussian operations, which exclude Gaussian measurements, such as homodyne detection (basically the projection onto the position or momentum basis), as well as nonunitary trace‐preserving Gaussian channels, such as amplitude damping, are the Gaussian unitary transformations. These are the most practical operations, because they can be realized via beam splitters, squeezers, and phase shifters. On the level of the correlation matrices, the Gaussian unitary transformations correspond to the symplectic transformations,

10.16

where . Those transformations which are both symplectic, , and orthogonal, , belong to the class of passive transformations. These transformations, realizable via beam splitters and phase shifters, are photon number preserving, as opposed to the active squeezing transformations. Among the simplest examples of passive and active symplectic transformations are the beam splitters,

10.17

and the one‐mode squeezers,

10.18

respectively. On the level of the mode operators, the symplectic transformations are reflected by linear transformations. Among these, the passive linear transformations are described by , with a unitary matrix . More general linear transformations, including both passive and active elements such as multimode squeezing, are expressed by . Here, the matrices and are, in general, not unitary. With these transformations, one can see that the position and momentum operators in are also linearly transformed, in agreement with the matrix transformation in Eq. 10.16. In the following, we will now discuss entangled Gaussian states. First, we will focus on the pure‐state case.

10.3.3 Pure Entangled Gaussian States

The prime example of an entangled Gaussian state is the pure two‐mode squeezed (vacuum) state, described by the Gaussian Wigner function

10.19

with . This Wigner function approaches in the limit of infinite squeezing , corresponding to the original EPR state. In spite of having a well‐defined relative position and total momentum for large squeezing, the two modes of the two‐mode squeezed vacuum state exhibit increasing uncertainties in their individual positions and momenta as the squeezing grows. In fact, upon tracing (integrating) out either mode of the Wigner function in Eq. 10.19, we obtain the “thermal state”

10.20

with the mean photon number . As the two‐mode squeezed state is the maximally entangled state at a given energy, the thermal state corresponds to the maximally mixed state at this energy. This is analogous to the finite‐dimensional discrete case, where tracing out one party of a maximally entangled state yields the maximally mixed state. The correlation matrix of the two‐mode squeezed state is given by

10.21

according to Eq. 10.9 and Eq. 10.19. By extracting the second moments from the correlation matrix in Eq. 10.21, we can verify that the individual quadratures become very noisy for large squeezing , whereas the relative position and the total momentum become very quiet,

10.22

However, what about arbitrarily small, but nonzero squeezing ? From Eq. 10.3, we can easily infer that the two‐mode squeezed state is entangled for any nonzero squeezing , even though this entanglement appears to be very bad for small squeezing values. In fact, we may even quantify the entanglement of the two‐mode squeezed state using the Schmidt decomposition in Eq. 10.3. A unique measure of bipartite entanglement for pure states is given by the partial von Neumann entropy (20). The von Neumann entropy, , of the reduced system after tracing out either subsystem is as follows: , using the Schmidt decomposition and , . Using Eq. 10.3, we can then quantify the entanglement of the two‐mode squeezed vacuum state via the partial von Neumann entropy (21),

10.23

However, all these results are based on the discrete Schmidt decomposition in the photon number basis rather than any nonclassical correlations in the continuous position and momentum variables.⁰²In the next section on mixed‐entangled Gaussian states and inseparability criteria, we will see how the presence of entanglement can be verified through correlations similar to those in Eq. 10.22. In the remainder of this section, we will now discuss some simple examples of transforming pure Gaussian two‐mode states into two‐mode squeezed states of the form Eq. 10.21. Finally, we will put these examples in a more general context.

Let us now first consider the case in which a separable Gaussian two‐mode state is transformed into an entangled Gaussian two‐mode state. Remarkably, a simple passive linear transformation corresponding to a beam splitter operation is sufficient to accomplish this. However, obviously, this operation is a nonlocal Gaussian transformation acting upon both input modes globally. Otherwise, through only local operations, a separable state cannot be turned into an entangled state. We use the separable input state , a product state of two one‐mode squeezed states, where the first one shall be squeezed in and the second one squeezed in . The correlation matrix of this Gaussian two‐mode state is given by , using Eq. 10.15. Now, applying the beam splitter operation from Eq. 10.17 to leads to the following transformation:

10.24

with the correlation matrix of a two‐mode squeezed state in Eq. 10.21. This example demonstrates how one can actually build an entangled state from one‐mode squeezed states using passive linear transformations. The corresponding method for creating Gaussian entanglement has been employed in many experiments around the world.

Our second example is even simpler. Again, we start with the separable two‐mode state . However, this time, we allow for local only Gaussian unitary transformations. In other words, assuming that the two modes are shared by two spatially separated people, Alice and Bob, both Alice and Bob can only act upon their own single mode. As mentioned above, via local operations, Alice and Bob will not be able to transform their shared state into an entangled two‐mode squeezed state. However, by applying local squeezers to it,

10.25

with and Eq. 10.18, they can convert their state locally into the two‐mode vacuum state corresponding to . This case is an almost trivial example of locally transforming a pure two‐mode Gaussian state into the form of Eq. 10.21. More interesting, however, is that any pure two‐mode Gaussian state, including entangled and separable ones, can be transformed into the form via local Gaussian unitary transformations (23,24). More generally, any bipartite pure multimode Gaussian state corresponds to a product of two‐mode squeezed states (with ) up to local Gaussian unitary transformations ( 23, 24). Thus, the two‐mode squeezed states represent a kind of standard form for pure Gaussian states. Let us now consider the case of mixed Gaussian states.

10.3.4 Mixed‐Entangled Gaussian States and Inseparability Criteria

A quantum state is called a mixture of some states if it can be written as a convex combination of these states:

10.26

where the real numbers form a probability distribution, that is, they are nonnegative, for all , and sum up to one

10.27

This definition is completely general and applicable to states with an arbitrary number of parts. For multipartite states, specific mixtures play an important role from a fundamental as well as a practical point of view.

A quantum state of a two‐party system is separable, if it is a mixture 10.26 of product states (25),

10.28

otherwise, it is inseparable. In general, it is a nontrivial question whether a given density operator is separable or inseparable. For states of tripartite systems, there are several notions of separability. A state is called ‐separable, if it is a mixture 10.26 of product states , where the states are bipartite states of the 23‐subsystem of the larger system:

10.29

The ‐ and ‐separability is defined analogously. If the state is a mixture of fully factorizable states, then the state is called fully separable:

10.30

Finally, a state which is a mixture of ‐, ‐, and ‐separable states is called biseparable. Explicitly, biseparable states can be written as follows:

10.31

where , , and are probability distributions and , . This is the most general notion of separability, since all others are just special cases of it.

For verifying the inseparability of a given two‐mode bipartite continuous‐variable state, Duan et al. derived an inequality in terms of the variances of position and momentum linear combinations (6), similar to those in Eq. 10.22. This inequality is satisfied by any separable state and is violated only by inseparable states. Thus, its violation is a sufficient, but not a necessary condition for the inseparability of arbitrary bipartite states, including non‐Gaussian ones. The corresponding inseparability criterion is a good example for applying the concept of “entanglement witnesses” to continuous variables. An entanglement witness is an observable that can detect the presence of entanglement of a quantum state . The state is entangled if there exists a Hermitian operator , such that , whereas for any separable state , holds. The Hermitian operator is then called an entanglement witness. Duan et al. proved that, for example, the sum of the variances of and can never drop below some nonzero bound for any separable state . However, for an inseparable state, this total variance may drop to zero. This is possible, because quantum mechanics allows the observables and to simultaneously take on arbitrarily well‐defined values due to the vanishing commutator

10.32

In fact, the EPR state from Eq. 10.1 is a simultaneous eigenstate of these two combinations.

Before we derive the Duan criterion, note that the variance of any Hermitian operator (observable) is concave, that is, it satisfies the following inequality:

10.33

where is given by the sum 10.26. This inequality easily follows from the Cauchy–Schwarz inequality. In fact, it is enough to demonstrate that

10.34

The Cauchy–Schwarz inequality reads as for all complex numbers and , provided that the sums on the right‐hand side converge. If we apply the Cauchy–Schwarz inequality to the numbers and and take into account the relation 10.27, we immediately get the inequality 10.34.

A good property of the concavity of the variance, expressed by the inequality 10.33, is that it is applicable to arbitrary states and arbitrary observables. If we can establish an inequality of the form for all states in some class, where is a positive number, then we can automatically extend this inequality for all convex combinations 10.26. For example, if we establish this inequality for all product states, then it will be automatically valid for all separable states. In many cases, it is much easier to establish an inequality of such a form for product states than for arbitrary separable states. The Duan condition is one example of such an inequality.

The proof of Duan's criterion works as follows. As we already know, it is enough to prove it only for product states. For such a state , we have

10.35

since , . It is at this step that the factorization property greatly simplifies the derivation, since for such a state we have, for example, . Applying the uncertainty relation

10.36

we find that the total variance itself is bounded below by 1 for all product states. Thus, the inequality

10.37

is a necessary condition for any separable state. Any violation of this proves the inseparability of the state in question. For example, the position and momentum correlations in Eq. 10.22 confirm that the two‐mode squeezed vacuum state is entangled for any nonzero squeezing . Note that the derivation of Eq. 10.37 does not depend on the assumption of Gaussian states. However, for two‐mode Gaussian states in a particular standard form, a condition very similar to that in Eq. 10.37 turns out to be necessary and sufficient for separability (6). This standard form can be obtained for any two‐mode Gaussian state via local Gaussian unitary transformations.

In the tripartite case, a similar approach allows one to distinguish between different kinds of separability. We show that the quantity , defined via

10.38

depends on the separability properties of the corresponding state. We split this quantity into three parts as , where

10.39

and the other two terms are defined analogously. We first find the minimal value of without any separability assumptions. Note that , and the same relation holds for the other two ‐operator combinations. We thus have , so we conclude that . It can be shown that a perfect equality cannot be achieved, but the lower bound can be approached arbitrarily closely by tripartite pure Gaussian states (26). This inequality, , is a tight physicality bound on in the sense that no physical state can violate it.

For separable states, we can now find a stronger bound. Due to the concavity of the variance, it is enough to consider only factorizable states. For example, in the case of a ‐factorizable state, we have

10.40

The first two terms on the right are bounded by , as in Eq. 10.36, and for the other two, we have , since . Therefore, we arrive at the inequality , which is valid for all ‐product states, and thus for all ‐separable states. The estimations for and cannot be improved compared to that done earlier. We see that the lower bound is now , so the inequality is valid for all ‐separable states. For a fully separable state, the inequality is valid not only for , but also for the other two terms, so we have the stronger inequality , valid for all fully separable states. Thus, we learn that the separability properties of states can be directly translated into the properties of – the more separability a state contains, the larger must be.

An important observation is that is symmetric with respect to the indexing of the subsystems. Thus, the inequality is satisfied not only by all ‐separable states, but also by all ‐ and ‐separable states. Combining these inequalities, from the definition of biseparability 10.31, we immediately find that all tripartite biseparable states satisfy the inequality . This bound is not tight. With a more sophisticated technique (42), it can be shown that the tight bound is . As we have already said, there are pure Gaussian states with arbitrarily close to , so all these Gaussian states are genuinely multipartite (i.e., tripartite) entangled.

The simplicity of proving these inequalities is based on relations such as 10.40, which are valid for factorizable states only, but it is the Cauchy–Schwarz inequality 10.34 that automatically implies an extension of these results from factorizable states to all separable states. It is applicable to the general multipartite case and allows one to simplify the given task by working only with factorizable states (for the class of inequalities considered). Not all separability conditions can be reduced so easily to the factorizable case, but convexity (or concavity) plays an important role in many of them due to the very nature of separability as a convex combination.

Apart from Duan's criterion, a necessary and sufficient condition for proving the inseparability of two‐mode bipartite Gaussian states is based on the continuous‐variable version of Peres' partial transpose criterion (27). In general, for any separable state as in Eq. 10.28, transposition of either subsystem's density matrix yields again a legitimate nonnegative density operator with unit trace, for example,

10.41

since corresponds to a legitimate density matrix. This is a necessary condition for a separable state, and hence a single negative eigenvalue of the partially transposed density matrix is a sufficient condition for inseparability. Transposition is a so‐called positive, but not completely positive map, which means that its application to a subsystem may yield an unphysical state when the subsystem is entangled to other subsystems. In general, for states of arbitrary dimension, negative partial transpose (npt) is sufficient only for inseparability (28). Entangled states with positive partial transpose (ppt) are the so‐called bound entangled states. However, the class of Gaussian states belongs to those classes where npt is indeed necessary and sufficient for inseparability ( 7, 8).

What does partial transposition applied to bipartite Gaussian or, in general, continuous‐variable states actually mean? Due to the hermiticity of a density operator, transposition corresponds to complex conjugation. Moreover, as for the time evolution of a quantum system described by the Schrödinger equation, complex conjugation is equivalent to time reversal, . Hence, intuitively, transposition of a density operator means time reversal, or, expressed in terms of continuous variables, sign change of the momenta. Thus, in phase‐space, transposition is described by , that is, by transforming the Wigner function as (7)

10.42

This general transposition rule is, in the case of ‐mode Gaussian states, reduced to the transformation

10.43

for the second‐moment correlation matrix (where again the first moments do not affect the entanglement). Now, the partial transposition of a bipartite Gaussian system can be expressed by . Here, means the block diagonal matrix with the matrices and as diagonal entries, and and are, respectively, and square matrices for modes at 's side and modes at 's side. According to Eq. 10.11, the condition that the partially transposed Gaussian state described by is unphysical,

10.44

is sufficient for the inseparability between and ( 7, 8). For Gaussian states with (7) and for those with and arbitrary (8), this condition is necessary and sufficient. For the general bipartite case of Gaussian states, however, there is also a necessary and sufficient condition: the correlation matrix corresponds to a separable state if and only if a pair of correlation matrices and exists such that (8)

10.45

Since it is in general hard to find such a pair of correlation matrices and for a separable state or to prove the nonexistence of such a pair for an inseparable state, this criterion in not very practical. A more practical solution was proposed by Giedke et al. (29). The operational criteria for Gaussian states there, computable and testable via a finite number of iterations, are entirely independent of the npt criterion. They rely on a nonlinear map between the correlation matrices rather than on a linear one such as the partial transposition. Moreover, as opposed to the npt criterion, these operational criteria also detect the inseparability of bound entangled states. Therefore, in principle, the separability problem for bipartite Gaussian states with arbitrarily many modes at each side is completely solved.

Let us now consider arbitrary bipartite two‐mode states. According to the definition of the ‐mode correlation matrix in Eq. 10.10, we can write the correlation matrix of an arbitrary bipartite two‐mode system in block form,

10.46

where , , and are real matrices. Any bipartite state satisfies the following inequality, a kind of physicality condition (7),

10.47

where is the matrix from Eq. 10.13. It turns out that the complicated expression on the left‐hand side has a simple structure

10.48

where and are the annihilation and creation operators defined by

10.49

and similarly for the second mode's operators and . The nonnegativity of the determinant 10.48 can be obtained from the nonnegativity of the quantity with

10.50

where , , are arbitrary complex numbers. By expanding as a quadratic form with respect to complex variables , we obtain the nonnegativity of the determinant 10.48.

Simon's continuous‐variable version of the Peres–Horodecki partial transpose criterion reads as follows (7):

10.51

Any separable bipartite state satisfies the inequality of Eq. 10.51, so that it represents a necessary condition for separability. Hence, its violation is sufficient for inseparability. Inequality Eq. 10.51 is a consequence of the fact that the two‐mode uncertainty relation, Eq. 10.11 with , is preserved under partial transpose, , provided the state is separable. A simple observation is that if , then the separability condition 10.51 is the same as the physicality condition 10.47, so no state with can violate this separability condition. Of course, this does not mean that any such state (with ) is separable, however, for Gaussian states, it is indeed the case: any bipartite Gaussian state with is separable (7).

So, violations are possible only if , and in this case, the Simon condition reads as

10.52

This condition also has a simple and regular structure, similar to Eq. 10.48:

10.53

It is easy to see that the determinants in 10.48 and 10.53 are related through partial transposition: the determinant of Eq. 10.53 is the same as that of Eq. 10.48 applied to the partially transposed state. It is easy to prove that the moments of the partially transposed state are expressed in terms of the original state via the relation

10.54

If we replace all the moments in Eq. 10.48 according to this relation, we will get exactly the inequality 10.53.

This inequality is a part of a larger hierarchy of separability conditions constructed from the nonnegativity of . Taking in the form

10.55

which is a generalization of Eq. 10.50, and applying the condition to the partially transposed state, we get an infinite hierarchy of conditions in terms of determinants of ever‐growing size. The inequality 10.53 is one of these conditions. Full details are given in (30).

As a next step, we may now define the following standard form for the correlation matrix of an arbitrary two‐mode Gaussian state:

10.56

This standard form is very useful and important, because it represents a compact description for analyzing the entanglement properties of arbitrary two‐mode Gaussian states in terms of only four parameters , , , and . Any two‐mode correlation matrix can be transformed into this standard form via appropriate local Gaussian unitary transformations (7). Simon's criterion does not rely on that specific standard form and can be applied to an arbitrary (even non‐Gaussian) state using Eq. 10.51. For Gaussian two‐mode states, however, Eq. 10.51 turns out to be a necessary and sufficient condition for separability (7). With the standard form from Eq. 10.56, the condition of Eq. 10.51 then simplifies to

10.57

Using Eq. 10.21, one can easily verify that Simon's separability condition in the form of Eq. 10.57 with Eq. 10.56 is violated by a two‐mode squeezed state for any .

As for the quantification of bipartite mixed‐state entanglement, various measures are available such as the entanglement of formation (EoF) and distillation (31). Only for pure states do these measures coincide and equal the partial von Neumann entropy. In general, the EoF is hard to compute. However, apart from the qubit case (32), also for symmetric two‐mode Gaussian states given by a correlation matrix in Eq. 10.56 with , the EoF can be calculated via the total variances in Eq. 10.37 (33). A Gaussian version of the EoF was proposed by Wolf et al. (34). Another computable measure of entanglement for any mixed state of an arbitrary bipartite system, including bipartite Gaussian states, is the “logarithmic negativity” based on the negativity of the partial transpose (35).

Exercises

1. 10.1 Pure entangled two‐mode Gaussian states
   1. Calculate the correlation matrix of the two‐mode state that emerges from a beam splitter operation applied to a momentum‐squeezed vacuum mode (with squeezing ) and a vacuum mode. Confirm that the resulting state is pure and determine for what squeezing values it is entangled. [Hint: use the inequality in Eq. 10.37.]
   2. Quantify the entanglement of the two‐mode state derived in (a). (Hint: use local squeezers to transform the state into the canonical two‐mode squeezed vacuum state.)
2. 10.2 Multipartite entanglement and mixed‐entangled two‐mode Gaussian states Consider the pure symmetric three‐mode Gaussian state described by the correlation matrix
   

   where , , , and . Show that for any , this is a genuinely three‐party entangled state, that is, a state that cannot be written as a product of a single mode with the remaining two modes. (Hint: look at the purity of the reduced states after tracing out one or two modes.) Further, check the inseparability properties of the two‐mode state after tracing out any one of the three modes.