Continuousvariable dense coding via a general Gaussian state: Monogamy relation
Abstract
We study a continuous variable (CV) densecoding protocol, originally proposed to employ a twomode squeezed state, using a general twomode Gaussian state as a quantum channel. We particularly obtain conditions to manifest quantum advantage by beating two wellknown singlemode schemes, namely, the squeezedstate scheme (best Gaussian scheme) and the numberstate scheme (optimal scheme achieving the Holevo bound). We then extend our study to a multipartite Gaussian state and investigate the monogamy of operational entanglement measured by the communication capacity under the densecoding protocol. We show that this operational entanglement represents a strict monogamy relation, by means of Heisenberg’s uncertainty principle among different parties, i.e., the quantum advantage for communication can be possible for only one pair of twomode systems among many parties.
I Introduction
Quantum correlation, especially entanglement RevModPhys.81.865 (), is a key resource for quantum information processing, e.g., quantum teleportation bib:PhysRevLett.70.1895 (); bib:PhysRevLett.80.869 () and quantum dense coding bib:PhysRevLett.69.2881 (); bib:PhysRevA.61.042302 (). Performance in such protocols, quantified by the output fidelity of teleportation and the communication capacity of dense coding, respectively, can be used as an operational measure of entanglement. When entanglement is shared by more than two parties, we may look into the multipartite entanglement structure and its usefulness by selecting a few parties among all and investigating their performance in quantum protocols for all such selected subsystems.
Numerous studies have so far demonstrated that a monogamy relation is one of the fundamental properties of multipartite entanglement, that is, entanglement between some subsystems puts limitation on their correlation with the other parties. For instance, if two quantum systems are maximally entangled, they cannot be correlated with a third party at all, even classically bib:PhysRevA.61.052306 (). The monogamy relation has been formulated quantitatively with a proper measure of entanglement for discrete variables bib:PhysRevA.61.052306 (); bib:PhysRevLett.96.220503 (), which was also extended to Gaussian CV systems bib:NewJPhys.8.15 (); bib:PhysRevA.73.032345 (). On the other hand, only a few studies have attempted to find such monogamy relations in view of useful entanglement for quantum communication protocols. In Ref. bib:PhysRevA.79.054309 (), it was shown that teleportation fidelity satisfies a monogamy relation for threequtrit pure states and for qubit pure states, but not in general. Recently, it was proved that a strict monogamy relation exists in dense coding for discrete variables bib:PhysRevA.87.052319 (), i.e., if a sender has some quantum advantage in dense coding with one receiver, (s)he can not have quantum advantage with any other receivers.
In this paper, we study a CV dense coding protocol and show a strict monogamy relation in this regime. Ever since quantum dense coding was first developed in qubit systems bib:PhysRevLett.69.2881 (), the dense coding capacity has been derived for arbitrary finitedimensional systems bib:QantumInfComput.1.70 () with arbitrary number of parties bib:PhysRevLett.93.210501 () for discretevariable systems. On the other hand, there have been few studies on CV dense coding with a general Gaussian state. The CV dense coding protocol was first proposed by Braunstein and Kimble bib:PhysRevA.61.042302 () in which a pure twomode squeezed vacuum (TMSV) state is employed as a quantum resource. A controlled dense coding was later developed bib:PhysRevA.66.032318 (), similar to the network quantum teleportation bib:PhysRevLett.84.3482 (), in which a threemode entangled state is prepared and information is transferred between two parties with the help of classical communication provided by a third party.
We here study the dense coding protocol employing an arbitrary twomode Gaussian state. We then compare this twomode scheme with two singlemode schemes, namely, the squeezedstate scheme known to be best among Gaussian schemes and the numberstate scheme known to be the ultimate optimal scheme achieving the Holevo bound. We derive conditions to beat those two schemes, i.e. to detect twomode Gaussian states that can manifest quantum advantage in CV dense coding. These conditions can be expressed in terms of the variances of two correlated quadratures as
(1) 
where and are the variances of and , respectively. The bound turns out to be and to beat the squeezedstate scheme and the numberstate scheme, respectively. These are more stringent than the sufficient condition to verify entanglement bib:PhysRevA.60.2752 (), where the bound is .
We also extend our study to the case of multipartite Gaussian states. We particularly show that, by means of Heisenberg’s uncertainty principle, the criterion (1) can be satisfied for only one pair of twomodes among all. Accordingly, we have a strict monogamy relation for CV dense coding, that is, a sender can have quantum advantage with one receiver only.
Ii Gaussian state description
We start with a brief description of Gaussian states and their transformations under Gaussian operations.
Nmode Gaussian States—To describe an mode CV system, we introduce a dimensional vector whose components are canonical variables of each mode, given by
(2) 
with their corresponding operators
(3) 
A Gaussian state is fully characterized by the firstorder moments and the secondorder moments of the position and the momentum operators. Its phasespace distribution (Wigner function) takes a form of Gaussian function
(4) 
where are the firstorder moments and is the covariance matrix (CM) whose elements are the secondorder moments,
(5) 
. Due to the uncertainty relation arising from , every CM must satisfy
(6) 
Gaussian Operations—An arbitrary Gaussian unitary operation is a linear transformation of canonical operators that can be represented as . Here is a dimensional real vector displacing the firstorder moments by and is a symplectic transformation satisfying bib:JPhys.45.471 (). Under a symplectic transformation, CM evolves as .
Standard form—Every twomode Gaussian state, pure and mixed, can be transformed into a standard form via a set of local symplectic transformations, which can be written as
(7) 
For the case of threemode systems, a pure threemode Gaussian state can be transformed to a standard form bib:PhysRevA.73.032345 ()
(8) 
Each offdiagonal element is shown in Appendix A. The coefficient ’s must satisfy the following triangular inequality due to the uncertainty principle:
(9) 
Iii CV dense coding and singlemode communication
iii.1 CV dense coding
In the CV densecoding scheme originally proposed by Braunstein and Kimble bib:PhysRevA.61.042302 (), Alice and Bob make use of an initially shared TMSV with squeezing parameter , of which Wigner function is
(10) 
where and . Alice encodes two classical variables {, } by displacing her mode in phase space with the amplitude . The probability distribution of the encoded amplitude can be taken as
(11) 
Then, Alice sends her mode to Bob, who combines it with his mode at a 50/50 beam splitter. Finally, Bob carries out two quadrature measurements, and , on each of the output modes, respectively. The measurement outcomes are related to the quadrature amplitudes before the beam splitter as and . Denoting Bob’s measured outcomes as , the probability distribution conditioned on the input is given by
(12) 
where is the Wigner function of the output state at Bob’s station, with the beamsplitting and the displacement acting on the input state . The achievable information in this scheme can be quantified by the mutual information between the two sets of variables, {, } and ,
(13) 
Here, denotes the Shannon entropy as
For a TMSV input, the mutual information turns out to be
(15) 
It increases with the squeezing parameter , for Bob can read information more accurately due to the decreased variances of the correlated quadrature . Another point to mention is that the mutual information becomes infinite with , where is the variance of encoded variable in Eq. (11). It is true regardless of squeezing , which is simply the result of communicating infinitely large amount of information. However, it would require an infinite energy for encoding, thus, the constraint of finiteenergy is typically introduced for a fair comparison between different schemes.
When the average number of photons passing through the channel (Alice mode’s energy plus the noise added by displacement) is restricted to , one may adjust in order to maximize the mutual information in Eq. (15) for a given . With for a TMSV, the optimal value occurs at the choice of and the CV dense coding capacity, i.e. optimized mutual information, is given by
(16) 
iii.2 singlemode schemes
On the other hand, there are three wellknown singlemode schemes under the energy constraint . The first one makes use of a coherent state, which is displaced by (encoding) and then measured via heterodyne detection, i.e. simultaneous measurements of two quadratures (decoding). This coherentstate scheme yields the capacity bib:RevModPhys.58.1001 ()
(17) 
We note that the CV densecoding scheme using a TMSV always beats the coherentstate scheme regardless of [Cf. Eqs. (16) and (17)].
The second scheme employs a squeezed state, which is displaced by the amount along the squeezed axis (encoding) and measured via homodyne detection (decoding). This Gaussian scheme, which can be a best strategy under the restriction of Gaussian states and Gaussian operations bib:RevModPhys.58.1001 (), yields the capacity
(18) 
Compared with Eq. (16), this bound for singlemode Gaussian communication can be beaten by the CV dense coding for and it was demonstrated experimentally in Ref. bib:PhysRevA.71.012304 ().
The ultimate capacity of singlemode communication, when nonGaussian operations are also available, is given by the maximum possible entropy of singlemode state under constraint. This bound, known as the Holevo bound, is achieved when Alice encodes information in number states according to a thermal distribution and Bob decodes the information by photon number counting bib:PhysRevLett.70.363 (), although this scheme requires highly demanding experimental tasks. In this case, the capacity is given by
(19) 
The CV dense coding can also beat this bound with a sufficiently large photon number, i.e., .
Iv CV dense coding with arbitrary twomode Gaussian states
iv.1 Mutual information
Now we study the CV dense coding protocol using an arbitrary twomode Gaussian state as a quantum channel beyond TMSVs. In the latter case, the conditions to beat the singlemode schemes are given only in terms of energy , just because there exists a monotonic relation between the degree of squeezing (entanglement) and , which is generally not the case. We here intend to derive general conditions to beat the squeezedstate scheme and the photonnumber scheme, respectively, for an arbitrary twomode Gaussian state.
We first assume that Alice and Bob share a twomode state having correlations between and and between and , respectively, with the condition . In the next section, we show that the scheme becomes optimal under the condition , which can always be met via certain local phaserotations for a given state.
For the case of TMSV, Alice encodes the same amount of information on both quadratures, which is reasonable because two correlated quadratures and have the same variances. In general, however, the communication capacity can be enhanced by encoding a different amount of information on each quadrature as
(20) 
as will be shown below. After Alice sends her mode to Bob, Bob measures two correlated quadratures and . The conditional probability distribution of Bob reading when Alice encodes is necessarily a Gaussian distribution that has variances and centered at , that is,
(21)  
The factor dividing emerges from the output signal reduced by the beam splitter interaction. Using Eqs. (20) and (21), we obtain the mutual information as
(22) 
Under the energy constraint , where is the mean photon number of Alice’s mode before encoding and is the average number of photons added through displacement, we may adjust the amount of information encoded on each quadrature. If we assume to be large enough, , we find that the optimal encoding is achieved for the choices and . It implies, e.g., that we need to encode more information on the quadrature () if the state is more strongly correlated with respect to quadratures (). The optimized mutual information can be written as
(23) 
On the other hand, if is small, , the singlequadrature encoding on the quadrature possessing a smaller variance is optimal, which is not the case of our interest.
iv.2 Criteria to beat singlemode communications
Now we examine when our CV dense coding protocol can beat the singlemode Gaussian communication by solving
(24)  
In Appendix B, we prove that, if , one can always find a range of satisfying , otherwise can never be positive. Therefore, a twomode Gaussian state can be useful beyond the squeezedstate scheme under the condition
(25) 
Note that this criterion is more stringent than the entanglement detection criterion in which the product of correlated variances is bounded by bib:PhysRevA.60.2752 ().
On the other hand, for the CV dense coding scheme to be truly dense coding, it must beat the ultimate singlemode communication with the capacity involving nonGaussian operations. In a large regime, in Eq. (19) asymptotically behaves as . Thus, looking into
(26)  
we find that the CV dense coding protocol beats any singlemode communications under the condition
(27) 
in a large regime, which is stricter than the condition in Eq. (25).
iv.3 Improving the mutual information via local unitary operations
We have previously identified the conditions to beat the singlemode schemes in terms of the product . This product is, however, not invariant under local sympletic operations, which makes it possible that the communication capacity can be further improved by local operations for a given state. We here investigate how local operations affect the performance of CV dense coding scheme and how one can enhance the mutual information, applying local Gaussian unitary operations, i.e., displacement, phase rotation, and local squeezing.
iv.3.1 Displacement
Displacement operation does not affect the second moments but only change the first moments . The only parameter changed by displacement in Eq. (23) is of Alice mode. To maximize the mutual information, Alice should thus adjust the first moments to zero, by a proper displacement, to have a minimum .
iv.3.2 Phase rotation
We have derived the mutual information in Eq. (23) by assuming the condition , which can now be relaxed. For an arbitrary twomode state, using Eqs. (12) and (20), we obtain the mutual information achieved by the CV dense coding as
(28) 
This is maximized with the optimal encoding and , which gives
(29) 
where .
Now let Alice and Bob perform a phase rotation on their modes so that
(30) 
(), with and . The second moments associated with and becomes
(31) 
We readily see that the two quantities and are invariant under the rotation in Eq. (IV.3.2). Furthermore, the optimized over the angle occurs at , which leads to . Therefore, by a phase rotation, the correlation can be made zero and an optimal communication capacity arises with the value
(32) 
Comparing Eq. (32) with Eq. (23), we now see that a general condition to beat the squeezedstate scheme is given by
(33) 
That is, if a given state satisfies Eq. (33), one readily sees that of Eq. (29) can be larger than of Eq. (18) in a large region. On the other hand, if the given state has the correlation as , the product is also obtained in a frame of , using the invariance under our rotation described above, where an optimized capacity arises. However, we have already shown in Appendix B that for a case of , the state with the condition cannot beat the squeezedstate scheme, nor does the given state before rotation.
iv.3.3 Local squeezing
In the previous subsections, we have shown that an optimal CV dense coding arises under the conditions of zero mean amplitudes and . Henceforth, we consider a twomode CM in standard form (7) with only local squeezing operations applied to a given state. A symplectic transformation corresponding to a local squeezing is represented by for each mode . The CM after local squeezings is written as
(34) 
First let us take an example of twomode states, which is a twomode reduced state out of a pure threemode state in (8), with . In Fig. 1(a), we plot the mutual information against squeezing parameters and .
A maximum value (red dot) is found at and , which means that the standard form before the squeezing operations is not optimal for CV dense coding. In general, it is a nontrivial task to find optimal squeezing parameters and that maximize the mutual information Eq. (23), because the quantities , , and are not invariant under local squeezing.
Instead we here optimize which is a key quantity in the criterion of Eq. (1), which can be written as
(35) 
with . Note that is independent of the sum . By solving , we can find an explicit expression of that minimizes , although the expression is lengthy. In Fig. 1(b), we plot against squeezing parameters for the case we considered above. We see that has the same value along the line . The minimum is achieved when , which is not the case of a standard form.
In Fig. 2(a), we plot the optimal for the twomode reduced state of a pure threemode state in standard form (8) with a fixed and varying and (scaled in terms of and ).
For a symmetric state with (), is minimized when so that the standard form is optimal. Otherwise, a nonzero local squeezing is required for optimization. By minimizing , we can modify a state which does not initially satisfy the criterion of Eq. (1) to a form satisfying it via local squeezing. In Fig. 2(b), we plot the region where the criterion (1) is satisfied with(without) local squeezing with a fixed and varying and . We find the region between the solid curve and the dashed curve, where is satisfied with optimization but not in the standard form.
V CV dense coding with multimode Gaussian states
In this section, we extend our study of CV dense coding to the case of multipartite Gaussian states, specifically employing different pairs of two modes out of multiparties. Let us first assume that a pure threemode Gaussian state is distributed to three parties, Alice, Bob, and Charlie in the standard form of Eq. (8). Alice here plays as an information sender and either Bob or Charlie is a receiver.
First we consider the case without optimization and obtain the mutual information and , respectively, for a given threemode state. In Fig. 3, we plot the region where the mutual information attained by CV dense coding protocol between Alice and Bob (Charlie), , surpasses the capacity of Gaussian singlemode schemes.
We see that the CV dense coding protocol beats the coherentstate communication in a broad region and that both and can surpass the capacity of coherentstate communication simultaneously in a certain region. However, we do not find the region where both and surpass the capacity of squeezedstate communication simultaneously. It means that Alice can not make a useful CV dense coding to beat the squeezedstate communication with both receivers simultaneously.
We now show that even if they attempt to optimize the mutual information by applying local unitary operations for the two pairs {A,B} and {A,C}, respectively, it is impossible to beat the squeezedstate communication with both receivers simultaneously. Alice and Bob (Charlie) now perform optimal local squeezing operations on their modes and have the optimized state (). We prove that the criterion to beat squeezedstate communication, Eq. (1), cannot be satisfied for both and because the product of variances is bounded, regardless of local squeezing parameters, as
(36) 
The main idea of proof is to apply Heisenberg’s uncertainty principle for noncommuting operators and and also for and (see Appendix C for details). Therefore, we find a strict monogamy relation for CV dense coding: if Alice has quantum advantage in CV dense coding with Bob, she can never have quantum advantage with Charlie. In Fig. 4, we plot the region where is satisfied for and .
We see that there is no overlap between two regions so that a strict monogamy relation is satisfied. As the condition to beat the Holevo bound (numberstate scheme) is even stricter as , we also conclude that a true densecoding is possible only for a single pair of two modes.
A general mixed threemode state may not be written in the standard form (8). However, if the threemode state has the condition for both pairs of {A,B} and {A,C}, the proof in Appendix C is still valid, which in fact applies to any two pairs of two modes selected out of many parties beyond threemode cases.
For a symmetric, mixed, mode state which is invariant under the permutation of modes, we generally obtain a strict monogamy relation without resort to the condition . In Sec. IV C 2, we have derived a general conditon to beat the squeezedstate scheme, in Eq. (33). As shown before, it is equivalent to eliminating by a rotation of each mode, respectively. Let and denote those angles of rotation for each pair of two modes {A,B} and {A,C}, respectively. For a symmetric state, we have and prove that is possible for only one pair in Appendix C.
Vi Conclusion
In this paper, we have studied the CV densecoding protocol employing an arbitrary twomode Gaussian state. We have particularly derived criteria, Eq. (1) or its generalized form Eq. (33), to detect twomode states that can be more useful for communication than singlemode schemes, namely the squeezedstate scheme (best among Gaussian schemes) and the numberstate scheme (the optimal scheme achieving the Holevo bound). We have also shown how to enhance the performance of CV dense coding for a given twomode state via local operations.
We have also extended our study to a multipartite Gaussian state and proved that there exists a strict monogamy relation among different parties in view of the operational entanglement, i.e., communication capacity via densecoding protocol. That is, a sender (Alice) can have quantum advantage over the singlemode schemes strictly with one receiver only, which was proved for the case of symmetric mode systems unconditionally and for the case of general multimode systems possessing  and correlations only.
For a future work, it will be interesting to study the CV densecoding and its monogamy relation beyond Gaussian regime. It has been known that some nonGaussian operations can enhance performance of quantum information tasks, e.g. teleportation PhysRevA.61.032302 (). Thus, it is of fundamental and practical interest whether the nonGaussian regime can manifest a different feature from the monogamy relation identified here. In addition, the study of monogamy relation can be further extended to other operational measures like the output fidelity of CV teleportation. These studies may contribute to our understanding of multipartite entanglement structure in CV systems providing an insight into quantum network communication using multimode CV systems.
acknowledgement
This work is supported by the NPRP grant 45541084 from Qatar National Research Fund.
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Appendix A Standard form CM of pure threemode Gaussian states
A pure threemode Gaussian state satisfies
(37) 
where () is the reduced twomode (singlemode) CM of with different from each other. From these conditions, after some algebra, one finds the expression of offdigonal elements of CM in Eq. (8) as
(38) 
Appendix B Proof for the criterion of Equation (1)
Arranging in Eq. (24), we have
(39) 

Case I:
For a given state, the last two terms in the righthand side of Eq. (39) are finite. Thus, we can always find with a sufficiently large .

Case II:
Above all, we derive the bound for . Without loss of generality, let us assume . A lower bound arises from Heisenberg’s uncertainty principle as
(40) Taking partial derivative of with respect to , we find
(41) which is negative under the twoquadrature encoding condition . Therefore becomes maximum when is minimum, i.e. , which is given by
(42) In order to beat squeezedstate communication, must be smaller than .
On the other hand, we can find the upper bound of under the encoding condition :
(43) Since is a decreasing function in the range , we find the range of as
(44) The upper bound satisfies
(45) where the Heisenberguncertainty inequality is used. By putting the upper bound of into Eq. (43), we obtain .
Appendix C Lower bound for the product of correlated variances
With operators and for the initial threemode state , we can write, in the Heisenberg picture, operators for the optimal state as , and similarly, for . Then we find
(46) 
In the case of applying local phase rotations, we have and