A Minimum Output Entropy Conjectures

Classical capacity of bosonic broadcast communication and a new minimum output entropy conjecture

Abstract

Previous work on the classical information capacities of bosonic channels has established the capacity of the single-user pure-loss channel, bounded the capacity of the single-user thermal-noise channel, and bounded the capacity region of the multiple-access channel. The latter is a multi-user scenario in which several transmitters seek to simultaneously and independently communicate to a single receiver. We study the capacity region of the bosonic broadcast channel, in which a single transmitter seeks to simultaneously and independently communicate to two different receivers. It is known that the tightest available lower bound on the capacity of the single-user thermal-noise channel is that channel’s capacity if, as conjectured, the minimum von Neumann entropy at the output of a bosonic channel with additive thermal noise occurs for coherent-state inputs. Evidence in support of this minimum output entropy conjecture has been accumulated, but a rigorous proof has not been obtained. In this paper, we propose a new minimum output entropy conjecture that, if proved to be correct, will establish that the capacity region of the bosonic broadcast channel equals the inner bound achieved using a coherent-state encoding and optimum detection. We provide some evidence that supports this new conjecture, but again a full proof is not available.

pacs:
03.67.Hk, 89.70.+c, 42.79.Sz

I Introduction

The past decade has seen several advances in evaluating classical information capacities of several important quantum communication channels (1)(5). Despite these advances (1), exact capacity results are not known for many important and practical quantum communication channels. Here we extend the line of research aimed at evaluating capacities of bosonic communication channels, which began with the capacity derivation for the input photon-number constrained lossless bosonic channel (2),(3). The capacity of the lossy bosonic channel was found in (4), where it was shown that a modulation scheme using classical light (coherent states) suffices to achieve ultimate communication rates over this channel. Subsequent attempts to evaluate the capacity of the lossy bosonic channel with additive Gaussian noise (5) led to a crucial conjecture on the minimum output entropy of a class of bosonic channels (6). Proving that conjecture would complete the capacity proof for the bosonic channel with additive Gaussian noise, and it would show that this channel’s capacity is achievable with classical-light modulation. More recent work that addressed bosonic multiple-access communication channels (7) revealed that modulation of information using non-classical states of light is necessary to achieve ultimate single-user rates in the multiple-access scenario.

In the present work, we study the classical information capacity of the bosonic broadcast channel. A broadcast channel is the congregation of communication media connecting a single transmitter to two or more receivers. In general, the transmitter encodes and sends out independent information to each receiver in a way that each receiver can reliably decode its respective information. We will show that when coherent-state encoding is employed in conjunction with coherent detection, the bosonic broadcast channel is equivalent to a classical degraded Gaussian broadcast channel whose capacity region is known, and known to be dual to that of the classical Gaussian multiple-access channel (8). Thus, under these coding and detection assumptions, the capacity region of the bosonic broadcast channel is dual to that of the multiple-access bosonic channel with coherent-state encoding and coherent detection. To treat more general transmitter and receiver conditions, we use a limiting argument to apply the degraded quantum broadcast-channel coding theorem for finite-dimensional state spaces (9) to the infinite-dimensional bosonic channel with an average photon-number constraint. We consider the two-receiver case in which Alice () simultaneously transmits to Bob (), via the transmissivity port of a lossless beam splitter, and to Charlie (), via that beam splitter’s reflectivity port, using arbitrary encoding and optimum measurement with an average photon number at the input. Given a new conjecture about the minimum output entropy of a lossy bosonic channel, we show that the ultimate capacity region is achieved by coherent-state encoding, and is given by

(1)

for , where is the von Neumann entropy of the Bose-Einstein distribution with mean . Interestingly, this capacity region is not dual to that of the bosonic multiple-access channel with coherent-state encoding and optimum measurement that was found in (7).

The remainder of this paper is organized as follows. Section II gives a brief introduction to the capacity region of classical broadcast channels. In Sec. III, we describe some recent work on the capacity region of the degraded quantum broadcast channel (9). In Sec. IV, we introduce the noiseless bosonic broadcast channel model, and derive its capacity region subject to a new minimum output entropy conjecture. In Sec. V we compare the capacity region obtained in Sec. IV with the classical Gaussian broadcast channel results that apply for coherent-state encoding and coherent (homodyne or heterodyne) detection. We also show that a recent duality result between capacity regions of classical multiple-input, multiple-output Gaussian multiple-access and broadcast channels (8) does not hold for bosonic channels with coherent-state encoding. Discussion of bosonic-channel minimum output entropy conjectures, and evidence supporting the conjecture associated with the bosonic broadcast channel, will be given in Appendix A.

Ii Classical Broadcast Channel

A two-user discrete, memoryless broadcast channel is modeled by a classical probability distribution, , where , , and are drawn from Alice’s input alphabet , and Bob and Charlie’s output alphabets , respectively. A broadcast channel is said to be memoryless if successive uses of the channel are independent, i.e., is the transition distribution for channel uses. A code for a broadcast channel consists of an encoder

(2)

and two decoders

(3)
(4)

The probability of error is the probability that the overall decoded message does not match the transmitted message, i.e.,

(5)

where the messages, , that are sent to Bob and Charlie, respectively, are assumed to be uniformly distributed over the possibilities. A rate pair is said to be achievable, for the broadcast channel, if there exists a sequence of codes with as . The capacity region of the broadcast channel is the closure of the set of achievable rates.

Determining the capacity region of a general broadcast channel is still an open problem. The capacity region is known, however, for degraded broadcast channels, in which one receiver (say ) is “downstream” from the first receiver (say ), so that always receives a degraded version of what observes. In other words, is a Markov chain, so that there exists a distribution, , such that

(6)

Degraded broadcast channels were first studied by Cover (10), who conjectured that the capacity region for Alice to send independent information to Bob and Charlie at rates and respectively over such a channel is the convex hull of the closure of all satisfying

(7)
(8)

for some joint distribution . Here, denotes the Shannon mutual information between ensembles and , and is an auxiliary random variable with cardinality . The achievability of the above capacity result was proved by Bergmans (11), and Gallager proved the converse (12).

Iii Quantum Degraded Broadcast Channel

A quantum channel from Alice to Bob is a trace-preserving completely positive map that transforms Alice’s single-use density operator into Bob’s: . The two-user quantum broadcast channel is a quantum channel from sender Alice () to two independent receivers, Bob () and Charlie (). The quantum channel from Alice to Bob is obtained by tracing out from the channel map, i.e., , with a similar definition for . We say that a broadcast channel is degraded if there exists a degrading channel, , from to satisfying The degraded broadcast channel describes a physical scenario in which for each successive uses of Alice communicates a randomly generated classical message to Bob and Charlie, where the message sets and have cardinalities and respectively. The messages are assumed to be uniformly distributed over . Because of the degraded nature of the channel, Bob receives both and , whereas Charlie only receives .

To convey the message , Alice prepares an -channel-use input state, with density operator , from , the tensor product space of her single-use input-state alphabet. After transmission through the channel, this state results in the bipartite density operator for Bob and Charlie. The reduced density operators for Bob and Charlie, and respectively, can be found by tracing out the other receiver. A code for this channel consists of an encoder

(9)

a positive operator-valued measure (POVM) on and a POVM on that satisfy (13)

(10)

for all . Its error probability therefore obeys . A rate-pair is achievable if there exists a sequence of codes with , so that for such a sequence. The classical capacity region of the degraded quantum broadcast channel is then the convex hull of the closure of all achievable rate pairs .

The classical capacity region of the two-user degraded quantum broadcast channel was recently derived by Yard et al. (9), and can be expressed in terms of the Holevo information (14),

(11)

where is a probability distribution associated with the density operators , and is the von Neumann entropy of the quantum state . Because may not be additive, the rate region of the degraded broadcast channel must be computed by maximizing over multiple channel uses. Thus, for channel uses we can achieve the rate region specified by

(12)
(13)

where is a collective index. The probabilities form a distribution over an auxiliary classical alphabet , of size , satisfying . The ultimate rate-region is computed by maximizing the region specified by Eqs. (12) and (13), over , , , and , subject to the cardinality constraint on . Figure 1 illustrates the setup of the two-user degraded quantum channel.

Figure 1: Schematic of the degraded quantum broadcast channel. The transmitter Alice () encodes her messages to Bob () and Charlie () in a classical index by preparing and in the bipartite state . The dashed line from to denotes the existence of a degrading channel, , whose -fold tensor product will transform into for all .

Iv Noiseless bosonic Broadcast Channel

The two-user noiseless bosonic broadcast channel consists of a collection of spatial and temporal bosonic modes at the transmitter (Alice) that interact with a minimal-quantum-noise environment and split into two sets of spatio-temporal modes en route to two independent receivers (Bob and Charlie). The multi-mode two-user bosonic broadcast channel is given by , where is the broadcast-channel map for the th mode, which can be obtained from the Heisenberg evolutions

(14)
(15)

where are Alice’s modal annihilation operators, and , are the corresponding modal annihilation operators for Bob and Charlie, respectively. The modal transmissivities satisfy , and the environment modes are in their vacuum states. We will limit our treatment here to the single-mode bosonic broadcast channel, as the capacity of the multi-mode channel can in principle be obtained by summing up capacities of all spatio-temporal modes and maximizing the sum capacity region subject to an overall input-power budget using Lagrange multipliers, cf. (5), where this was done for the capacity of the multi-mode single-user lossy bosonic channel.

The principal result we have for the single-mode degraded bosonic broadcast channel depends on a minimum output entropy conjecture (strong conjecture 2, see Appendix A). Assuming this conjecture to be true, we will show that the ultimate capacity region of the single-mode noiseless bosonic broadcast channel (see Fig. 2) with a mean input photon-number constraint is

(16)
(17)

Here, is a parameter that represents the fraction of Alice’s average photon number that is used to convey information to Bob, with remainder to be used to communicate information to Charlie. The boundary of the broadcast channel’s capacity region is traced out by varying from 0 to 1.

It is worth noting, at this point, that our assumption of a lossless beam splitter—in Eqs. (14) (15), and Fig. 2—is not essential. In particular, if the coupling coefficients from to and to in Fig. 2 were and , respectively, with , and , then we still have a degraded quantum broadcast channel, and, assuming strong conjecture 2 is correct, its capacity region is given by Eqs. (16) and (17), with and replacing and , respectively. For simplicity, in all that follows, we shall presume that the lossless beam splitter model applies.

The rate region from Eqs. (16) and (17) is additive and achievable with single channel use coherent-state encoding using the distributions

(18)
(19)
Figure 2: (Color online) A single-mode noiseless bosonic broadcast channel can be envisioned as a beam splitter with transmissivity . With , the bosonic broadcast channel is a degraded quantum broadcast channel, where Bob () is the less-noisy receiver and Charlie () is the more-noisy receiver.

The first step toward proving that Eqs. (16) and (17) do indeed specify the bosonic broadcast channel’s capacity region is to show that Eqs. (18) and (19) achieve these rates. It is straightforward to show that if , the bosonic broadcast channel is a degraded quantum broadcast channel, in which Bob’s is the less-noisy receiver and Charlie’s is the more-noisy receiver. To do so we merely recognize that, when , Charlie’s reduced density operator can be obtained from Bob’s by applying to the input of a lossless beam splitter that has transmissivity to output modes , and whose other input port is driven by vacuum-state modes. The Yard et al. capacity region in Eqs. (12) and (13) requires finite-dimensional Hilbert spaces for the channel’s input and outputs. Nevertheless, we will use their result for the bosonic broadcast channel, which has an infinite-dimensional state space, by extending it to infinite-dimensional state spaces through a limiting argument (15). The = 1 rate-region for the bosonic broadcast channel using a coherent-state encoding is thus:

(20)
(21)

where we need to maximize the bounds for and over all joint distributions subject to . Note that and are complex-valued random variables, and the second term in the bound (12) vanishes, because the von Neumann entropy of a pure state is zero. Substituting Eqs. (18) and (19) into Eqs. (20) and (21) shows that the rate region in Eqs. (16) and (17) is achievable with single-use coherent-state encoding.

For the converse, assume that the rate pair is achievable. Let and the POVMs , comprise any code in the achieving sequence. Suppose that Bob and Charlie store their decoded messages in the classical registers and respectively. Let us use to denote the joint probability mass function of the independent message registers and . As is an achievable rate-pair, there must exist , such that

(22)

where gives the Shannon mutual information in terms of the Shannon entropy for a probability distribution , and . The second line follows from Fano’s inequality (16) and the third line follows from Holevo’s bound (17). Similarly, for , we can bound as follows:

(23)

where the three lines above follow from Fano’s inequality, Holevo’s bound and the concavity of Holevo information.

To complete the converse proof, we need only show that there exists a , such that

(24)
(25)

From the non-negativity of the von Neumann entropy , it follows that , as the second term of the Holevo information above is non-negative. Because von Neumann entropy is subadditive, and the maximum von Neumann entropy of a single-mode bosonic state with is given by , we have that

(26)

where, , and is the mean photon number of the th channel use for the state . Therefore, because and is monotonically increasing for , we see that for each there is a such that

(27)

We know that is Alice’s maximum average photon number per channel use, where the averaging is over the entire codebook. Thus, the mean photon number of the -use average codeword at Bob, , is . Hence, we get

(28)

where the second inequality follows from the convexity of von Neumann entropy. Again invoking the monotonicity of we have that there is a , such that whence

(29)

This proves the first inequality that we need for the capacity region’s converse statement.

To prove the second inequality needed for that converse, we start from Eq. (27) and use strong conjecture 2 (see Appendix A) to get

(30)

Next, we use the uniform distribution to obtain

(31)

Using (31), the convexity of and , we have shown (see Appendix B) that

(32)

From Eq. (32), and Eq. (30) summed over , we then obtain

(33)

Finally, writing Charlie’s Holevo information as

(34)

we can use Eq. (33) to get

(35)

which completes proof of the converse, given that strong conjecture 2 is true.

V Discussion

Without a proof of strong conjecture 2, we cannot assert that Eqs. (16) and (17) define the capacity region of the bosonic broadcast channel. However, because the rate region specified by these equations is achievable—with single-use coherent-state encoding—we know that they comprise an inner bound on that capacity region. In this regard it is instructive to examine how the rate region defined by Eqs. (16) and (17) compares with what can be realized by conventional, coherent-detection optical communications. Suppose Alice sends a coherent state, , to the beam splitter shown in Fig. 2. Bob and Charlie will then receive coherent states and , respectively. Moreover, if Bob and Charlie employ homodyne-detection receivers, with local oscillator phases set to observe the real-part quadrature, their post-measurement data will be for Bob and for Charlie, where and are independent, identically distributed, real-valued Gaussian random variables that are zero mean and have variance 1/4 (18). Similarly, if Bob and Charlie employ heterodyne-detection receivers, their post-measurement data will be and , where and are independent, identically-distributed, complex-valued Gaussian random variables that are zero mean, isotropic, and have variance 1/2 (18). These results imply that the bosonic broadcast channel with coherent-state encoding and homodyne detection is a classical degraded scalar-Gaussian broadcast channel, whose capacity region is known to be (19)

(36)
(37)

for . Likewise, the bosonic broadcast channel with coherent-state encoding and heterodyne detection is a classical degraded vector-Gaussian broadcast channel, whose capacity region is known to be

(38)
(39)

for . In Fig. 3 we compare the capacity regions attained by a coherent-state input alphabet using homodyne detection, heterodyne detection, and optimum reception. As is known for single-user bosonic communications, homodyne detection performs better than heterodyne detection when the transmitters are starved for photons, because it has lower noise. Conversely, heterodyne detection outperforms homodyne detection when the transmitters are photon rich, because it has a factor-of-two bandwidth advantage. To bridge the gap between the coherent-detection capacity regions and the ultimate capacity region, one must use joint detection over long codewords. Future investigation will be needed to develop receivers that can approach the ultimate communication rates over the bosonic broadcast channel.

Figure 3: (Color online) Comparison of bosonic broadcast channel capacity regions, in bits per channel use, achieved by coherent-state encoding with homodyne detection (the capacity region lies inside the boundary marked by circles), heterodyne detection (the capacity region lies inside the boundary marked by dashes), and optimum reception (the capacity region lies inside the region bounded by the solid curve), for , and , , and .

Recently, Vishwanath et al. (8) established the duality between the dirty-paper achievable rate region—recently proved to be the ultimate capacity region (25)—for the classical multiple-input, multiple-output (MIMO) Gaussian broadcast channel and the capacity region of the classical MIMO Gaussian multiple-access channel (MAC). Their duality result states that if we evaluate the capacity regions of the MIMO Gaussian MACs—with fixed total received power and channel-gain values—over all possible power allocations between the users, the corners of those capacity regions trace out the capacity region of the MIMO Gaussian broadcast channel with transmitter power and the same channel-gain values. Of course, the bosonic broadcast channel and the bosonic multiple-access channel satisfy this duality when they employ coherent-state encoding and coherent detection, because under these conditions these quantum channels reduce to classical additive Gaussian-noise channels. However, it turns out that the capacity region of the bosonic broadcast channel using coherent-state inputs and optimum reception is not equal to of the envelope of the MAC capacity regions using coherent-state inputs. The capacity region of the bosonic MAC using coherent-state inputs was first computed by Yen (7). In Fig. 4 we compare the envelope of coherent-state MAC capacities to the capacity region of the coherent-state broadcast channel. This figure shows that with a fixed beam splitter and identical average photon number budgets, more collective classical information can be sent when the beam splitter is used as a multiple-access channel as opposed to when it is used as a broadcast channel if coherent-state encoding is employed.

Figure 4: (Color online) Comparison of bosonic broadcast and multiple-access channel capacity regions, in bits per channel use, for , and with coherent-state encoding. The dashed curve is the outer boundary of the broadcast capacity region. It lies below the solid curve, which is the outer envelope of the MAC capacity regions.

Acknowledgment

This research was supported by the Defense Advanced Research Projects Agency and by the W. M. Keck Foundation Center for Extreme Quantum Information Theory.

Appendix A Minimum Output Entropy Conjectures

In general, the evolution of a quantum state resulting from the state’s propagation through a quantum communication channel is not unitary, so that a pure state loses some coherence in its transit through the channel. The minimum von Neumann entropy at the channel’s output provides a useful measure of the channel’s ability to preserve the coherence of its input state. In particular, the output entropy associated with a pure state measures the entanglement that such a state establishes with the environment during propagation through the channel. Because the state of the environment is not accessible, this entanglement is responsible for the loss of quantum coherence and hence for the injection of noise into the channel output. The study of minimum output entropy yields important information about channel capacities, viz., an upper bound on the classical capacity derives from a lower bound on the output entropy of multiple channel uses, see, e.g., (5). Also, the additivity of the minimum output entropy implies the additivity of the classical capacity and of the entanglement of formation (20).

In this appendix we first briefly review previous work on a minimum output entropy conjecture that arose in conjunction with the channel capacity analysis of the single-user bosonic channel with additive Gaussian noise (5),(6),(21)(23). Then we will turn our attention to the minimum output entropy conjecture that we employed in our capacity theorem for the degraded bosonic broadcast channel. Both conjectures have weak (single-use) and strong (multiple-use) versions.

Let and denote the two input modes of a lossless beam splitter of transmissivity , that has output modes and . In (6), the following minimum output entropy conjecture was made.

Conjecture 1 — Let the input mode be in a thermal state with average photon number (hence von Neumann entropy ). Then the von Neumann entropy of the output mode is minimized when the input mode is in the vacuum state. The resulting minimum von Neumann output entropy is .

The above conjecture is a special case of the following strong (multi-mode) version whose proof would establish the ultimate capacity of the single-user bosonic channel with thermal noise.

Strong Conjecture 1 — Let the input modes be in a product state of thermal states with total von Neumann entropy . Then the von Neumann entropy of the output modes is minimized when the input modes are in their vacuum states. The resulting minimum von Neumann output entropy is .

Neither strong conjecture 1 nor its weak (single-use) form have been proven yet, but considerable evidence in support of their validity has been developed. For example, strong conjecture 1 has been shown to be true when the input states are restricted to be Gaussian (23). It has also been proven that the vacuum state provides a local minimum for the output entropy (6). Strong conjecture 1 has been shown to be true when Rényi entropy of integer order is employed in lieu of von Neumann entropy (21). Similarly, conjecture 1 has been proven when Wehrl entropy—the continuous Boltzmann-Gibbs entropy of the Husimi probability function (24)—is used instead of von Neumann entropy (21). Additional evidence in support of conjecture 1 can be found in (6).

In proving the converse to the bosonic broadcast channel’s capacity theorem we assumed the validity of the following conjecture.

Strong Conjecture 2 — Let the input modes be in their vacuum states, and let the von Neumann entropy of the input modes be . Then, putting the modes in a product state of mean-photon-number thermal states minimizes the von Neumann entropy of the output modes . The resulting minimum von Neumann output entropy is .

The weaker, single-use version of this conjecture is also of interest.

Conjecture 2 — Let the input mode be in its vacuum state, and the let the von Neumann entropy of the input mode be . Then the von Neumann entropy of the output mode is minimized when is in a thermal state with average photon number . The resulting minimum von Neumann output entropy is .

We have yet to develop proofs for either strong conjecture 2 or conjecture 2. In the rest of this appendix we will present evidence that supports their validity. Toward that end, we first show that strong conjecture 2 is true when Wehrl entropy is used instead of von Neumann entropy.

a.1 Strong Conjecture 2 for Wehrl Entropy

Strong Conjecture 2: Wehrl — Let the input modes be in their vacuum states, and let the Wehrl entropy of the input modes be . Then, putting the modes in a product state of mean-photon-number thermal states minimizes the Wehrl entropy of the output modes . The resulting minimum Wehrl output entropy is .

Proof — The Wehrl entropy for an -mode density operator is

(40)

where , with a coherent state, is the Husimi distribution, i.e., the joint probability density function for multi-mode heterodyne detection. The Wehrl entropy provides a measurement of the state in phase space and its minimum value is achieved on coherent states (24).

Our proof of strong conjecture 2 for Wehrl entropy relies on the antinormally-ordered characteristic function, , associated with the -mode density operator , namely

(41)

where is a column vector of complex numbers, , and . The antinormally-ordered characteristic function and the Husimi function are a 2-D Fourier transform pair:

(42)
(43)

As the two -use input modes and are in a product state, Eq. (41) implies that the output-state characteristic function is a product of the input-state characteristic functions with appropriately scaled arguments,

(44)

From Eq. (44), the multiplication-convolution and scaling properties of Fourier-transforms pairs, and the fact that is in the -mode vacuum state, we find that

(45)

where denotes convolution.

Suppose that the state of the input modes is a product of thermal states, each with mean photon number , i.e.,

(46)

The Wehrl entropy for the modes is then

(47)

which satisfies the hypothesis of strong conjecture 2 for Wehrl entropy. Using Eq. (45), we can now show that the Husimi function and the Wehrl entropy for the state of the output modes are

(48)
(49)

providing an upper-bound to the minimum Wehrl output entropy.

To show that the expression in Eq. (47) is also a lower bound for the Wehrl output entropy, we use Theorem 6 of (26), which states that for two probability distributions and over -dimensional complex vectors we have

(50)

for all , where the Wehrl entropy of a probability distribution is found from Eq. (40) by replacing with the given probability distribution. Choosing

(51)
(52)

we get

(53)

The Wehrl entropy of a scaled distribution is easily shown to satisfy

(54)

for any . From Eqs. (54) and (53) we then obtain

(55)