On the Second-Order Asymptotics for
The entanglement-assisted classical capacity of a quantum channel is known to provide the formal quantum generalization of Shannon’s classical channel capacity theorem, in the sense that it admits a single-letter characterization in terms of the quantum mutual information and does not increase in the presence of a noiseless quantum feedback channel from receiver to sender. In this work, we investigate second-order asymptotics of the entanglement-assisted classical communication task. That is, we consider how quickly the rates of entanglement-assisted codes converge to the entanglement-assisted classical capacity of a channel as a function of the number of channel uses and the error tolerance. We define a quantum generalization of the mutual information variance of a channel in the entanglement-assisted setting. For covariant channels, we show that this quantity is equal to the channel dispersion, and thus completely characterize the convergence towards the entanglement-assisted classical capacity when the number of channel uses increases. Our results also apply to entanglement-assisted quantum communication, due to the equivalence between entanglement-assisted classical and quantum communication established by the teleportation and super-dense coding protocols.
Let us consider the transmission of classical information through a memoryless quantum channel. If the sender and receiver initially share entangled states which they may use in their communication protocol, then the information transmission is said to be entanglement-assisted. The entanglement-assisted classical capacity of a quantum channel is defined to be the maximum rate at which a sender and receiver can communicate classical information with vanishing error probability by using the channel as many times as they wish and by using an arbitrary amount of shared entanglement of an arbitrary form. For a noiseless quantum channel, the entanglement-assisted classical capacity is twice as large as its unassisted one, an enhancement realized by the super-dense coding protocol [BW92]. This is in stark contrast to the setting of classical channels where additional shared randomness or entanglement does not increase the capacity.
Similarly, for a noisy quantum channel, the presence of entanglement as an auxiliary resource can also lead to an enhancement of its classical capacity [BSST99, BSST02]. Somewhat surprisingly, entanglement assistance is advantageous even for some entanglement-breaking channels [HSR03], such as depolarizing channels with sufficiently high error probability. Bennett, Shor, Smolin and Thapliyal [BSST02] proved that the entanglement-assisted classical capacity of a quantum channel is given by a remarkably simple, single-letter formula in terms of the quantum mutual information (defined in the following section). This is in contrast to the unassisted classical capacity of a quantum channel [Hol02b, SW97], for which the best known general expression involves a regularization of the Holevo formula over infinitely many instances of the channel [Has09]. The regularization renders the explicit evaluation of the capacity for a general quantum channel intractable. The formula for the entanglement-assisted capacity is formally analogous to Shannon’s well-known formula [Sha48] for the capacity of a discrete memoryless classical channel, which is given in terms of the mutual information between the channel’s input and output. The entanglement-assisted capacity does not increase under the presence of a noiseless quantum feedback channel from receiver to sender [Bow04], much like the capacity of a classical channel does not increase in the presence of a noiseless classical feedback link [Sha56].
The formula for derived in [BSST99], however, is only relevant if the channel is available for as many uses as the sender and receiver wish, with there being no correlations in the noise acting on its successive inputs.111In other words, the channel is assumed to be memoryless. To see this, let us consider the practical scenario in which a memoryless channel is used a finite number times, and let . Let denote the maximum number of bits of information that can be transmitted through uses of the channel via an entanglement-assisted communication protocol, such that the average probability of failure is no larger than . Then [BSST99] and the strong converse [BDH14, BCR11] imply that
The strong converse from [GW15] implies that
for all . The results of [CMW14] imply that this same expansion holds even when noiseless quantum feedback communication is allowed from receiver to sender.
We are interested in investigating the behavior of for large but finite as a function of . In this paper, we derive a lower bound on , for any fixed value of and large enough, of the following form:
The coefficient that we identify in this paper constitutes a second-order coding rate. The second-order coding rate obtained here depends on the channel as well as on the allowed error threshold , and we obtain an explicit expression for it in Theorem 3. In addition, we conjecture that in fact for all quantum channels. We show that this conjecture is true for the class of covariant channels [Hol02b].
Our lower bound on resembles the asymptotic expansion for the maximum number of bits of information which can be transmitted through uses of a generic discrete, memoryless classical channel , with an average probability of error no larger than , denoted . The latter was first derived by Strassen in 1962 [Str62] and refined by Hayashi [Hay09] as well as Polyanskiy, Poor and Verdú [PPV10]. It is given by
where denotes uses of the channel, is its capacity given by Shannon’s formula [Sha48], is the inverse of the cumulative distribution function of a standard normal random variable, and is a property of the channel (which depends on ) called its -dispersion [PPV10]. The right hand side of (1.4) is called the Gaussian approximation of . This result has recently been generalized to classical coding for quantum channels [TT15] and it was shown that a formula reminiscent of (1.4) also holds for the classical capacity of quantum channels with product inputs. In fact, the Gaussian approximation is a common feature of the second-order asymptotics for optimal rates of many other quantum information processing tasks such as data compression, communication, entanglement manipulation and randomness extraction (see, e.g., [TH13, KH13, TT15, DL15] and references therein).
Even though we focus our presentation throughout on entanglement-assisted classical communication, we would like to point out that all of the results established in this paper apply to entanglement-assisted quantum communication as well. This is because the protocols of teleportation [BBC93] and super-dense coding [BW92] establish an equivalence between entanglement-assisted classical and quantum communication. This equivalence was noted in early work on entanglement-assisted communication [BSST99]. That this equivalence applies at the level of individual codes is a consequence of the development, e.g., in Appendix B of [LM15], and as a result, the equivalence applies to second-order asymptotics as well. This point has also been noted in [TBR15].
Finally, we note that a one-shot lower bound on has already been derived in [DH13]. Moreover, in [MW14] a one-shot upper bound was obtained. Even though these bounds converge in first order to the formula for the capacity obtained by Bennett et al. [BSST02], neither of these works deals with characterizing second-order asymptotics.
This paper is organized as follows. Section 2 introduces the necessary notation and definitions. Section 3 presents our main theorem and our conjecture. The proof of the theorem is given in Section 4. In Section 4, we also provide a proof of our conjecture for the case of covariant channels. We end with a discussion of open questions in Section 5, summarizing the problems encountered when trying to prove the converse for general channels.
2 Notations and Definitions
Let denote the algebra of linear operators acting on a finite-dimensional Hilbert space . Let be the set of positive semi-definite operators, and let denote the set of quantum states (density matrices), . We denote the dimension of a Hilbert space by and write when and are isomorphic, i.e., if . A quantum state is called pure if it is rank one; in this case, we associate with it an element such that . The set of pure quantum states is denoted .
For a bipartite operator , let denote its restriction to the subsystem , where denotes the partial trace over . Let denote the identity operator on , and let be the completely mixed state in .
A positive operator-valued measure (POVM) is a set such that , where denotes any index set. We use the convention that refers to a completely positive trace-preserving (CPTP) map . We call such maps quantum channels in the following. The identity map on is denoted .
We employ the cumulative distribution function for a standard normal random variable:
and its inverse .
2.1 Entanglement-Assisted Codes
We consider entanglement-assisted classical (EAC) communication through a noisy quantum channel, given by a CPTP map . The sender (Alice) and the receiver (Bob) initially share an arbitrary pure state , where without loss of generality we assume that , the system being with Alice and the system with Bob. The goal is to transmit classical messages from Alice to Bob, labelled by the elements of an index set , through .
Without loss of generality, any EAC communication protocol can be assumed to have the following form: Alice encodes her classical messages into states of the system in her possession. Let the encoding CPTP map corresponding to message be denoted by . Alice then sends the system through to Bob. Subsequently, Bob performs a POVM on the system in his possession. This yields a classical register which contains his inference of the message sent by Alice.
The above defines an EAC code for the quantum channel , which consists of a quadruple
The size of a code is denoted as . The average probability of error for on is
The following quantity describes the maximum size of an EAC code for transmitting classical information through a single use of with average probability of error at most .
Let and be a quantum channel. We define
where is a code as prescribed in (2.2).
We are particularly interested in the quantity , where and is the -fold memoryless repetition of .
2.2 Information Quantities
For a pair of positive semi-definite operators and with , the quantum relative entropy and the relative entropy variance [Li14, TH13] are respectively defined as follows:222All logarithms in this paper are taken to base two.
For a bipartite state , let us define the mutual information . Similarly, we define the mutual information variance .
The EAC capacity of a quantum channel is defined as
Bennett, Shor, Smolin and Thapliyal [BSST02] established that the EAC capacity for a quantum channel satisfies
In analogy with [TT15], the following definitions will be used to characterize our lower bounds on the second-order asymptotic behavior of .
Our main result is stated in the following theorem, which provides a second-order lower bound on the maximum number of bits of classical message which can be transmitted through independent uses of a noisy channel via an entanglement-assisted protocol, for any given allowed error threshold.
Let and let be a quantum channel. Then,
The proof of Theorem 3 is split into two parts, Proposition 11 in Section 4.2 and Proposition 14 in Section 4.3. We first derive a one-shot lower bound on using a coding scheme that is a one-shot version of the coding scheme given in [HDW08] and reviewed in [Wil13, Sec. 20.4]. Our one-shot bound is expressed in terms of an entropic quantity called the hypothesis testing relative entropy [WR12], which has its roots in early work on the quantum Stein’s lemma [HP91] (see Section 4.1 for a definition). The relation between classical coding over a quantum channels and binary quantum hypothesis testing was first pointed out by Hayashi and Nagaoka [HN03].
An asymptotic expansion for this quantity for product states was derived independently by Tomamichel and Hayashi [TH13] and Li [Li14], and we make use of this to obtain our lower bound on in the second step.
This leads us to the following conjecture:
We conjecture that (3.1) is an equality with . In particular, we conjecture that the EAC -channel dispersion satisfies
and, thus, .
4.1 Technical Preliminaries
For given orthonormal bases and in isomorphic Hilbert spaces of dimension , we define a maximally entangled state of Schmidt rank to be
Note that if then is a product state. We often make use of the following identity (“transpose trick”): for any operator ,
where denotes the transpose.
4.1.1 Distance Measures
The trace distance between two states and is given by
where denotes the projector onto the subspace where the operator is positive semi-definite. The fidelity of and is defined as
For a pair of pure states and , the trace distance and fidelity satisfy the following relation:
4.1.2 Relative Entropies for One-Shot Analysis
We will phrase our one-shot bounds in terms of the following relative entropy.
Let , and . Then, the hypothesis testing relative entropy [WR12] is defined as
Note that when , has an interpretation as the smallest type-II error of a hypothesis test between and , when the type-I error is at most . The following lemma lists some properties of .
Let . The hypothesis testing relative entropy has the following properties:
For any , we have .
For any , , , we have .
For any classical-quantum state
for any (with and probability distributions on ), and for any , we have
For any , with , and , we have .
Properties 1–3 can be verified by close inspection and we omit their proofs.
Proof of Property 4.
Consider to be the operator achieving the minimum in the definition of , i.e.
From (4.3), we have Hence, , and
which concludes the proof. ∎
Two other generalized relative entropies which are relevant for our analysis are the collision relative entropy and the information-spectrum relative entropy [TH13, Def. 8]. For any pair of positive semi-definite operators and satisfying the condition , they are respectively defined as follows:
and, for any ,
where we write if is positive semidefinite. The following result, proved in [BG14, Thm. 4], relates these quantities.
Lemma 9 ([Bg14]).
Let and . Then,
Finally, the following lemma provides a useful relation between the hypothesis testing relative entropy and the information spectrum relative entropy [TH13, Lm. 12].
Lemma 10 ([Th13]).
Let , , , and . Then,
4.2 One-Shot Achievability
Our protocol is modeled after [BSST02]. Consider a quantum channel and introduce an auxiliary Hilbert space . Let
be a decomposition of , and set . We assume that can be written as a superposition of maximally entangled states:
where denotes a maximally entangled state of Schmidt rank in and is some probability distribution so that .
Let , and let be a quantum channel. Then for any of the form (4.17), we have
where , , and is the maximally mixed state on .
Note that the hypothesis testing relative entropy on the right hand side is not reminiscent of a mutual information type quantity since the second argument is not a product state.
Consider the set
where the index labels the Hilbert spaces of the decomposition in (4.16). For any , consider the following unitary operator in :
where and are the Heisenberg-Weyl operators defined in Appendix A.
For any , we now construct a random code as follows. Let . We set (i.e., we use the labels interchangeably), and . We consider the resource state . For each message , choose a codeword, , uniformly at random from the set . The encoding operation, , is then given by the (random) unitary as prescribed above. In particular,
We denote the corresponding channel output state by and use “pretty good” measurements for decoding. These are given by the POVM , where
Then, following Beigi and Gohari [BG14, Thm. 5], we find that the average probability of successfully inferring the sent message can be expressed as
Moreover employing both the data-processing inequality and joint convexity of the collision relative entropy as in [BG14], we establish the following lower bound on the expected value of with respect to the randomly chosen codewords:
and and are the corresponding reduced states on the systems and , respectively. In particular, defining , using the decomposition (4.17) of the state and the definition (4.20) of the unitary operators , we find that
can be written as the sum of a diagonal () and an off-diagonal () term. It can be verified (see, e.g., [Wil13, pp. 504–505]) that the off-diagonal term vanishes and in fact
where and are completely mixed states. The above identity follows from the fact that applying a Heisenberg-Weyl operator uniformly at random completely randomizes a quantum state, yielding a completely mixed state.
Hence, for any , we have
where the last line follows from Lemma 9. Thus, provided that
the random code satisfies . In particular, there exists a (deterministic) code which satisfies . Hence, we conclude that
where we require that and use
Further, from (4.34) it follows that