The fully quantum reverse Shannon theorem establishes the optimal rate of noiseless classical communication required for simulating the action of many instances of a noisy quantum channel on an arbitrary input state, while also allowing for an arbitrary amount of shared entanglement of an arbitrary form. Turning this theorem around establishes a strong converse for the entanglement-assisted classical capacity of any quantum channel. The present work proves the strong converse for entanglement-assisted capacity by a completely different approach. Namely, we exploit the recent entanglement-assisted “meta-converse” theorem of Matthews and Wehner, several properties of the recently established sandwiched Rényi relative entropy (also referred to as the quantum Rényi divergence), and the multiplicativity of completely bounded p-norms due to Devetak et al. The proof here demonstrates the extent to which the Arimoto approach can be helpful in proving strong converse theorems, it provides an operational relevance for the multiplicativity result of Devetak et al., and it adds to the growing body of evidence that the sandwiched Rényi relative entropy is the correct quantum generalization of the classical concept for all α > 1.
The fully quantum reverse Shannon theorem establishes the optimal rate of noiseless classical communication required for simulating the action of many instances of a noisy quantum channel on an arbitrary input state, while also allowing for an arbitrary amount of shared entanglement of an arbitrary form. Turning this theorem around establishes a strong converse for the entanglement-assisted classical capacity of any quantum channel. The present work proves the strong converse for entanglement-assisted capacity by a completely different approach. Namely, we exploit the recent entanglement-assisted “meta-converse” theorem of Matthews and Wehner, several properties of the recently established sandwiched Rényi relative entropy (also referred to as the quantum Rényi divergence), and the multiplicativity of completely bounded p-norms due to Devetak et al. The proof here demonstrates the extent to which the Arimoto approach can be helpful in proving strong converse theorems, it provides an operational relevance for the multiplicativity result of Devetak et al., and it adds to the growing body of evidence that the sandwiched Rényi relative entropy is the correct quantum generalization of the classical concept for all α > 1.
Quantum entanglement can be used in a communication scheme to establish a correlation between successive channel inputs that is impossible by classical means. It is known that the classical capacity of quantum channels can be enhanced by such entangled encoding schemes, but this is not always the case. In this paper, we prove that a strong converse theorem holds for the classical capacity of an entanglement-breaking channel even when it is assisted by a classical feedback link from the receiver to the transmitter. In doing so, we identify a bound on the strong converse exponent, which determines the exponentially decaying rate at which the success probability tends to zero, for a sequence of codes with communication rate exceeding capacity. Proving a strong converse, along with an achievability theorem, shows that the classical capacity is a sharp boundary between reliable and unreliable communication regimes. One of the main tools in our proof is the sandwiched Rényi relative entropy. The same method of proof is used to derive an exponential bound on the success probability when communicating over an arbitrary quantum channel assisted by classical feedback, provided that the transmitter does not use entangled encoding schemes.
We determine the exact strong converse exponent for entanglement-assisted classical communication of a quantum channel. Our main contribution is the derivation of an upper bound for the strong converse exponent which is characterized by the sandwiched Rényi divergence. It turns out that this upper bound coincides with the lower bound of Gupta and Wilde (Commun. Math. Phys. 334:867-887, 2015). Thus, the strong converse exponent follows from the combination of these two bounds. Our result has two implications. Firstly, it implies that the exponential bound for the strong converse property of quantum-feedback-assisted classical communication, derived by Cooney, Mosonyi and Wilde (Commun. Math. Phys. 344:797-829, 2016), is optimal. This answers their open question in the affirmative. Hence, we have determined the exact strong converse exponent for this problem as well. Secondly, due to an observation of Leung and Matthews, it can be easily extended to deal with the transmission of quantum information under the assistance of entanglement or quantum feedback, yielding similar results. The above findings provide, for the first time, a complete operational interpretation to the channel’s sandwiched Rényi information of order α > 1.
Abstract In their seminal work, Bennett et al. [1] showed that, with sufficient shared randomness, one noisy channel can simulate another at a rate equal to the ratio of their capacities. We establish that when coding above this channel interconversion capacity, the exact strong converse exponent is characterized by a simple optimization involving the difference of the corresponding Rényi channel capacities with Hölder dual parameters. We further extend this result to the entanglement-assisted interconversion of classical–quantum channels, showing that the strong converse exponent is likewise determined by differences of sandwiched Rényi channel capacities. The converse bound is obtained by relaxing to non-signaling assisted codes and applying Hölder duality together with the data processing inequality for Rényi divergences. Achievability is proven by concatenating refined channel coding and simulation protocols that go beyond first-order capacities, attaining an exponentially small conversion error, remaining robust under small variations in the input distribution, and tolerating a sublinear gap between the conversion rates.
The Quantum Reverse Shannon Theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel’s entanglement assisted classical capacity. In this paper, we provide a new proof of this theorem, which has previously been proved by Bennett, Devetak, Harrow, Shor, and Winter. Our proof has a clear structure being based on two recent information-theoretic results: one-shot Quantum State Merging and the Post-Selection Technique for quantum channels.
Dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one, reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones, and more generally the use of one noisy channel to simulate another. For channels of nonzero capacity, this simulation is always possible, but for it to be efficient, auxiliary resources of the proper kind and amount are generally required. In the classical case, shared randomness between sender and receiver is a sufficient auxiliary resource, regardless of the nature of the source, but in the quantum case the requisite auxiliary resources for efficient simulation depend on both the channel being simulated, and the source from which the channel inputs are coming. For tensor power sources (the quantum generalization of classical IID sources), entanglement in the form of standard ebits (maximally entangled pairs of qubits) is sufficient, but for general sources, which may be arbitrarily correlated or entangled across channel inputs, additional resources, such as entanglement-embezzling states or backward communication, are generally needed. Combining existing and new results, we establish the amounts of communication and auxiliary resources needed in both the classical and quantum cases, the tradeoffs among them, and the loss of simulation efficiency when auxiliary resources are absent or insufficient. In particular we find a new single-letter expression for the excess forward communication cost of coherent feedback simulations of quantum channels (i.e. simulations in which the sender retains what would escape into the environment in an ordinary simulation), on non-tensor-power sources in the presence of unlimited ebits but no other auxiliary resource. Our results on tensor power sources establish a strong converse to the entanglement-assisted capacity theorem.
We use a Rényi entropy method to prove a strong converse theorem for the task of quantum state redistribution. More precisely, we establish the strong converse property for the boundary of the entire achievable rate region in the (e, q)-plane, where the entanglement cost e and quantum communication cost q are the operational rates describing a state redistribution protocol. The strong converse property is deduced from explicit bounds on the fidelity of the protocol in terms of a Rényi generalization of the optimal rates. Hence, we identify candidates for the strong converse exponents for entanglement cost e and quantum communication cost q, respectively. To prove our results, we establish various new entropic inequalities, which might be of independent interest. These involve conditional entropies and mutual information derived from the sandwiched Rényi divergence. In particular, we obtain novel bounds relating these quantities to the fidelity of two quantum states.