Additive Classical Capacity of Quantum Channels Assisted by Noisy Entanglement

Abstract

We give a capacity formula for the classical information transmission over a noisy quantum channel, with separable encoding by the sender and limited resources provided by the receiver's preshared ancilla. Instead of a pure state, we consider the signal-ancilla pair in a mixed state, purified by a "witness." Thus, the signalwitness correlation limits the resource available from the signal-ancilla correlation. Our formula characterizes the utility of different forms of resources, including noisy or limited entanglement assistance, for classical communication. With separable encoding, the sender's signals across multiple channel uses are still allowed to be entangled, yet our capacity formula is additive. In particular, for generalized covariant channels, our capacity formula has a simple closed form. Moreover, our additive capacity formula upper bounds the general coherent attack's information gain in various two-way quantum key distribution protocols. For Gaussian protocols, the additivity of the formula indicates that the collective Gaussian attack is the most powerful. DOI: 10.1103/PhysRevLett.118.200503 Communication channels model the physical medium for information transmission between the sender (Alice) and the receiver (Bob). Classical information theory [1,2] says that a channel is essentially characterized by a single quantity-the (classical) channel capacity, i.e., its maximum (classical) information transmission rate. However, quantum channels [3] can transmit information beyond the classical. Formally, a (memoryless) quantum channel is a time-invariant completely positive trace preserving (CPTP) linear map between quantum states. Various types of information lead to various capacities, e.g., classical capacity C [4,5] for classical information transmission encoded in quantum states, and quantum capacity Q [6-8] for quantum information transmission. For both cases, implicit constraints on the input Hilbert space, e.g., fixed dimension or energy, quantify the resources. Resources can also be in the form of assistance: given unlimited entanglement, one has the entanglement-assisted classical capacity C E [9]. References [10,11] provide a capacity formula for the trade-off of classical and quantum information transmission and entanglement generation (or consumption).With the trade-off capacity formula in hand, it appears that the picture of communication over quantum channels is complete. However, our understanding about the trade-off is plagued by the "nonadditivity" issue [3], best illustrated by the example of C. The Holevo-Schumacher-Westmoreland (HSW) theorem [4,5] gives the one-shot capacity C ð1Þ ðΨÞ of channel Ψ, which assumes product-state input in multiple channel uses. Consider An exception is the (unlimited) entanglement-assisted classical capacity C E [9]. Since it has the form of quantum mutual information [14,15], C E is additive [9,16]. One immediately hopes that the additivity can be extended to classical communication assisted by imperfect entanglement since entanglement is fragile...

Security-proof framework for two-way Gaussian quantum-key-distribution protocols

“…III, we prove that the capacity formula in Ref. [40] provides an upper bound on Eve's information gain from her most general coherent attack. There, we also describe the pair-wise sum constraints and permutation-invariant sum constraints that lead to efficiently calculable bounds on Eve's coherent-attack information gain, and we show that the resulting upper bound can be achieved by a collective attack.…”

“…Here we present a detailed proof that Ref. [40]'s capacity formula provides an upper bound on the the most general blockwise coherent attack's information gain. Our proof applies to all TW-QKD protocols in which Bob performs the key map, not just to the Gaussian special case.…”

“…To accomplish the reduction, we will make use of the tools from Ref. [40]. Consider a multiple-block QKD session in which Alice and Bob spend some small amount of pre-shared key to randomly discard some of the size-M B blocks, during post-processing of their raw keys.…”

“…Reference [40] developed a way to bound Eve's information gain for a TW-QKD protocol, but that reference focused on a rigorous formulation for noisy entanglementassisted classical capacity, and did not present in full detail a security proof for TW-QKD protocols. Here we present a detailed proof that Ref.…”

“…We will use the recently developed noisy-entanglementassisted classical capacity formula [14,40] to resolve the preceding single-letterization dilemma for Gaussian TW-QKD protocols, i.e., protocols that employ Gaussianstate sources and Gaussian operations [41]. We consider two different constraints-the pair-wise sum constraints and the permutation-invariant sum constraintsthat provide intrusion parameters which suffice to establish tractable (single-letter) upper bounds on Eve's coherent-attack information gain.…”