For many information processing tasks, de Finetti-style theorems can often simplify the analysis in worst-case input scenarios for which the task exhibits some permutation-invariance symmetry, as they can allow for a reduction from an analysis on worst-case inputs to that of i.i.d. inputs. If further information is available on the inputs, it might be advantageous to reflect this information in the de Finetti reduction. In our work, we focus on a form of such constraint, based on the type of the input. This allows us to obtain a conceptually simple proof of a new de Finetti reduction for classical probability distributions, derived from elementary properties from the method of types. We apply our constrained de Finetti reduction to the compression of quantum interactive communication protocols with classical inputs, and prove that the prior-free quantum information cost equals the worst-case input amortized quantum communication cost.
For many information processing tasks, de Finetti-style theorems can often simplify the analysis in worst-case input scenarios for which the task exhibits some permutation-invariance symmetry, as they can allow for a reduction from an analysis on worst-case inputs to that of i.i.d. inputs. If further information is available on the inputs, it might be advantageous to reflect this information in the de Finetti reduction. In our work, we focus on a form of such constraint, based on the type of the input. This allows us to obtain a conceptually simple proof of a new de Finetti reduction for classical probability distributions, derived from elementary properties from the method of types. We apply our constrained de Finetti reduction to the compression of quantum interactive communication protocols with classical inputs, and prove that the prior-free quantum information cost equals the worst-case input amortized quantum communication cost.
De Finetti theorems show how sufficiently exchangeable states are well-approximated by convex combinations of independent identically distributed states. Recently, it was shown that in many quantum information applications, a more relaxed de Finetti reduction (i.e., only a matrix inequality between the symmetric state and one of the de Finetti forms) is enough and that it leads to more concise and elegant arguments. Here we show several uses and general flexible applicability of a constrained de Finetti reduction in quantum information theory, which was recently discovered by Duan, Severini, and Winter. In particular, we show that the technique can accommodate other symmetries commuting with the permutation action and permutation-invariant linear constraints. We then demonstrate that, in some cases, it is also fruitful with convex constraints, in particular separability in a bipartite setting. This is a constraint particularly interesting in the context of the complexity class QMA(2) of interactive quantum Merlin-Arthur games with unentangled provers, and our results relate to the soundness gap amplification of QMA(2) protocols by parallel repetition. It is also relevant for the regularization of certain entropic channel parameters. As an aside, we present an alternative way of attacking this problem, relying on an entanglement measure theory rather than the de Finetti approach. Finally, we explore an extension to infinite-dimensional systems, which usually pose inherent problems to de Finetti techniques in the quantum case.
Correlations across channel uses can improve quantum communication rates, making optimization over arbitrarily large blocks a central difficulty in determining quantum capacity. We show how full permutation invariance limits this advantage for every finite-dimensional memoryless channel. The optimized coherent information per use converges to the single-use maximum and consequently attains its supremum at a finite block length. Thus finite-block superadditivity remains possible, while the asymptotic optimization reduces to a single channel use. We also establish a strong converse for pure-source entanglement generation with permutation-invariant inputs: at any fixed rate above the single-use coherent-information maximum, the fidelity tends to zero. Beyond these communication bounds, we identify a common structure underlying completely bounded norms and the sandwiched Rényi channel quantities governing discrimination against a replacer and entanglement-assisted communication. De Finetti reduction and permutation covariance yield systematic alternative proofs of their known multiplicativity and additivity results within a shared variational framework. We further show that, for permutation-invariant inputs, correlations cannot sustain an asymptotic reduction in output entropy per use below the single-use minimum. This entropy limit holds for Rényi orders greater than one and for the von Neumann entropy, extending the role of symmetry from communication bounds to the control of output entropy.
de Finetti-type theorems enable a substantially simplified analysis of information-processing tasks in various areas, such as quantum cryptography and quantum tomography. The idea is that instead of carrying out the analysis for any possible state it is sufficient to consider one particular de Finetti state, that is, a convex combination of i.i.d. states. It is thus interesting to see whether such de Finetti-type theorems are unique for quantum theory or whether they hold for more general theories. Here we prove a de Finetti-type theorem in the framework of conditional probability distributions. In this framework, a system is described by a conditional probability distribution P_A|X where X denotes the measurement and A the outcome. We show that any permutation invariant system P_A|X can be reduced to a de Finetti system. We then demonstrate how the theorem can be applied to security proofs of device independent cryptographic protocols.
We define a new notion of information cost for quantum protocols, and a corresponding notion of quantum information complexity for bipartite quantum channels, and then investigate the properties of such quantities. These are the fully quantum generalizations of the analogous quantities for bipartite classical functions that have found many applications recently, in particular for proving communication complexity lower bounds. Our definition is strongly tied to the quantum state redistribution task. Previous attempts have been made to define such a quantity for quantum protocols, with particular applications in mind; our notion differs from these in many respects. First, it directly provides a lower bound on the quantum communication cost, independent of the number of rounds of the underlying protocol. Secondly, we provide an operational interpretation for quantum information complexity: we show that it is exactly equal to the amortized quantum communication complexity of a bipartite channel on a given state. This generalizes a result of Braverman and Rao to quantum protocols, and even strengthens the classical result in a bounded round scenario. Also, this provides an analogue of the Schumacher source compression theorem for interactive quantum protocols, and answers a question raised by Braverman. We also discuss some potential applications to quantum communication complexity lower bounds by specializing our definition for classical functions and inputs. Building on work of Jain, Radhakrishnan and Sen, we provide new evidence suggesting that the bounded round quantum communication complexity of the disjointness function is Ω(n/M + M), for M-message protocols. This would match the best known upper bound.
Winter's measurement compression theorem stands as one of the most penetrating insights of quantum information theory. In addition to making an original and profound statement about measurement in quantum theory, it also underlies several other general protocols used for entanglement distillation and local purity distillation. The theorem provides for an asymptotic decomposition of any quantum measurement into noise and information . This decomposition leads to an optimal protocol for having a sender simulate many independent instances of a quantum measurement and send the measurement outcomes to a receiver, using as little communication as possible. The protocol assumes that the parties have access to some amount of common randomness, which is a strictly weaker resource than classical communication. In this review, we provide a second look at Winter's measurement compression theorem, detailing the information processing task, giving examples for understanding it, reviewing Winter's achievability proof, and detailing a new approach to its single-letter converse theorem. We prove an extension of the theorem to the case in which the sender is not required to receive the outcomes of the simulated measurement. The total cost of common randomness and classical communication can be lower for such a 'non-feedback' simulation, and we prove a single-letter converse theorem demonstrating optimality. We then review the Devetak–Winter theorem on classical data compression with quantum side information, providing new proofs of its achievability and converse parts. From there, we outline a new protocol that we call 'measurement compression with quantum side information,' announced previously by two of us in our work on triple trade-offs in quantum Shannon theory. This protocol has several applications, including its part in the 'classically-assisted state redistribution' protocol, which is the most general protocol on the static side of the quantum information theory tree, and its role in reducing the classical communication cost in a task known as local purity distillation. We also outline a connection between measurement compression with quantum side information and recent work on entropic uncertainty relations in the presence of quantum memory. Finally, we prove a single-letter theorem characterizing measurement compression with quantum side information when the sender is not required to obtain the measurement outcome.
In the context of two-party interactive quantum communication protocols, we study a recently defined notion of quantum information cost (QIC), which possesses most of the important properties of its classical analogue. Although this definition has the advantage to be valid for fully quantum inputs and tasks, its interpretation for classical tasks remained rather obscure. Also, the link between this new notion and other notions of information cost for quantum protocols that had previously appeared in the literature was not clear, if existent at all. We settle both these issues: for quantum communication with classical inputs, we provide an alternate characterization of QIC in terms of information about the input registers, avoiding any reference to the notion of a purification of the classical input state. We provide an exact operational interpretation of this alternative characterization as the sum of the cost of transmitting information about the classical inputs and the cost of forgetting information about these inputs. To obtain this characterization, we prove a general lemma, the Information Flow Lemma, assessing exactly the transfer of information in general interactive quantum processes. Furthermore, we clarify the link between QIC and IC of classical protocols by simulating quantumly classical protocols. Finally, we apply these concepts to argue that any quantum protocol that does not forget information solves Disjointness on n-bits in Omega (n) communication, completely losing the quadratic quantum speedup. This provides a specific sense in which forgetting information is a necessary feature of interactive quantum protocols. We also apply these concepts to prove that QIC at zero-error is exactly n for the Inner Product function, and n (1 - o(1)) for a random Boolean function on n+n bits.