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arXiv:1605.09013 · Journal of Mathematical Physics

Flexible constrained de Finetti reductions and applications

Cécilia Lancien, Andreas Winter

Published Sep 1, 2017• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.5003633

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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.
De Finetti theorems convert permutation symmetry into approximate mixtures of product states and thereby justify a wide range of reductions in classical and quantum statistics. In this work we study de Finetti hierarchies in which the feasible states are additionally constrained to be fixed points of quantum channels, a condition that subsumes invariance under arbitrary compact symmetry groups. Combining the mean-ergodic theorem with the structure theory of conditional expectations, we prove a tight bound on the entanglement-assisted classical capacity of the dual of a conditional expectation, block-wise distortion bounds for informationally complete measurements adapted to fixed point algebras, and an exact type-based refinement of the chain rule for permutation-invariant states. From these tools we derive several de Finetti theorems: a double-sided extension theorem with O(log⁡n/n)O\left(\sqrt{\log n}/n\right)O(logn​/n) convergence, an interpolation theorem whose dimension dependence is governed solely by the block structure of the fixed point algebras and which recovers the dimension-independent classical behavior for maximal tori, and a Bose-symmetric variant. Exploiting Schur-Weyl duality and Gelfand-Tsetlin bases, we further show that the rounding scheme producing certifiably good separable inner approximations for (constrained) separability problems can be implemented in time polynomial in 1/ε1/ε1/ε for fixed local dimensions, complementing the known efficient outer hierarchies. Applications to bilinear optimization under symmetries and to approximate quantum error correction are discussed.
When analysing quantum information processing protocols, one has to deal with large entangled systems, each consisting of many subsystems. To make this analysis feasible, it is often necessary to identify some additional structures. de Finetti theorems provide such a structure for the case where certain symmetries hold. More precisely, they relate states that are invariant under permutations of subsystems to states in which the subsystems are independent of each other. This relation plays an important role in various areas, e.g., in quantum cryptography or state tomography, where permutation invariant systems are ubiquitous. The known de Finetti theorems usually refer to the internal quantum state of a system and depend on its dimension. Here, we prove a different de Finetti theorem where systems are modelled in terms of their statistics under measurements. This is necessary for a large class of applications widely considered today, such as device independent protocols, where the underlying systems and the dimensions are unknown and the entire analysis is based on the observed correlations.
Standard finite quantum de Finetti theorems approximate the kkk-system marginals of permutation-invariant states of nnn-systems by mixtures of independent and identically distributed (iid) states, usually in trace distance. We prove both standard and Renner's exponential de Finetti theorems in the stronger form of operator inequalities, implying bounds in every Schatten norm and for every quantum Rényi divergence satisfying data processing. In the standard case, for fixed local dimension, our k/nk/nk/n error bound in max-relative entropy improves on the previously best known k2/nk^2/nk2/n scaling, even in the classical setting. The operator-inequality approach is particularly suited to study channel de Finetti representations because operator order between Choi states is equivalent to completely positive (CP) order between the underlying channels. For permutation-covariant channels N(n):A⊗n→B⊗nN^{(n)}:A^{\otimes n}\to B^{\otimes n}N(n):A⊗n→B⊗n, where dA=dim⁡Ad_A=\dim AdA​=dimA, we prove that the kkk-system reduced channel is CP-dominated by a mixture of tensor-power channels with error O(k/n)O(k/\sqrt{n})O(k/n​) and polynomial dependence on the local dimensions, addressing a question raised by Berta et al. [Math. Program. 194, 781-829 (2022)]. Under the no-signalling condition, we also prove an exponential channel de Finetti theorem where the approximating mixture consists of Choi-almost-iid channels, whose normalized Choi states are almost-iid in the sense of Mazzola-Sutter-Renner. In the case of rrr defects, the representation error is at most poly(n)(2dA4k3/(nr2))(r+1)/2\mathrm{poly}(n)\bigl(2d_A^4k^3/(nr^2)\bigr)^{(r+1)/2}poly(n)(2dA4​k3/(nr2))(r+1)/2 and decays exponentially in nnn for a suitable choice of parameters rrr and kkk.
The de Finetti representation theorem for continuous variable quantum system is first developed to approximate an N-partite continuous variable quantum state with a convex combination of independent and identical subsystems, which requires the original state to obey permutation symmetry conditioned on successful experimental verification on k of N subsystems. We generalize the de Finetti theorem to include asymmetric bounds on the variance of canonical observables and biased basis selection during the verification step. Our result thereby enables application of infinite-dimensional de Finetti theorem to situations where two conjugate measurements obey different statistics, such as the security analysis of quantum key distribution protocols based on squeezed state against coherent attack.
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.
The de Finetti theorem and its extensions concern the structure of multipartite probability distributions with certain symmetry properties, the paradigmatic original example being permutation symmetry. These theorems assert that such symmetric distributions are well approximated by convex combinations of uncorrelated ones. In this article, we apply de Finetti theorems to quantum gravity theories, such as the Sachdev-Ye-Kitaev (SYK) model or large-NNN vector and gauge theories. For SYK we put recent studies of information/entanglement dynamics in a general and rigorous basis. For vector and gauge theories, we describe the classicality of the modular Hamiltonian, its modular flows, and the entanglement entropy. These results can be unified through a generic statement about the nature of Schmidt decompositions and decoherence in large-NNN theories. In the reverse direction, we extend de Finetti theorems in various ways and provide an independent approach to the theorems only based on the large-NNN properties of the gauge invariant coherence group.