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arXiv:1705.03048 · Physical review. D/Physical review. D.

De Finetti theorems and entanglement in large- N theories and gravity

Javier M. Magán

Published Oct 5, 2017• 14 citations
arXivHTMLPDF

Abstract

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-$N$ 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-$N$ 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-$N$ properties of the gauge invariant coherence group.

DOI
10.1103/physrevd.96.086002

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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.
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.
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.
Exchangeability is a fundamental concept in probability theory and statistics. It allows to model situations where the order of observations does not matter. The classical de Finetti's theorem provides a representation of infinitely exchangeable sequences of random variables as mixtures of independent and identically distributed variables. The quantum de Finetti theorem extends this result to symmetric quantum states on tensor product Hilbert spaces. It is well known that both theorems do not hold for finitely exchangeable sequences. The aim of this work is to investigate two lesser-known representation theorems, which were developed in classical probability theory to extend de Finetti's theorem to finitely exchangeable sequences by using quasi-probabilities and quasi-expectations. With the aid of these theorems, we illustrate how a de Finetti-like representation theorem for finitely exchangeable sequences can be formulated through a mathematical representation which is formally equivalent to quantum theory (with boson-symmetric density matrices). We then show a promising application of this connection to the challenge of defining entanglement for indistinguishable bosons.
Symmetries are of fundamental interest in many areas of science. In quantum information theory, if a quantum state is invariant under permutations of its subsystems, it is a well-known and widely used result that its marginal can be approximated by a mixture of tensor powers of a state on a single subsystem. Applications of this quantum de Finetti theorem range from quantum key distribution (QKD) to quantum state tomography and numerical separability tests. Recently, it has been discovered by Gross, Nezami and Walter that a similar observation can be made for a larger symmetry group than permutations: states that are invariant under stochastic orthogonal symmetry are approximated by tensor powers of stabilizer states, with an exponentially smaller overhead than previously possible. This naturally raises the question if similar improvements could be found for applications where this symmetry appears (or can be enforced). Here, two such examples are investigated.