We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form $\mathrm{Tr}\big[M(X\otimes Y)\big]$, maximized with respect to semidefinite constraints on $X$ and $Y$. Applied to the problem of quantum error correction this gives hierarchies of efficiently computable outer bounds on the optimal fidelity for any message dimension and error model. The first level of our hierarchies corresponds to the non-signalling assisted fidelity previously studied by [Leung & Matthews, IEEE Trans.~Inf.~Theory 2015], and positive partial transpose constraints can be added and used to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo's asymptotic de Finetti theorem for quantum channels. Finally, our proof methods also allow us to answer an open question from [Brand\~ao & Harrow, STOC 2013] by improving the approximation factor of de Finetti theorems with no symmetry from $O(d^{k/2})$ to $\mathrm{poly}(d,k)$, where $d$ denotes local dimension and $k$ the number of copies.
We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form Tr[M(X⊗Y)], maximized with respect to semidefinite constraints on X and Y. Applied to the problem of quantum error correction this gives hierarchies of efficiently computable outer bounds on the optimal fidelity for any message dimension and error model. The first level of our hierarchies corresponds to the non-signalling assisted fidelity previously studied by [Leung & Matthews, IEEE Trans.~Inf.~Theory 2015], and positive partial transpose constraints can be added and used to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo's asymptotic de Finetti theorem for quantum channels. Finally, our proof methods also allow us to answer an open question from [Brand~ao & Harrow, STOC 2013] by improving the approximation factor of de Finetti theorems with no symmetry from O(dk/2) to poly(d,k), where d denotes local dimension and k the number of copies.
Abstract We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form Tr[H(D⊗E)] Tr [ H ( D ⊗ E ) ] , maximized with respect to semidefinite constraints on D and E. Applied to the problem of approximate error correction in quantum information theory, this gives hierarchies of efficiently computable outer bounds on the success probability of approximate quantum error correction codes in any dimension. The first level of our hierarchies corresponds to a previously studied relaxation (Leung and Matthews in IEEE Trans Inf Theory 61(8):4486, 2015) and positive partial transpose constraints can be added to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our sum-of-squares hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo’s asymptotic de Finetti theorem for quantum channels. Finally, our proof methods answer a question of Brandão and Harrow (Proceedings of the forty-fourth annual ACM symposium on theory of computing, STOC’12, p 307, 2012) by improving the approximation factor of de Finetti theorems with no symmetry from O(dk/2) O ( d k / 2 ) to poly(d,k) poly ( d , k ) , where d denotes local dimension and k the number of copies.
We derive converging hierarchies of efficiently computable semidefinite programming outer bounds on the optimal fidelity for the transmission of quantum information over noisy quantum channels. Based on positive partial transpose conditions we give a sufficient criterion for the exact convergence at any given level of the hierarchies. The worst case convergence speed of our hierarchies is quantified via positive semidefinite representable outer approximations on the set of separable Choi states, which are based on novel finite de Finetti theorems for quantum channels.
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(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/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.
We study optimization programs given by a bilinear form over noncommutative variables subject to linear inequalities. Problems of this form include the entangled value of two-prover games, entanglement-assisted coding for classical channels, and quantum-proof randomness extractors. We introduce an asymptotically converging hierarchy of efficiently computable semidefinite programming (SDP) relaxations for this quantum optimization. This allows us to give upper bounds on the quantum advantage for all of these problems. Compared to previous work of Pironio, Navascués, and Acín [SIAM J. Optim., 20 (2010), pp. 2157--2180], our hierarchy has additional constraints. By means of examples, we illustrate the importance of these new constraints both in practice and for analytical properties. Moreover, this allows us to give a hierarchy of SDP outer approximations for the completely positive semidefinite cone introduced by Laurent and Piovesan.
We study zero-error communication and exact simulation of quantum channels assisted by quantum no-signalling correlations. For a channel with Kraus operator space K, we introduce a semidefinite programming parameter Γ(K) that is exactly multiplicative under tensor products and upper-bounds ΘNS(K), where log2ΘNS(K)=C0,NS(K) and C0,NS(K) denotes the asymptotic quantum no-signalling-assisted zero-error capacity. Among known upper bounds on ΘNS given by semidefinite programs, Γ is the tightest. We also prove that the previously introduced simulation lower bound Σ−(K) is exactly multiplicative and satisfies ΘNS(K)≤Γ(K)≤Σ−(K)≤Γ(K)2. Using this multiplicativity, we show that neither S0,NS(K)=log2Σ−(K) nor its regularized extension S0,NS(K)=limn→∞n−1log2Σ−(K⊗n) holds in general. This answers an open question about the asymptotic simulation cost S0,NS(K). For classical-quantum channels, we obtain sharp bounds. If r is the effective output dimension, then ΘNS≤Γ≤(1+r1/2)ΘNS/2 and Γ≤Σ−≤(1+r1/2)Γ/2. If k is the minimum rank of the output support projections, then Σ−/ΘNS≤[(1+r1/2)/(1+k1/2)]2, and this bound is sharp for every 1≤k≤r. Thus, we obtain the sharp bound S0,NS−C0,NS≤2log2[(1+r1/2)/(1+k1/2)].
Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of ε in poly(1/ε,input dimension) time.