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.
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.
Standard finite quantum de Finetti theorems approximate the k-system marginals of permutation-invariant states of n-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/n error bound in max-relative entropy improves on the previously best known k2/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⊗n, where dA=dimA, we prove that the k-system reduced channel is CP-dominated by a mixture of tensor-power channels with error 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 r defects, the representation error is at most poly(n)(2dA4k3/(nr2))(r+1)/2 and decays exponentially in n for a suitable choice of parameters r and k.
Quantum de Finetti theorem indicates that a surprising connection exists between the permutation-invariant property of a large system and the independent, identical distribution that describes its subsystems. The theorem is highly nontrivial in that it allows one to infer independence---within a finite error---from symmetry alone, both of which are important and fundamental concepts in physics and mathematics. Here, we present an improved quantum de Finetti theorem that provides a lower error bound, thus establishing an even closer relationship between symmetry and independence than before. As application, we study the security of quantum key distribution under the most general attacks, and show that the key rate is higher than previous ones obtained by existing trace-norm-based de Finetti theorems. Therefore, the application of the quantum de Finetti theorem in quantum information processing, as well as developing the theorem itself, is advanced here.
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.
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.
We prove an exponential variant of the Gaussian de Finetti theorem: the subsystems of permutation-invariant, free-fermionic Gaussian states are well approximated by convex combinations of almost-i.i.d. states that are Gaussian on subsets of their parts. Our result provides an error bound between the original state and its approximants that decays exponentially in the number of unconstrained parts, becoming super-exponential when the subsystem under consideration is small. The dependence of our bound on the local Hilbert space dimension is polylogarithmic, giving an exponential improvement over the standard de Finetti theorem of [Nat. Phys. 3, 645-649]. Previous works considered Gaussian-symmetric states, which are supported on the trivial irrep of the tensor matchgate representation. We extend these results to the broader class of Gaussian-invariant states, which contains, for example, i.i.d. copies of single-replica mixed Gaussian states. We show that Gaussian-invariant states are precisely the partial traces of Gaussian-symmetric states on locally enlarged replicas and always admit a purification into a larger Gaussian-symmetric state. As an application, we derive explicit convergence bounds for a hierarchy of convex relaxations of polynomial optimisation over fermionic Gaussian states. For variational energy minimisation, the hierarchy gives lower bounds converging to the global Gaussian minimum. Combined with upper bounds from Gaussian trial states, these bounds quantify how far a variational calculation can be from the best energy attainable within the Gaussian family.
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.