One-and-a-Half Quantum de Finetti Theorems

By Matthias Christandl, Robert T. König, Graeme J. Mitchison, Renato Renner. 2006

218 citations recorded by the source corpus.

When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by $${2\frac{kd^2}{n}}$$ , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation $${U_{\mu +\nu} \subset U_\mu \otimes U_\nu}$$ of the unitary group U(d), for highest weights μ, ν and μ + ν. Let ξμ be the state obtained by tracing out Uν. Then ξμ is close to a convex combination of the coherent states $${U_\mu(g)|{v_\mu\rangle}}$$ , where $${g\in U(d)}$$ and $${|v_\mu\rangle}$$ is the highest weight vector in Uμ.For the class of symmetric Werner states, which are invariant under both the permutation and unitary groups, we give a second de Finetti-style theorem (our “half” theorem). It arises from a combinatorial formula for the distance of certain special symmetric Werner states to states of fixed spectrum, making a connection to the recently defined shifted Schur functions [1]. This formula also provides us with useful examples that allow us to conclude that finite quantum de Finetti theorems (unlike their classical counterparts) must depend on the dimension d. The last part of this paper analyses the structure of the set of symmetric Werner states and shows that the product states in this set do not form a polytope in general.

Subfields: Artificial Intelligence, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/66bf0a8744ed9cf56d98051cdcc63c8dbf90b4c3

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Citations in the corpus

Derivation of Hartree's theory for generic mean-field Bose systems (2013, background): “1 was proved in [65, 31] (see [26, 12] for related content).”
A Generalization of Quantum Stein’s Lemma (2009, background): “There have been several interesting recent developments on quantum versions [20, 21, 59, 60] of the seminal result by Bruno de Finetti on the characterization of exchangeable probability distributions [61].”
The Theory of Quantum Information (2018, result): “28 were proved by Christandl, König, Mitchison, and Renner [51], who improved on the error bounds and generalized the results obtained by König and Renner.”
Quantum Conditional Mutual Information and Approximate Markov Chains (2014, methodology): “The second is an extension of the method of de Finetti reductions [14,15,45,46] (Sect.”
Practical challenges in quantum key distribution (2016, unknown): “…device independent (MDI) QKD. 21 New theory that is due to practical advances in QKD also includes, for instance, the quantum de Finetti theorem, 22 while security loopholes in QKD are closely related to loopholes in Bell inequality tests 23 — a key subject in the foundations of quantum…”
Symmetry of large physical systems implies independence of subsystems (2007, methodology): “The ﬁrst can be seen as a variant of Winter’s gentle measurement lemma [Win99] and is implicitly used in related work [CKMR07].”
Faithful Squashed Entanglement (2010, background): “Watrous has conjectured that this is not the case and that for every , δ 45 ].”
Advances in quantum cryptography (2019, unknown): “[506] M.”