Quantum compression relative to a set of measurements

Abstract

In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of classical side information. We find that the latter can be bounded, which implies that the minimal compression dimension is stable in the sense that it cannot be decreased by allowing for small errors. Various bounds on the minimal compression dimension are proven and an SDP-based algorithm for its computation is provided. The results are based on two independent approaches: an operator algebraic method using a fixed point result by Arveson and an algebro-geometric method that relies on irreducible polynomials and Bézout's theorem. The latter approach allows lifting the results from the single copy level to the case of multiple copies and from completely positive to merely positive maps.

40 pages. Minor clarifications in Section 9.2 and Section 7.2

References in corpus (3)

• A lower bound on the dimension of a quantum system given measured data
• The Lax conjecture is true
• Extending quantum operations

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• Quantum State Reduction: Generalized Bipartitions from Algebras of Observables
• Quantum measurement incompatibility in subspaces
• Application of Shemesh theorem to quantum channels
• Reductions in finite-dimensional quantum mechanics: from symmetries to operator algebras and beyond