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Some upper and lower bounds on PSD-rank

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Abstract

Positive semidefinite rank (PSD-rank) is a relatively new complexity measure on matrices, with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for showing lower bounds on the PSD-rank. All of these bounds are based on viewing a positive semidefinite factorization of a matrix M as a quantum communication protocol. These lower bounds depend on the entries of the matrix and not only on its support (the zero/nonzero pattern), overcoming a limitation of some previous techniques. We compare these new lower bounds with known bounds, and give examples where the new ones are better. As an application we determine the PSD-rank of (approximations of) some common matrices.

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Notes

  1. \(\mathrm {Tr}_B(\rho \otimes \sigma )=\mathrm {Tr}(\sigma )\rho \), which is extended linearly to states that are not tensor products.

  2. Even though a nonnegative matrix has the same PSD-rank as its transposition, the bounds given by \(B_3\) (or \(B_4\)) can be quite different, for instance for the matrix A of Example 2.

  3. A Latin square is an n-by-n matrix in which each row and each column is a permutation of \(\{0,\ldots ,n-1\}\).

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Acknowledgments

We would like to thank Rahul Jain for helpful discussions, and Hamza Fawzi, Richard Robinson, and Rekha Thomas for sharing their results on the derangement matrix. Troy Lee and Zhaohui Wei are supported in part by the Singapore National Research Foundation under NRF RF Award No. NRF-NRFF2013-13. Ronald de Wolf is partially supported by ERC Consolidator Grant QPROGRESS and by the EU STREP project QALGO (Grant Agreement No. 600700).

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Authors and Affiliations

  1. School of Physics and Mathematical Sciences, Nanyang Technological University and Centre for Quantum Technologies, Singapore, Singapore

    Troy Lee & Zhaohui Wei

  2. QuSoft, CWI and University of Amsterdam, Amsterdam, The Netherlands

    Ronald de Wolf

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  1. Troy Lee
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  2. Zhaohui Wei
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  3. Ronald de Wolf
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Correspondence to Zhaohui Wei.

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Lee, T., Wei, Z. & de Wolf, R. Some upper and lower bounds on PSD-rank. Math. Program. 162, 495–521 (2017). https://doi.org/10.1007/s10107-016-1052-0

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  • DOI: https://doi.org/10.1007/s10107-016-1052-0

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