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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 quantity 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.
Forward citations
Cited by 2 Pith papers
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Quantum Communication Lower Bounds for Search Problems via Matrix Discrepancy
A matrix-discrepancy argument proves tight one-way quantum lower bounds for collision finding (Ω(N^{1/4})) and for streaming triangle finding (Ω(√Δ_V)) where Boolean-Hidden-Matching reductions fail.
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Minimal Help, Maximal Gain: Environmental Assistance Unlocks Encoding Strength
Certain quantum channels with suboptimal environment-assisted capacity can still have their full encoding strength unlocked by minimal environment assistance, measured through the psd rank of channel matrices.
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