The Non-m-Positive Dimension of a Positive Linear Map

Nathaniel Johnston1,2, Benjamin Lovitz3, and Daniel Puzzuoli4,5

1Department of Mathematics & Computer Science, Mount Allison University, Sackville, NB, Canada E4L 1E4
2Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1
3Institute for Quantum Computing, Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1
4Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada K1N 6N5
5School of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada K1S 5B6

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We introduce a property of a matrix-valued linear map $\Phi$ that we call its ``non-m-positive dimension'' (or ``non-mP dimension'' for short), which measures how large a subspace can be if every quantum state supported on the subspace is non-positive under the action of $I_m \otimes \Phi$. Equivalently, the non-mP dimension of $\Phi$ tells us the maximal number of negative eigenvalues that the adjoint map $I_m \otimes \Phi^*$ can produce from a positive semidefinite input. We explore the basic properties of this quantity and show that it can be thought of as a measure of how good $\Phi$ is at detecting entanglement in quantum states. We derive non-trivial bounds for this quantity for some well-known positive maps of interest, including the transpose map, reduction map, Choi map, and Breuer--Hall map. We also extend some of our results to the case of higher Schmidt number as well as the multipartite case. In particular, we construct the largest possible multipartite subspace with the property that every state supported on that subspace has non-positive partial transpose across at least one bipartite cut, and we use our results to construct multipartite decomposable entanglement witnesses with the maximum number of negative eigenvalues.

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Cited by

[1] Ion Nechita, "How good is a positive map at detecting quantum entanglement?", Quantum Views 3, 24 (2019).

[2] Felix Huber, "Positive maps and trace polynomials from the symmetric group", Journal of Mathematical Physics 62 2, 022203 (2021).

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