The Non-m-Positive Dimension of a Positive Linear Map
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
|Published:||2019-08-12, volume 3, page 172|
|Citation:||Quantum 3, 172 (2019).|
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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.
► BibTeX data
 O. Gühneand G. Tóth ``Entanglement detection'' Physics Reports 474, 1-75 (2009).
 R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, ``Quantum entanglement'' Reviews of Modern Physics 81, 865–942 (2009).
 M. B. Plenioand S. Virmani ``An introduction to entanglement measures'' Quantum Information and Computation 7, 1–51 (2007).
 G. Vidal ``Entanglement monotones'' Journal of Modern Optics 47, 355–376 (2000).
 M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, ``Optimization of entanglement witnesses'' Physical Review A 62, 052310 (2000).
 M. A. Nielsenand I. L. Chuang ``Quantum computation and quantum information'' Cambridge University Press (2000).
 J. Watrous ``The Theory of Quantum Information'' Cambridge University Press (2018).
 M.-D. Choi ``Completely positive linear maps on complex matrices'' Linear Algebra and Its Applications 10, 285–290 (1975).
 R. F. Werner ``Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model'' Physical Review A 40, 4277–4281 (1989).
 V. I. Paulsen ``Completely bounded maps and operator algebras'' Cambridge University Press (2003).
 J. Watrous ``Notes on super-operator norms induced by Schatten norms'' Quantum Information and Computation 5, 58–68 (2005).
 A. Yu. Kitaev ``Quantum computations: Algorithms and error correction'' Russian Mathematical Surveys 52, 1191–1249 (1997).
 N. Johnston ``Non-Positive Partial Transpose Subspaces Can be as Large as Any Entangled Subspace'' Physical Review A 87, 064302 (2013).
 S. Boydand L. Vandenberghe ``Convex optimization'' Cambridge University Press (2004).
 Ł. Skowronek, E. Størmer, and K. Życzkowski, ``Cones of positive maps and their duality relations'' Joural of Mathematical Physics 50, 062106 (2009).
 K. R. Parthasarathy ``On the maximal dimension of a completely entangled subspace for finite level quantum systems'' Proceedings of the Indian Academy of Sciences (Mathematical Sciences) 114, 365–374 (2004).
 T. S. Cubitt, A. Montanaro, and A. Winter, ``On the dimension of subspaces with bounded Schmidt rank'' Journal of Mathematical Physics 49, 022107 (2008).
 B. M. Terhaland P. Horodecki ``Schmidt number for density matrices'' Physical Review A 61, 040301(R) (2000).
 N. Johnston ``How to compute hard-to-compute matrix norms'' http://www.njohnston.ca/2016/01/how-to-compute-hard-to-compute-matrix-norms/ (2016) Accessed: May 22, 2019.
 J. Watrous ``Semidefinite programs for completely bounded norms'' Theory of Computing 5, 217–238 (2009).
 J. Watrous ``Simpler semidefinite programs for completely bounded norms'' Chicago Journal of Theoretical Computer Science 1–19 (2013).
 J. Tomiyama ``On the geometry of positive maps in matrix algebras II'' Linear Algebra and Its Applications 69, 169–177 (1985).
 N. J. Cerf, C. Adami, and R. M. Gingrich, ``Reduction criterion for separability'' Physical Review A 60, 898–909 (1999).
 M. Horodeckiand P. Horodecki ``Reduction criterion of separability and limits for a class of distillation protocols'' Physical Review A 59, 4206–4216 (1999).
 M.-D. Choi ``Positive semidefinite biquadratic forms'' Linear Algebra and Its Applications 12, 95–100 (1975).
 H.-P. Breuer ``Optimal entanglement criterion for mixed quantum states'' Physical Review Letters 97, 080501 (2006).
 W. Hall ``A new criterion for indecomposability of positive maps'' Journal of Physics A: Mathematical and General 39, 14119 (2006).
 K.-C. Ha ``Notes on extremality of the Choi map'' Linear Algebra and Its Applications 439, 3156–3165 (2013).
 R. Sengupta, Arvind, and A. I. Singh, ``Entanglement properties of positive operators with ranges in completely entangled subspaces'' Physical Review A 90, 062323 (2014).
 Benjamin Lovitz and Nathaniel Johnston, "Entangled subspaces and generic local state discrimination with pre-shared entanglement", Quantum 6, 760 (2022).
 Ion Nechita, "How good is a positive map at detecting quantum entanglement?", Quantum Views 3, 24 (2019).
 K.V. Antipin, "Construction of genuinely entangled multipartite subspaces from bipartite ones by reducing the total number of separated parties", Physics Letters A 445, 128248 (2022).
 Felix Huber, "Positive maps and trace polynomials from the symmetric group", Journal of Mathematical Physics 62 2, 022203 (2021).
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