Characterization of linear maps on $M_n$ whose multiplicity maps have maximal norm, with an application in quantum information

Daniel Puzzuoli

Department of Applied Mathematics and Institute for Quantum Computing
University of Waterloo, Waterloo, Ontario, Canada

Given a linear map $\Phi : M_n \rightarrow M_m$, its multiplicity maps are defined as the family of linear maps $\Phi \otimes \textrm{id}_{k} : M_n \otimes M_k \rightarrow M_m \otimes M_k$, where $\textrm{id}_{k}$ denotes the identity on $M_k$. Let $\|\cdot\|_1$ denote the trace-norm on matrices, as well as the induced trace-norm on linear maps of matrices, i.e. $\|\Phi\|_1 = \max\{\|\Phi(X)\|_1 : X \in M_n, \|X\|_1 = 1\}$. A fact of fundamental importance in both operator algebras and quantum information is that $\|\Phi \otimes \textrm{id}_{k}\|_1$ can grow with $k$. In general, the rate of growth is bounded by $\|\Phi \otimes \textrm{id}_{k}\|_1 \leq k \|\Phi\|_1$, and matrix transposition is the canonical example of a map achieving this bound. We prove that, up to an equivalence, the transpose is the unique map achieving this bound. The equivalence is given in terms of complete trace-norm isometries, and the proof relies on a particular characterization of complete trace-norm isometries regarding preservation of certain multiplication relations.
We use this result to characterize the set of single-shot quantum channel discrimination games satisfying a norm relation that, operationally, implies that the game can be won with certainty using entanglement, but is hard to win without entanglement. Specifically, we show that the well-known example of such a game, involving the Werner-Holevo channels, is essentially the unique game satisfying this norm relation. This constitutes a step towards a characterization of single-shot quantum channel discrimination games with maximal gap between optimal performance of entangled and unentangled strategies.

► BibTeX data

► References

[1] R. Smith. Completely Bounded Maps between C$^*$-Algebras. Journal of the London Mathematical Society, s2-27 (1): 157-166, 1983. 10.1112/​jlms/​s2-27.1.157.

[2] J. Tomiyama. Recent Development of the Theory of Completely Bounded Maps between C$^*$-Algebras. Publications of the Research Institute for Mathematical Sciences, 19 (3): 1283-1303, 1983a. 10.2977/​prims/​1195182030.

[3] V. Paulsen. Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge, 2003. 10.1017/​CBO9780511546631.

[4] M. Horodecki, P. Horodecki, and R. Horodecki. Separability of mixed states: necessary and sufficient conditions. Physics Letters A, 223 (1): 1-8, 1996. 10.1016/​S0375-9601(96)00706-2.

[5] G. Vidal and R. F. Werner. Computable measure of entanglement. Physical Review A, 65 (3): 032314, 2002. 10.1103/​PhysRevA.65.032314.

[6] J. Tomiyama. On the transpose map of matrix algebras. Proceedings of the American Mathematical Society, 88 (4): 635-638, 1983b. 10.1090/​S0002-9939-1983-0702290-4.

[7] M.-D. Choi. A Schwarz inequality for positive linear maps on C$^*$-algebras. Illinois Journal of Mathematics, 18 (4): 565-574, 1974. ISSN 0019-2082. URL https:/​/​​euclid.ijm/​1256051007.

[8] C.-K. Li, Y.-T. Poon, and N.-S. Sze. Isometries for Ky Fan Norms between Matrix Spaces. Proceedings of the American Mathematical Society, 133 (2): 369-377, 2005. 10.1090/​S0002-9939-04-07510-0.

[9] J. Watrous. The Theory of Quantum Information. https:/​/​​ watrous/​TQI, 2017.

[10] M.-D. Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Applications, 10 (3): 285-290, 1975. 10.1016/​0024-3795(75)90075-0.

[11] D. Puzzuoli and J. Watrous. Ancilla Dimension in Quantum Channel Discrimination. Annales Henri Poincaré, 18 (4): 1153-1184, 2017. 10.1007/​s00023-016-0537-y.

[12] A. Jenc̆ová. Reversibility conditions for quantum operations. Reviews in Mathematical Physics, 24 (07): 1250016, 2012. 10.1142/​S0129055X1250016X.

[13] D. Sutter, M. Berta, and M. Tomamichel. Multivariate Trace Inequalities. Communications in Mathematical Physics, 352 (1): 37-58, 2017. 10.1007/​s00220-016-2778-5.

[14] R. Kadison. Isometries of Operator Algebras. Annals of Mathematics, 54 (2): 325-338, 1951. 10.2307/​1969534.

[15] W.-S. Cheung, C.-K. Li, and Y.-T. Poon. Isometries between matrix algebras. Journal of the Australian Mathematical Society, 77 (1): 1-16, 2004. 10.1017/​S1446788700010119.

[16] C.-K. Li and S. Pierce. Linear Preserver Problems. The American Mathematical Monthly, 108 (7): 591-605, 2001. 10.2307/​2695268.

[17] J.-T. Chan, C.-K. Li, and N.-S. Sze. Isometries for unitarily invariant norms. Linear Algebra and its Applications, 399: 53-70, 2005. 10.1016/​j.laa.2004.05.017.

[18] A. Kitaev. Quantum computations: algorithms and error correction. Russian Mathematical Surveys, 52 (6): 1191-1249, 1997. 10.1070/​RM1997v052n06ABEH002155.

[19] A. Kitaev, A. Shen, and M. Vyalyi. Classical and Quantum Computation, volume 47 of Graduate Studies in Mathematics. American Mathematical Society, 2002. 10.1090/​gsm/​047.

[20] M. Sacchi. Optimal discrimination of quantum operations. Physical Review A, 71 (6): 062340, 2005a. 10.1103/​PhysRevA.71.062340.

[21] M. Sacchi. Entanglement can enhance the distinguishability of entanglement-breaking channels. Physical Review A, 72 (1): 014305, 2005b. 10.1103/​PhysRevA.72.014305.

[22] B. Rosgen. Distinguishing short quantum computations. In Proceedings of the 25th International Symposium on Theoretical Aspects of Computer Science, pages 597-608, 2008. 10.4230/​LIPIcs.STACS.2008.1322.

[23] J. Watrous. Distinguishing quantum operations having few Kraus operators. Quantum Information and Computation, 8 (9): 819-833, 2008. ISSN 1533-7146. URL http:/​/​​journals/​qiconline.html#v8n89.

[24] M. Piani and J. Watrous. All Entangled States are Useful for Channel Discrimination. Physical Review Letters, 102 (25): 250501, 2009. 10.1103/​PhysRevLett.102.250501.

[25] A. Jenc̆ová and M. Plávala. Conditions for optimal input states for discrimination of quantum channels. Journal of Mathematical Physics, 57 (12): 122203, 2016. 10.1063/​1.4972286.

[26] C. Helstrom. Detection theory and quantum mechanics. Information and Control, 10 (3): 254-291, 1967. 10.1016/​S0019-9958(67)90302-6.

[27] A. Holevo. An analog of the theory of statistical decisions in noncommutative probability theory. Transactions of the Moscow Mathematical Society, 26: 133-149, 1972. ISSN 0077-1554. URL https:/​/​​?q=an:0289.62007.

[28] R. F. Werner and A. S. Holevo. Counterexample to an additivity conjecture for output purity of quantum channels. Journal of Mathematical Physics, 43 (9): 4353-4357, 2002. 10.1063/​1.1498491.

► Cited by (beta)

Crossref's cited-by service has no data on citing works. Unfortunately not all publishers provide suitable citation data.