LOW ENTROPY OUTPUT STATES FOR PRODUCTS OF RANDOM UNITARY CHANNELS
Abstract
In this paper, we study the behavior of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings [Superadditivity of communication capacity using entangled inputs, Nat. Phys.5 (2009) 255–257] and Hayden–Winter [Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1, Comm. Math. Phys.284(1) (2008) 263–280] to the additivity problems. In particular, we study in depth the difference of behavior between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space, we show that the eigenvalue distribution converges to a limiting shape that we characterize with free probability tools. In order to perform the required computations, we need a systematic way of dealing with moment problems for random matrices whose blocks are i.i.d. Haar distributed unitary operators. This is achieved by extending the graphical Weingarten calculus introduced in [B. Collins and I. Nechita, Random quantum channels I: Graphical calculus and the Bell state phenomenon, Comm. Math. Phys.297(2) (2010) 345–370].
References
- Comm. Math. Phys. 288 , 1103 ( 2009 ) . Crossref, Google Scholar
- Invent. Math. 190(3), 647 (2012). Crossref, Google Scholar
-
R. Bhatia , Matrix Analysis ,Graduate Texts in Mathematics 169 ( Springer-Verlag , New York , 1997 ) . Crossref, Google Scholar - Open Syst. Inf. Dyn. 17(1), 31 (2010). Link, Google Scholar
- Int. Math. Res. Not. 2003(17), 953 (2003). Crossref, Google Scholar
- J. Math. Phys. 53 , 032203 ( 2012 ) . Crossref, Google Scholar
- B. Collins, M. Fukuda and I. Nechita, On the geometry of the image of large random quantum channels, in preparation . Google Scholar
- Comm. Math. Phys. 297(2), 345 (2010). Crossref, Google Scholar
- Entropy 12(6), 1612 (2010). Crossref, Google Scholar
- Adv. Math. 226 , 1181 ( 2011 ) . Crossref, Google Scholar
- Ann. Appl. Probab. 21(3), 1136 (2011). Crossref, Google Scholar
- J. Phys. A: Math. Theor. 43 , 275303 ( 2010 ) . Crossref, Google Scholar
- Comm. Math. Phys. 264(3), 773 (2004). Crossref, Google Scholar
- J. Math. Phys. 51 , 042201 ( 2010 ) . Crossref, Google Scholar
- Comm. Math. Phys. 296(1), 111 (2010). Crossref, Google Scholar
- Nature Phys. 5 , 255 ( 2009 ) . Crossref, Google Scholar
- Comm. Math. Phys. 250 , 371 ( 2004 ) . Crossref, Google Scholar
- Comm. Math. Phys. 284(1), 263 (2008). Crossref, Google Scholar
-
F. Hiai and D. Petz , The Semicircle Law, Free Random Variables and Entropy ( AMS Press , 2006 ) . Crossref, Google Scholar - Probab. Theory Appl. 51 , 133 ( 2005 ) . Google Scholar
- Markov Process. Related Fields 13(2), 391 (2007). Google Scholar
-
A. Nica and R. Speicher , Lectures on the Combinatorics of Free Probability ,London Mathematical Society Lecture Note Series 335 ( Cambridge University Press , Cambridge , 2006 ) . Crossref, Google Scholar -
R. Speicher , Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory ,Memoirs of the American Mathematical Society 132 ( American Mathematical Society , Providence, RI , 1998 ) . Google Scholar
