Mutually unbiased product bases for multiple qudits

We investigate the interplay between mutual unbiasedness and product bases for multiple qudits of possibly different dimensions. A product state of such a system is shown to be mutually unbiased to a product basis only if each of its factors is mutually unbiased to all the states which occur in the corresponding factors of the product basis. This result implies both a tight limit on the number of mutually unbiased product bases which the system can support and a complete classification of mutually unbiased product bases for multiple qubits or qutrits. In addition, only maximally entangled states can be mutually unbiased to a maximal set of mutually unbiased product bases.

[1]  Gary L. Mullen,et al.  Finite Fields and Applications , 2007, Student mathematical library.

[2]  Stefan Weigert,et al.  All mutually unbiased product bases in dimension 6 , 2012 .

[3]  W. Wootters,et al.  Optimal state-determination by mutually unbiased measurements , 1989 .

[4]  I. D. Ivonovic.  Geometrical description of quantal state determination , 1981 .

[5]  William Hall,et al.  Numerical evidence for the maximum number of mutually unbiased bases in dimension six , 2007 .

[6]  A. Zeilinger,et al.  Entanglement in mutually unbiased bases , 2011, 1102.2080.

[7]  Stefan Weigert,et al.  The limited role of mutually unbiased product bases in dimension 6 , 2011, 1111.3849.

[8]  M. Grassl.  On SIC-POVMs and MUBs in Dimension 6 , 2004, quant-ph/0406175.

[9]  H. D. Watson.  At 14 , 1979 .

[10]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[11]  J. van Leeuwen,et al.  Finite Fields and Applications , 2004, Lecture Notes in Computer Science.

[12]  P. Jaming,et al.  A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6 , 2009, 0902.0882.

[13]  Andreas Klappenecker,et al.  Constructions of Mutually Unbiased Bases , 2003, International Conference on Finite Fields and Applications.

[14]  A. Zeilinger,et al.  Mutually Unbiased Quantum Bases : Existence , Entanglement , Information , 2015 .

[15]  Stefan Weigert,et al.  Maximal sets of mutually unbiased quantum states in dimension 6 , 2008, 0808.1614.

[16]  Pawel Wocjan,et al.  New construction of mutually unbiased bases in square dimensions , 2005, Quantum Inf. Comput..

[17]  Dave Cliff,et al.  In/Proceedings of the 15th IEEE International Conference on the Engineering of Complex Computer Systems/ (ICECCS 2010), Oxford , 2010 .

[18]  Stefan Weigert,et al.  ON THE IMPOSSIBILITY TO EXTEND TRIPLES OF MUTUALLY UNBIASED PRODUCT BASES IN DIMENSION SIX , 2012, 1203.6887.

[19]  Stefan Weigert,et al.  All mutually unbiased bases in dimensions two to five , 2009, Quantum Inf. Comput..