Informationally complete joint measurements on finite quantum systems

We show that there are informationally complete joint measurements of two conjugated observables on a finite quantum system, meaning that they enable to identify all quantum states from their measurement outcome statistics. We further demonstrate that it is possible to implement a joint observable as a sequential measurement. If we require minimal noise in the joint measurement, then the joint observable is unique. If the dimension d is odd, then this observable is informationally complete. But if d is even, then the joint observable is not informationally complete and one has to allow more noise in order to obtain informational completeness.

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

[2]  E. Stachow.  An Operational Approach to Quantum Probability , 1978 .

[3]  J. Schwinger.  UNITARY OPERATOR BASES. , 1960, Proceedings of the National Academy of Sciences of the United States of America.

[4]  Paul Busch,et al.  PROBABILITY STRUCTURES FOR QUANTUM STATE SPACES , 1995 .

[5]  Adam Paszkiewicz,et al.  On quantum information , 2012, ArXiv.

[6]  Teiko Heinosaari,et al.  Coexistence of qubit effects , 2008, 0802.4248.

[7]  S. T. Ali,et al.  Systems of imprimitivity and representations of quantum mechanics on fuzzy phase spaces , 1977 .

[8]  Bruce C. Berndt,et al.  The determination of Gauss sums , 1981 .

[9]  D. M. Appleby.  SIC-POVMs and the Extended Clifford Group , 2004 .

[10]  Joseph M. Renes,et al.  Symmetric informationally complete quantum measurements , 2003, quant-ph/0310075.

[11]  Teiko Heinosaari,et al.  Non-disturbing quantum measurements , 2010, 1005.5659.

[12]  Paul Busch,et al.  The determination of the past and the future of a physical system in quantum mechanics , 1989 .

[13]  Li Li,et al.  Joint measurement of two unsharp observables of a qubit , 2008, 0805.1538.

[14]  M. Sentís.  Quantum theory of open systems , 2002 .

[15]  Claudio Carmeli,et al.  Commutative POVMs and Fuzzy Observables , 2009, 0903.0523.

[16]  Claudio Carmeli,et al.  Sequential measurements of conjugate observables , 2011, 1105.4976.

[17]  E. Prugovec̆ki.  Information-theoretical aspects of quantum measurement , 1977 .

[18]  M. Ozawa.  Quantum measuring processes of continuous observables , 1984 .

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

[20]  P. Busch,et al.  Unsharp reality and joint measurements for spin observables. , 1986, Physical review. D, Particles and fields.

[21]  H. Imai,et al.  Heisenberg's uncertainty principle for simultaneous measurement of positive-operator-valued measures , 2008, 0809.1714.

[22]  Position and momentum observables on R and on R 3 , 2004, quant-ph/0403108.

[23]  Marian Grabowski,et al.  Operational Quantum Physics , 2001 .

[24]  Tomographically complete sets of orthonormal bases in finite systems , 2011 .

[25]  W. D. Muynck.  Foundations of Quantum Mechanics, an Empiricist Approach , 2002 .

[26]  On the quantum theory of sequential measurements , 1990 .

[27]  Propensities in discrete phase spaces: Q function of a state in a finite-dimensional Hilbert space. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[28]  A. J. Scott,et al.  Symmetric informationally complete positive-operator-valued measures: A new computer study , 2010 .

[29]  Paul Busch,et al.  Informationally complete sets of physical quantities , 1991 .

[30]  Paul Busch,et al.  Coexistence of qubit effects , 2008, Quantum Inf. Process..

[31]  D. M. Appleby.  Symmetric informationally complete–positive operator valued measures and the extended Clifford group , 2005 .

[32]  Phase space observables and isotypic spaces , 2000, quant-ph/0004032.

[33]  A. Vourdas.  Phase space methods for finite quantum systems , 1997 .

[34]  Sylvia Pulmannová,et al.  Coexistent observables and effects in quantum mechanics , 1997 .

[35]  William K. Wootters,et al.  Quantum mechanics without probability amplitudes , 1986 .

[36]  Werner Stulpe,et al.  Phase‐space representations of general statistical physical theories , 1992 .

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

[38]  Paul Busch,et al.  Approximate joint measurements of qubit observables , 2007, Quantum Inf. Comput..

[39]  Kraus.  Complementary observables and uncertainty relations. , 1987, Physical review. D, Particles and fields.

[40]  G. D’Ariano,et al.  Informationally complete measurements and group representation , 2003, quant-ph/0310013.

[41]  P. Raynal,et al.  Mutually unbiased bases in six dimensions: The four most distant bases , 2011, Physical Review A.