Boolean approach to dichotomic quantum measurement theories

Recently, a new measurement theory based on truth values was proposed by Nagata and Nakamura [Int. J. Theor. Phys. 55, 3616 (2016)], that is, a theory where the results of measurements are either 0 or 1. The standard measurement theory accepts a hidden variable model for a single Pauli observable. Hence, we can introduce a classical probability space for the measurement theory in this particular case. Additionally, we discuss in the present contribution the fact that projective measurement theories (the results of which are either +1 or −1) imply the Bell, Kochen, and Specker (BKS) paradox for a single Pauli observable. To justify our assertion, we present the BKS theorem in almost all the two-dimensional states by using a projective measurement theory. As an example, we present the BKS theorem in two-dimensions with white noise. Our discussion provides new insight into the quantum measurement problem by using this measurement theory based on the truth values.

[1]  N. Mermin.  Quantum mysteries revisited , 1990 .

[2]  Jiangfeng Du,et al.  Implementation of a quantum algorithm to solve the Bernstein-Vazirani parity problem without entanglement on an ensemble quantum computer , 2001 .

[3]  A. Suarez.  On Bell, Suarez-Scarani, and Leggett Experiments: Reply to a Comment by Marek Żukowski in [Found. Phys. 38:1070, 2008] , 2009 .

[4]  A. Zeilinger,et al.  Going Beyond Bell’s Theorem , 2007, 0712.0921.

[5]  L. Ballentine,et al.  Quantum Theory: Concepts and Methods , 1994 .

[6]  N. Mermin.  Whats Wrong with those Epochs , 1990 .

[7]  R. Morrow,et al.  Foundations of Quantum Mechanics , 1968 .

[8]  S. Massar,et al.  Fiber-optics implementation of the Deutsch-Jozsa and Bernstein-Vazirani quantum algorithms with three qubits. , 2002, Physical review letters.

[9]  Li Yang,et al.  A quantum algorithm for approximating the influences of Boolean functions and its applications , 2014, Quantum Information Processing.

[10]  A. Peres.  Incompatible results of quantum measurements , 1990 .

[11]  A. Shimony,et al.  Bell’s theorem without inequalities , 1990 .

[12]  Anthony J Leggett,et al.  Nonlocal Hidden-Variable Theories and Quantum Mechanics: An Incompatibility Theorem , 2006 .

[13]  K J Resch,et al.  Experimental application of decoherence-free subspaces in an optical quantum-computing algorithm. , 2003, Physical review letters.

[14]  Yoon-Ho Kim,et al.  Single-photon two-qubit entangled states: preparation and measurement , 2005, 2005 Quantum Electronics and Laser Science Conference.

[15]  T. Paterek,et al.  Experimental test of nonlocal realistic theories without the rotational symmetry assumption. , 2007, Physical review letters.

[16]  F. Schmidt-Kaler,et al.  Implementation of the Deutsch–Jozsa algorithm on an ion-trap quantum computer , 2003, Nature.

[17]  Michael Redhead,et al.  The breakdown of quantum non-locality in the classical limit , 1991 .

[18]  D. Deutsch.  Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[19]  J. Bell.  On the Einstein-Podolsky-Rosen paradox , 1964 .

[20]  Mermin Nd.  Simple unified form for the major no-hidden-variables theorems. , 1990 .

[21]  Umesh V. Vazirani,et al.  Quantum Complexity Theory , 1997, SIAM J. Comput..

[22]  J. Neumann.  Mathematical Foundations of Quantum Mechanics , 1955 .

[23]  Christian Kurtsiefer,et al.  Experimental falsification of Leggett's nonlocal variable model. , 2007, Physical review letters.

[24]  J. A. Jones,et al.  Implementation of a quantum algorithm on a nuclear magnetic resonance quantum computer , 1998, quant-ph/9801027.

[25]  R. Cleve,et al.  Quantum algorithms revisited , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[26]  Tadao Nakamura,et al.  Measurement Theory in Deutsch’s Algorithm Based on the Truth Values , 2015 .

[27]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[28]  T. Paterek,et al.  An experimental test of non-local realism , 2007, Nature.

[29]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[30]  M. Żukowski.  Comment on: Nonlocal “Realistic” Leggett Models Can be Considered Refuted by the Before-Before Experiment , 2008, 0809.0072.

[31]  Stephen P. Walborn,et al.  Implementing the Deutsch algorithm with polarization and transverse spatial modes , 2005 .

[32]  I. Chuang,et al.  Quantum Computation and Quantum Information: Bibliography , 2010 .

[33]  N. Mermin.  What's Wrong with these Elements of Reality? , 1990 .

[34]  J. A. Jones,et al.  Implementation of a quantum algorithm on a nuclear magnetic resonance quantum computer , 1998, quant-ph/9801027.

[35]  A. Suarez.  Nonlocal “Realistic” Leggett Models Can be Considered Refuted by the Before-Before Experiment , 2008 .

[36]  M S Kim,et al.  Experimental realization of Deutsch's algorithm in a one-way quantum computer. , 2007, Physical review letters.