A complementarity-based approach to phase in finite-dimensional quantum systems

We develop a comprehensive theory of phase for finite-dimensional quantum systems. The only physical requirement we impose is that phase is complementary to amplitude. To implement this complementarity we use the notion of mutually unbiased bases, which exist for dimensions that are powers of a prime. For a d-dimensional system (qudit) we explicitly construct d+1 classes of maximally commuting operators, each one consisting of d?1 operators. One of these classes consists of diagonal operators that represent amplitudes (or inversions). By finite Fourier transformation, it is mapped onto ladder operators that can be appropriately interpreted as phase variables. We discuss examples of qubits and qutrits, and show how these results generalize previous approaches.

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

[2]  Apostolos Vourdas,et al.  Quantum systems with finite Hilbert space , 2004 .

[3]  D. Ellinas.  Phase operators via group contraction , 1991 .

[4]  Chiara Macchiavello.  Optimal estimation of multiple phases , 2003 .

[5]  Caslav Brukner,et al.  Mutually unbiased binary observable sets on N qubits , 2002 .

[6]  Discrete phase space based on finite fields , 2004, quant-ph/0401155.

[7]  Quantum theory of rotation angles: The problem of angle sum and angle difference , 1998 .

[8]  A. Vourdas,et al.  SU(2) and SU(1,1) phase states. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[9]  Arvind,et al.  A generalized Pancharatnam geometric phase formula for three-level quantum systems , 1996, quant-ph/9605042.

[10]  P. Kok,et al.  Quantum lithography, entanglement and Heisenberg-limited parameter estimation , 2004, quant-ph/0402083.

[11]  Paul,et al.  Canonical and measured phase distributions. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[12]  B. Sanders,et al.  Quantum encodings in spin systems and harmonic oscillators , 2001, quant-ph/0109066.

[13]  M. Keyl.  Fundamentals of quantum information theory , 2002, quant-ph/0202122.

[14]  L. Sánchez-Soto,et al.  Multicomplementary operators via finite Fourier transform , 2004, quant-ph/0410155.

[15]  Howard Mark Wiseman,et al.  Optimal input states and feedback for interferometric phase estimation , 2001 .

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

[17]  Arthur O. Pittenger,et al.  Mutually Unbiased Bases, Generalized Spin Matrices and Separability , 2003 .

[18]  J. Preskill,et al.  Encoding a qubit in an oscillator , 2000, quant-ph/0008040.

[19]  A. I. Lvovsky,et al.  Iterative maximum-likelihood reconstruction in quantum homodyne tomography , 2003, quant-ph/0311097.

[20]  L. L. Sanchez-Soto,et al.  Quantum phases of a qutrit , 2003 .

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

[22]  Description of entanglement in terms of quantum phase , 2002, quant-ph/0202100.

[23]  G. M. D'Ariano,et al.  Phase-covariant quantum cloning , 1999, quant-ph/9909046.

[24]  Jinhyoung Lee,et al.  Operationally invariant measure of the distance between quantum states by complementary measurements. , 2003, Physical review letters.

[25]  Milburn,et al.  Optimal quantum measurements for phase estimation. , 1995, Physical review letters.

[26]  J. Wheeler,et al.  Quantum theory and measurement , 1983 .

[27]  Y. Aharonov,et al.  The mean king's problem: Prime degrees of freedom , 2001, quant-ph/0101134.

[28]  H. Weyl.  The Theory Of Groups And Quantum Mechanics , 1931 .

[29]  A. Galindo,et al.  Information and computation: Classical and quantum aspects , 2001, quant-ph/0112105.

[30]  D. Bouwmeester,et al.  The Physics of Quantum Information , 2000 .

[31]  G. J. Milburn,et al.  Generation of eigenstates using the phase-estimation algorithm , 2001 .

[32]  S. Chaturvedi,et al.  Aspects of mutually unbiased bases in odd-prime-power dimensions , 2001, quant-ph/0109003.

[33]  G. D’Ariano,et al.  Optimal quantum estimation of the coupling between two bosonic modes , 2001, quant-ph/0103080.

[34]  C. cohen-tannoudji,et al.  Quantum Mechanics: , 2020, Fundamentals of Physics II.

[35]  H. Zassenhaus,et al.  The Pauli matrices in n dimensions and finest gradings of simple Lie algebras , 1988 .

[36]  P. Oscar Boykin,et al.  A New Proof for the Existence of Mutually Unbiased Bases , 2002, Algorithmica.

[37]  A. Calderbank,et al.  Z4‐Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line‐Sets , 1997 .

[38]  D. Galetti,et al.  An extended Weyl-Wigner transformation for special finite spaces , 1988 .

[39]  H. Guise,et al.  Coherent state realizations of su(n+1) on the n-torus , 2002 .

[40]  Jaehyun Kim,et al.  Implementation of phase estimation and quantum counting algorithms on an NMR quantum-information processor , 2002 .

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

[42]  A. Luis,et al.  6 – Quantum phase difference, phase measurements and stokes operators , 2000 .

[43]  Christian Kurtsiefer,et al.  Ascertaining the values of sigma x, sigma y, and sigma z of a polarization qubit. , 2003, Physical review letters.

[44]  Testing operational phase concepts in quantum optics , 1999, quant-ph/9910092.

[45]  W. Wootters.  A Wigner-function formulation of finite-state quantum mechanics , 1987 .