Purity of Gaussian states: Measurement schemes and time evolution in noisy channels

We present a systematic study of the purity for Gaussian states of single-mode continuous variable systems. We prove the connection of purity to observable quantities for these states, and show that the joint measurement of two conjugate quadratures is necessary and sufficient to determine the purity at any time. The statistical reliability and the range of applicability of the proposed measurement scheme are tested by means of Monte Carlo simulated experiments. We then consider the dynamics of purity in noisy channels. We derive an evolution equation for the purity of general Gaussian states both in thermal and in squeezed thermal baths. We show that purity is maximized at any given time for an initial coherent state evolving in a thermal bath, or for an initial squeezed state evolving in a squeezed thermal bath whose asymptotic squeezing is orthogonal to that of the input state.

[1]  Simón.  Peres-horodecki separability criterion for continuous variable systems , 1999, Physical review letters.

[2]  J. L. Kelly,et al.  B.S.T.J. briefs: On the simultaneous measurement of a pair of conjugate observables , 1965 .

[3]  J. Cirac,et al.  Characterization of Gaussian operations and distillation of Gaussian states , 2002, quant-ph/0204085.

[4]  Jeffrey H. Shapiro,et al.  Degenerate four-wave mixing as a possible source of squeezed-state light , 1984 .

[5]  V. V. Dodonov,et al.  Purity- and entropy-bounded uncertainty relations for mixed quantum states , 2002 .

[6]  C. Caves.  Quantum Mechanical Noise in an Interferometer , 1981 .

[7]  N. Cerf,et al.  Quantum key distribution using gaussian-modulated coherent states , 2003, Nature.

[8]  Horace P. Yuen,et al.  Classical noise-based cryptography similar to two-state quantum cryptography , 1998 .

[9]  H. Haus,et al.  Characteristic functions and quantum measurements of optical observables , 1989 .

[10]  Paul,et al.  Phase measurement and Q function. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[11]  P. Marian,et al.  Squeezed states with thermal noise. II. Damping and photon counting. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[12]  W. Power,et al.  A scheme for the simultaneous measurement of atomic position and momentum , 1997 .

[13]  J Eisert,et al.  Distilling Gaussian states with Gaussian operations is impossible. , 2002, Physical review letters.

[14]  J. Fiurášek.  Gaussian transformations and distillation of entangled Gaussian states. , 2002, Physical review letters.

[15]  Jeffrey H. Shapiro,et al.  Phase and amplitude uncertainties in heterodyne detection , 1984 .

[16]  R. Filip.  Complementarity, entanglement and quantum erasing in continuous-variable quantum nondemolition experiments , 2002 .

[17]  Matteo G. A. Paris,et al.  Small amount of squeezing in high-sensitive realistic interferometry , 1995 .

[18]  Vogel,et al.  Quantum-state homodyne measurement with vacuum ports. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[19]  Jinhyoung Lee,et al.  Experimentally realizable characterizations of continuous-variable Gaussian states , 2002 .

[20]  Jeffrey H. Shapiro,et al.  Optical communication with two-photon coherent states-Part III: Quantum measurements realizable with photoemissive detectors , 1980, IEEE Trans. Inf. Theory.

[21]  J. Carroll,et al.  Multiport homodyne detection near the quantum noise limit , 1986 .

[22]  Phase space distributions from three-port couplers , 1996, quant-ph/9611034.

[23]  Giacomo Mauro D'Ariano,et al.  Information Gain in Quantum Communication Channels , 1995 .

[24]  N. G. Walker,et al.  Quantum Theory of Multiport Optical Homodyning , 1987 .

[25]  Martin B Plenio,et al.  Conditions for the local manipulation of Gaussian states. , 2002, Physical review letters.

[26]  Sudarshan,et al.  Gaussian-Wigner distributions in quantum mechanics and optics. , 1987, Physical review. A, General physics.

[27]  G. D’Ariano,et al.  Parameter estimation in quantum optics , 2000, quant-ph/0004066.

[28]  Cirac,et al.  Inseparability criterion for continuous variable systems , 1999, Physical review letters.

[29]  Klauder,et al.  SU(2) and SU(1,1) interferometers. , 1986, Physical review. A, General physics.

[30]  P. Grangier,et al.  Continuous variable quantum cryptography using coherent states. , 2001, Physical review letters.

[31]  G. Adam.  Density Matrix Elements and Moments for Generalized Gaussian State Fields , 1995 .

[32]  Stephen M. Barnett,et al.  Methods in Theoretical Quantum Optics , 1997 .

[33]  Schleich,et al.  Photon counting, quantum phase, and phase-space distributions. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[34]  G Leuchs,et al.  Continuous variable quantum cryptography: beating the 3 dB loss limit. , 2002, Physical review letters.

[35]  Stig Stenholm,et al.  Simultaneous measurement of conjugate variables , 1992 .

[37]  Vijay Chickarmane,et al.  Squeezing and dual recycling in laser interferometric gravitational wave detectors , 1994 .

[38]  K. Banaszek,et al.  Direct measurement of optical quasidistribution functions: Multimode theory and homodyne tests of Bell's inequalities , 2002, quant-ph/0204103.

[39]  Marian.  Squeezed states with thermal noise. I. Photon-number statistics. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[40]  Paweł Horodecki,et al.  Direct estimations of linear and nonlinear functionals of a quantum state. , 2002, Physical review letters.

[41]  Properties of a beam-splitter entangler with Gaussian input states , 2002, quant-ph/0204082.

[42]  Radim Filip.  Overlap and entanglement-witness measurements , 2002 .