Tightening Quantum Speed Limits for Almost All States.

Conventional quantum speed limits perform poorly for mixed quantum states: They are generally not tight and often significantly underestimate the fastest possible evolution speed. To remedy this, for unitary driving, we derive two quantum speed limits that outperform the traditional bounds for almost all quantum states. Moreover, our bounds are significantly simpler to compute as well as experimentally more accessible. Our bounds have a clear geometric interpretation; they arise from the evaluation of the angle between generalized Bloch vectors.

[1]  Kavan Modi,et al.  Quantacell: powerful charging of quantum batteries , 2015, 1503.07005.

[2]  E. Study.  Kürzeste Wege im komplexen Gebiet , 1905 .

[3]  A. Davie,et al.  Improved bound for complexity of matrix multiplication , 2013, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[4]  Tatsuhiko Koike,et al.  Time-optimal quantum evolution. , 2006, Physical review letters.

[5]  Robert W. Spekkens,et al.  Quantum speed limits, coherence, and asymmetry , 2015, 1510.06474.

[6]  Andreas Frommer,et al.  Verified Computation of Square Roots of a Matrix , 2009, SIAM J. Matrix Anal. Appl..

[7]  S. Lloyd.  Ultimate physical limits to computation , 1999, Nature.

[8]  S. Lloyd,et al.  Advances in quantum metrology , 2011, 1102.2318.

[9]  Arun Kumar Pati,et al.  Quantum speed limit for mixed states using an experimentally realizable metric , 2014, 1403.5182.

[10]  Connection between entanglement and the speed of quantum evolution (5 pages) , 2005, quant-ph/0507030.

[11]  C. Ross.  Found , 1869, The Dental register.

[12]  N. Khaneja,et al.  Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation , 2003, quant-ph/0302024.

[13]  G. N. Fleming.  A unitarity bound on the evolution of nonstationary states , 1973 .

[14]  M. M. Taddei,et al.  Quantum speed limit for physical processes. , 2012, Physical review letters.

[15]  Lev Vaidman,et al.  Minimum time for the evolution to an orthogonal quantum state , 1992 .

[16]  E. Lutz,et al.  Generalized clausius inequality for nonequilibrium quantum processes. , 2010, Physical review letters.

[17]  N. Margolus,et al.  The maximum speed of dynamical evolution , 1997, quant-ph/9710043.

[18]  Wei Han,et al.  Quantum speed limit for arbitrary initial states , 2013, Scientific Reports.

[19]  Tommaso Calarco,et al.  Communication at the quantum speed limit along a spin chain , 2010, 1004.3445.

[20]  E. Lutz,et al.  Quantum speed limit for non-Markovian dynamics. , 2013, Physical review letters.

[21]  Armin Uhlmann,et al.  An energy dispersion estimate , 1992 .

[22]  Sebastian Deffner,et al.  Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control , 2017, 1705.08023.

[23]  Gerardo Adesso,et al.  Generalized Geometric Quantum Speed Limits , 2015, 1507.05848.

[24]  D. Brody,et al.  Solution to the quantum Zermelo navigation problem. , 2014, Physical review letters.

[25]  L. Biedenharn,et al.  On the Representations of the Semisimple Lie Groups. II , 1963 .

[26]  W. Wootters.  Statistical distance and Hilbert space , 1981 .

[27]  B. M. Fulk.  MATH , 1992 .

[28]  A. Plastino,et al.  Entanglement and the quantum brachistochrone problem , 2008, 0801.3393.

[29]  R. Werner,et al.  Estimating the spectrum of a density operator , 2001, quant-ph/0102027.

[30]  Kavan Modi,et al.  Enhancing the Charging Power of Quantum Batteries. , 2016, Physical review letters.

[31]  K. Funo,et al.  Universal Work Fluctuations During Shortcuts to Adiabaticity by Counterdiabatic Driving. , 2016, Physical review letters.

[32]  T. Toffoli,et al.  Fundamental limit on the rate of quantum dynamics: the unified bound is tight. , 2009, Physical review letters.

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

[34]  Sk Sazim,et al.  Quantum coherence sets the quantum speed limit for mixed states , 2015, 1506.03199.

[35]  Sebastian Deffner,et al.  Trade-Off Between Speed and Cost in Shortcuts to Adiabaticity. , 2016, Physical review letters.

[36]  P. A. Mello,et al.  The time-energy uncertainty relation , 1978 .

[37]  Andrew G. Glen,et al.  APPL , 2001 .

[38]  S. Lloyd,et al.  The speed limit of quantum unitary evolution , 2004 .

[39]  Christiane P Koch,et al.  Monotonically convergent optimization in quantum control using Krotov's method. , 2010, The Journal of chemical physics.

[40]  Jozef B Uffink.  The rate of evolution of a quantum state , 1993 .

[41]  D. Bohm,et al.  Time in the Quantum Theory and the Uncertainty Relation for Time and Energy , 1961 .

[42]  Susan Stepney,et al.  The Geometry of Speed Limiting Resources in Physical Models of Computation , 2016, Int. J. Found. Comput. Sci..

[43]  L. P. S. Singh,et al.  Time Operators, Partial Stationarity, and the Energy-Time Uncertainty Relation , 1973 .

[44]  J. Hilgevoord,et al.  Uncertainty principle and uncertainty relations , 1985 .

[45]  S Montangero,et al.  Optimal control at the quantum speed limit. , 2009, Physical review letters.

[46]  Aharonov,et al.  Geometry of quantum evolution. , 1990, Physical review letters.

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

[48]  N. Moiseyev,et al.  Time-dependent Hamiltonians with 100% evolution speed efficiency , 2012, 1207.5373.

[49]  Sebastian Deffner.  Geometric quantum speed limits: a case for Wigner phase space , 2017, 1704.03357.

[50]  Wood,et al.  New form of the time-energy uncertainty relation. , 1985, Physical review. A, General physics.

[51]  J. Bekenstein.  Energy Cost of Information Transfer , 1981 .

[52]  M B Plenio,et al.  Quantum speed limits in open system dynamics. , 2012, Physical review letters.

[53]  Claudia Zander,et al.  Entanglement and the speed of evolution of multi-partite quantum systems , 2007 .

[54]  Masoud Mohseni,et al.  Quantum brachistochrone curves as geodesics: obtaining accurate minimum-time protocols for the control of quantum systems. , 2014, Physical review letters.

[55]  Sebastian Deffner,et al.  Energy–time uncertainty relation for driven quantum systems , 2011, 1104.5104.