Measurement-induced randomness and structure in controlled qubit processes.

When an experimentalist measures a time series of qubits, the outcomes constitute a classical stochastic process. We show that projective measurement induces high complexity in these processes in two specific senses: They are inherently random (finite Shannon entropy rate) and they require infinite memory for optimal prediction (divergent statistical complexity). We identify nonorthogonality of the quantum states as the mechanism underlying the resulting complexities and examine the influence that measurement choice has on the randomness and structure of measured qubit processes. We introduce quantitative measures of this complexity and provide efficient algorithms for their estimation.

[1]  Stanislav Straupe,et al.  Experimental neural network enhanced quantum tomography , 2019, npj Quantum Information.

[2]  Physics Letters , 1962, Nature.

[3]  J. Yorke,et al.  Chaotic behavior of multidimensional difference equations , 1979 .

[4]  Jens Eisert,et al.  Guaranteed recovery of quantum processes from few measurements , 2017, Quantum.

[5]  L. C. G. Govia,et al.  Bootstrapping quantum process tomography via a perturbative ansatz , 2019, Nature Communications.

[6]  S. Lloyd,et al.  Quantum metrology. , 2005, Physical review letters.

[7]  James P. Crutchfield,et al.  The Ambiguity of Simplicity , 2016, ArXiv.

[8]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[9]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[10]  James P. Crutchfield,et al.  A Closed-Form Shave from Occam's Quantum Razor: Exact Results for Quantum Compression , 2015, ArXiv.

[11]  Benson,et al.  Regulated and entangled photons from a single quantum Dot , 2000, Physical review letters.

[12]  H. Rabitz,et al.  Time-resolved quantum process tomography using Hamiltonian-encoding and observable-decoding , 2013 .

[13]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[14]  James P. Crutchfield,et al.  Nearly Maximally Predictive Features and Their Dimensions , 2017, Physical review. E.

[15]  Christopher Granade,et al.  Practical Bayesian tomography , 2015, 1509.03770.

[16]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[17]  H. Peitgen,et al.  Functional Differential Equations and Approximation of Fixed Points , 1979 .

[18]  J. Crutchfield,et al.  The ambiguity of simplicity in quantum and classical simulation , 2017 .

[19]  J. Yorke,et al.  The liapunov dimension of strange attractors , 1983 .

[20]  M. Paternostro,et al.  Non-Markovian quantum processes: Complete framework and efficient characterization , 2015, 1512.00589.

[21]  J. Crutchfield,et al.  Regularities unseen, randomness observed: levels of entropy convergence. , 2001, Chaos.

[22]  M. B. Plenio,et al.  Scalable reconstruction of unitary processes and Hamiltonians , 2014, 1411.6379.

[23]  J. Bechhoefer.  Hidden Markov models for stochastic thermodynamics , 2015, 1504.00293.

[24]  James P. Crutchfield,et al.  Anatomy of a Bit: Information in a Time Series Observation , 2011, Chaos.

[25]  S. Polyakov,et al.  : Single-photon sources and detectors , 2011 .

[26]  James P. Crutchfield,et al.  Occam’s Quantum Strop: Synchronizing and Compressing Classical Cryptic Processes via a Quantum Channel , 2015, Scientific Reports.

[27]  N. Bohr.  II - Can Quantum-Mechanical Description of Physical Reality be Considered Complete? , 1935 .

[28]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[29]  R Raussendorf,et al.  A one-way quantum computer. , 2001, Physical review letters.

[30]  Rainer Michalzik,et al.  Ultrafast spin-lasers , 2018, Nature.

[31]  Matison,et al.  Experimental Test of Local Hidden-Variable Theories , 1972 .

[32]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[33]  J. Crutchfield.  Between order and chaos , 2011, Nature Physics.

[34]  Lawrence R. Rabiner,et al.  A tutorial on hidden Markov models and selected applications in speech recognition , 1989, Proc. IEEE.

[35]  B. Englert,et al.  Cavity quantum electrodynamics , 2006 .

[36]  Karoline Wiesner,et al.  Quantum mechanics can reduce the complexity of classical models , 2011, Nature Communications.

[37]  James P. Crutchfield,et al.  Strong and Weak Optimizations in Classical and Quantum Models of Stochastic Processes , 2018, Journal of Statistical Physics.

[38]  C. Galland,et al.  Two types of luminescence blinking revealed by spectroelectrochemistry of single quantum dots , 2011, Nature.

[39]  D. Nesbitt,et al.  Origin and control of blinking in quantum dots. , 2016, Nature nanotechnology.

[40]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[41]  Young,et al.  Inferring statistical complexity. , 1989, Physical review letters.

[42]  C. Wu,et al.  The Angular Correlation of Scattered Annihilation Radiation , 1950 .

[43]  J. D. Trimmer,et al.  THE PRESENT SITUATION IN QUANTUM MECHANICS: A TRANSLATION OF SCHR6DINGER'S "CAT PARADOX" PAPER , 2014 .

[44]  A. D. Boozer,et al.  Trapped atoms in cavity QED: coupling quantized light and matter , 2005 .

[45]  R. Mcweeny.  On the Einstein-Podolsky-Rosen Paradox , 2000 .

[46]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[47]  Shih,et al.  New high-intensity source of polarization-entangled photon pairs. , 1995, Physical review letters.

[48]  James P. Crutchfield,et al.  Prediction, Retrodiction, and the Amount of Information Stored in the Present , 2009, ArXiv.

[49]  J Fan,et al.  Invited review article: Single-photon sources and detectors. , 2011, The Review of scientific instruments.

[50]  J. Rogers.  Chaos , 1876 .

[51]  De-Jun Feng,et al.  Dimension theory of iterated function systems , 2009, 1002.2036.

[52]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[53]  L. Rabiner,et al.  An introduction to hidden Markov models , 1986, IEEE ASSP Magazine.