Uncertainty principle guarantees genuine source of intrinsic randomness

The Born’s rule introduces intrinsic randomness to the outcomes of a measurement performed on a quantum mechanical system. But, if the system is prepared in the eigenstate of an observable, then the measurement outcome of that observable is completely predictable, and hence, there is no intrinsic randomness. On the other hand, if two incompatible observables are measured (either sequentially on a particle or simultaneously on two identical copies of the particle), then uncertainty principle guarantees intrinsic randomness in the subsequent outcomes independent of the preparation state of the system. In this article, we show that this is true not only in quantum mechanics but for any no-signaling probabilistic theory. Also the minimum amount of intrinsic randomness that can be guaranteed for arbitrarily prepared state of the system is quantified by the amount of (un)certainty.

[1]  Markus P. Mueller,et al.  A derivation of quantum theory from physical requirements , 2010, 1004.1483.

[2]  Roger Colbeck,et al.  Free randomness can be amplified , 2011, Nature Physics.

[3]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[4]  H. P. Robertson.  The Uncertainty Principle , 1929 .

[5]  P. Busch,et al.  Heisenberg's uncertainty principle , 2006, quant-ph/0609185.

[6]  Rodrigo Gallego,et al.  Full randomness from arbitrarily deterministic events , 2012, Nature Communications.

[7]  S. Popescu,et al.  Quantum nonlocality as an axiom , 1994 .

[8]  A. Uchida,et al.  Fast physical random bit generation with chaotic semiconductor lasers , 2008 .

[9]  Jonathan Barrett.  Information processing in generalized probabilistic theories , 2005 .

[10]  A. Winter,et al.  Entropic uncertainty relations—a survey , 2009, 0907.3704.

[11]  G. D’Ariano,et al.  Informational derivation of quantum theory , 2010, 1011.6451.

[12]  Robert König,et al.  The Operational Meaning of Min- and Max-Entropy , 2008, IEEE Transactions on Information Theory.

[13]  L. Hardy.  Quantum Theory From Five Reasonable Axioms , 2001, quant-ph/0101012.

[14]  N. Mermin.  Hidden variables and the two theorems of John Bell , 1993, 1802.10119.

[15]  N. Gisin,et al.  Optical quantum random number generator , 1999, quant-ph/9907006.

[16]  Robert W. Spekkens,et al.  Einstein, Incompleteness, and the Epistemic View of Quantum States , 2007, 0706.2661.

[17]  J. F. Dynes,et al.  A high speed , postprocessing free , quantum random number generator , 2008 .

[18]  Sibasish Ghosh,et al.  Degree of Complementarity Determines the Nonlocality in Quantum Mechanics , 2012, 1206.6054.

[19]  S. Wehner,et al.  The Uncertainty Principle Determines the Nonlocality of Quantum Mechanics , 2010, Science.

[20]  I. Bialynicki-Birula,et al.  Uncertainty relations for information entropy in wave mechanics , 1975 .

[21]  Maassen,et al.  Generalized entropic uncertainty relations. , 1988, Physical review letters.

[22]  Kraus.  Complementary observables and uncertainty relations. , 1987, Physical review. D, Particles and fields.

[23]  W. Heisenberg.  Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , 1927 .

[24]  G. D’Ariano,et al.  Spooky action-at-a-distance in general probabilistic theories , 2011, 1108.3681.

[25]  D. Deutsch.  Uncertainty in Quantum Measurements , 1983 .

[26]  H. Barnum,et al.  Teleportation in General Probabilistic Theories , 2008, 0805.3553.

[27]  R. Prevedel,et al.  Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement , 2010, 1012.0332.

[29]  W. Louisell.  Amplitude and phase uncertainty relations , 1963 .

[30]  Stefano Pironio,et al.  Randomness versus nonlocality and entanglement. , 2011, Physical review letters.

[31]  Paul Benioff,et al.  Possible strengthening of the interpretative rules of quantum mechanics , 1973 .

[32]  J. Bell.  On the Problem of Hidden Variables in Quantum Mechanics , 1966 .

[33]  H. Barnum,et al.  Entropy and information causality in general probabilistic theories , 2009, 0909.5075.

[34]  H. Barnum,et al.  Generalized no-broadcasting theorem. , 2007, Physical review letters.

[35]  Stefano Pironio,et al.  Random numbers certified by Bell’s theorem , 2009, Nature.

[36]  H. Barnum,et al.  Ensemble Steering, Weak Self-Duality, and the Structure of Probabilistic Theories , 2009, 0912.5532.

[37]  Max Born,et al.  Das Adiabatenprinzip in der Quantenmechanik , 1927 .

[38]  Adam Brandenburger,et al.  A classification of hidden-variable properties , 2007, 0711.4650.

[39]  Howard Barnum,et al.  Information Processing in Convex Operational Theories , 2009, QPL/DCM@ICALP.

[40]  M. Born.  Quantenmechanik der Stoßvorgänge , 1926 .

[41]  Paul Busch,et al.  Complementarity and Uncertainty in Mach-Zehnder Interferometry and beyond , 2006, quant-ph/0609048.

[42]  Michael Martin Nieto,et al.  Phase and Angle Variables in Quantum Mechanics , 1968 .

[43]  J. Wrench.  Table errata: The art of computer programming, Vol. 2: Seminumerical algorithms (Addison-Wesley, Reading, Mass., 1969) by Donald E. Knuth , 1970 .

[44]  R. Spekkens.  Evidence for the epistemic view of quantum states: A toy theory , 2004, quant-ph/0401052.

[45]  Corsin Pfister.  One simple postulate implies that every polytopic state space is classical , 2012, 1203.5622.