Quantum measurements: status and problems Michael B. Mensky P.N.Lebedev Physical Institute Moscow, Russia MARKOV READINGS Moscow, May 12, 2005

M.A.Markov and Bryce DeWitt 3d Intern. Seminar on Quantum Gravity Moscow, 1984 Quantum Gravity and Quantum Measurements M.A.Markov on Qu Meas Nature of physical knowledge (1947) Three interpretations of QM (1991)

Message of the talk Physics of Qu Meas: – Entanglement (  Qu Informatics) Phenomenology of Qu Meas: – Open quantum systems and decoherence Meta-physics of Qu Meas: – Everett’s interpretation and consciousness

Plan of the talk Physics: Entanglement and decoherence Continuous measurements: open quantum systems and dissipation Quantum informatics Bell’s theorem Conceptual problems (M.A.Markov 1947) Everett interpretation (M.A.Markov 1991)

Literature on decoherence H.D.Zeh, Found. Phys. 1, 69 (1970); 3, 109 (1973) W.H.Zurek, Phys. Rev. D 24, 1516 (1981); D 26, 1862 (1982) D.Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, & H.D. Zeh, Decoherence and the appearance of a classical world in quantum theory, Springer, Berlin etc., 1996 M.M.

Reduction postulate Von Neumann reduction postulate  c 1 |a 1  + c 2 |a 2   |a 1  p 1 =| c 1 | 2  |a 2  p 2 =| c 2 | 2 With projectors P 1 = |a 1  a 1  P 2 = |a 2  a 2   P 1  p 1 =  P 1   P 2  p 2 =  P 2 

Generalization of reduction postulate Many alternatives (  P i = 1 ) i  P i  p i =  P i  Fuzzy measurement (  dx R x † R x = 1 ) x  R x  p(x) =  R x † R x 

Open systems and continuous measurements Decoherence and dissipation from interaction with environment System Environment Measurement (phenomenology) Open quantum systems = continuously measured ones

Measuring as an interaction: evolution U |a 1  0   U |a 1  0  |a 1    |a 2  0   U |a 2  0  |a 2    Entanglement     c 1 |a 1  +c 2 |a 2    0  c 1 |a 1  0  +c 2 |a 2  0   U  c 1 |a 1  0  +c 2 |a 2  0    c 1 |a 1    +c 2 |a 2    Entanglement Entangled state

Entanglement       c 1 |a 1  + c 2 |a 2    0   c 1 |a 1    + c 2 |a 2    Decoherence  0 =   c 1 |a 1  + c 2 |a 2   c 1  a 1 | + c 2  a 1 |     Tr   |c 1 | 2 |a 1  a 1  |c 2 | 2 |a 2  a 2  Decoherence Reduction interpretated

Irreversible and reversible decoherence System Reservoir Meter info deco Macroscopic uncontrollable environment  practically irreversible decoherence Environment Reversion: U  U -1 Microscopic or mesoscopic environment  reversible decoherence

Restricted Path Integrals (RPI) Continuous measurements presented by RPI Monitoring an observable  decoherence Non-minimally disturbing monitoring  dissipation

Ideology of Feynman paths Feynman: path integral over all paths Propagator : U t  q'',q' ) =  d[q] exp { (i/  ) S[q] } =  d[p] d[q] exp { (i/  )  0 t (pdq-Hdt) } Evolution :  t  U t    t = U t  0 U t † t q q’q’ q”q”

Restricted Path Integral: the paths, compatible with the readout Partial propagator : U t   q'',q' ) = =  d[p]d[q] w   [p,q] exp { (i/ћ)  0 t (p dq - H dt) } t q q’q’ q”q”  Weight functional Evolution :  t   U t     t  = U t   0 ( U t  ) † Restricting Feynman path integral

Probabilities of measurement readouts Probability of the result: P  =Tr  t  =Tr [ U t   0 ( U t  ) † ] Non-selective description:   t =  d   t  =  d  U t   0 ( U t  ) † Generalized unitarity:  d  ( U t  ) † U t  = 1 t q q’q’ q”q” 

Monitoring an observable Gaussian weight functional w [a] [p,q] = exp { -   0 t [ A(t) - a(t) ] 2 dt } Why Gaussian? Quantum Central Limit Theorem! t A A”A” [a] A’A’ Observable A=A(p,q,t) Measurement readout: [a] = {a(t) | 0  t  t}

Effective Schroedinger equation Restricted Path Integral for monitoring A U t  a  (q'',q')=  d[p]d[q] exp { ( i/ћ )  0 t (p dq - H dt) -   0 t [ A(t) - a(t) ] 2 dt } Effective Hamiltonian  H [a] (p,q,t) = H(p,q,t) - i  ћ ( A(p,q,t) - a(t) ) 2 Effective Schroedinger equation  t  a   /  t = [ - ( i/ћ ) H -  ( A - a(t) ) 2 ]  t  a   Imaginary potential

Density matrix and master equation Selective description:  t  a    U t  a      non-selective (total density matrix):  t =  d [a]  t  a  =  d [a] U t  a   0 ( U t  a  ) † Density matrix  t satisfies master equation:   t /  t = - ( i/ћ ) [ H,  t ] - (  /2 ) [A, [A,  t ] ]  decoherence !

Non-minimally disturbing monitoring Imaginary terms in the exponent w[a] = exp {  dt [-  (A-a(t)) 2 - (i/ћ) a(t) B]} Disturbed evolution conditioned by the observation of a(t): U t  a  =  d[p]d[q] exp {  0 t [ (i/ћ) ( p dq-H(q,p) dt) -  (A(q,p)-a(t)) 2 - (i/ћ) a(t) B(q,p) ]}

Master equation Calculate the selective density matrix  t [a] =U t [a]  0 (U t [a] ) † and the total density matrix  t =  d[a]  t [a] The resulting  t satisfies the master equation   /  t = - (i/ћ) [ H,  ] - (i /2 ћ) [ B, [ A,  ] + ] - ( 2 /8  ћ 2 ) [ B, [ B,  ] ] - (  /2) [ A, [ A,  ] ] Dissipation Decoherence Correction to C&L

Lindblad form of the master equation Introduce the Lindblad operator L = A- i( /2  ћ)B The equation takes then the Lindblad form   /  t = - (i/ћ) [ H - i(  ћ /4) ( (L † ) 2 - L 2 ),  ] - (  /2) ( L † L  - 2 L  L † +  L L† ) –Hamiltonian is shifted by the measurement –Lindblad form  positivity of  No positivity in C&L –Dissipation results from continuous measurements

Dissipative harmonic oscillator Hamiltonian of an oscillator: H = P 2 /2 +  2 Q 2 /2 Momentum is monitored: A=P, B=  Q   /  t = - (i/ћ) [ H,  ] - (i  /2 ћ ) [ Q, [ P,  ] + ] - ( 2  2 /8  ћ 2 ) [ Q, [ Q,  ] ] - (  /2) [ P, [ P,  ] ] –Both momentum and position are monitored –Brownian motion of the oscillator is interpreted as an effect of monitoring its momentum by an environment No such term in Caldeira & Leggett

Dynamical role of information Von Neumann's projection: final state depends on the information RPI: projecting process Dynamics of a measured system depends on the information escaping from it The role for quantum informatic devices: the processed information not escaping

Quantum informatics Qubits Quantum computer Quantum cryptography Quantum teleportation

Qubits Two-level system |0  |1  Superposition  |0  |1   quantum parallelism (entangled states) ( |0  |1     |00  |01  |10  |11   ( |0  |1           |x 

Quantum computer Quantum parallelism  ( |0  |1           |x  Calculation time t  P(N) instead of t  e N Quantum algorithms Factorization in prime numbers = finding the period of a periodic function (digital Fourier decomposition)  Cryptography

Quantum cryptography Quantum cloning  |  |A   |  |  |A’  impossible |  1  |A   |  1  |  1  |A 1 , |  2  |A   |  2  |  2  |A 2  Linearity:  |  1  |  2   |A    |  1  |  1  |  2  |  2   |A’’  not  |  1  |  1  |  2  |  2  |  1  |  2  |  2  |  1   |A’’  Sequence of states: |1  |0  |1  |1  Eavesdropping discovered  |0  and  |1  non-orthogonal  Distribution of code sequences (factorization in prime numbers used)

Quantum teleportation Correlation takes no time (pre-arranged) Communication with light speed AB |  A =  |0  |1  |  B |  A Meas Qu correlation (entanglement) Meas Result i  U i |  B = |  B

Quantum teleportation Arbitrary state |  A =  |0  |1  in A: Qubit |  B and |  A’ quantum correlated |0  A |1  B - |1  A |0  B (entangled) Measurement of |  A |  A result i = 1,2,3,4 Communicating the measurement result i to B Unitary transformation |  B  U i |  B  |  A teleported: U i |  B = |   B =  |0  |1 

Bell’s theorem EPR effect Local realism Bell’s inequality Aspect’s experiment

EPR effect Maximal entanglement: |  |  |  |  =  |A +    |A -    |A -    | A +   anticorrelation of spin projections  Correlation of projections on different axes S=0 S=1/2

Local realism Anticorrelation: |A +    |A -    |A -    | A +   Assumtion of local realism means: – If |A -   , then really |A +   – If | A +   , then really |A -   Then measurement is interpreted as |A m  1  | B n  2  |A m  1  | B -n  1  (same particle)

Bell inequality Given P(A ± B ± C ± ) for a single particle and local realism From probability sum rule: P( A - B + ) = P( A - B + C + ) + P( A - B + C - ) P( A + C - ) = P( A + B + C - ) + P( A + B - C - ) P( B + C - ) = P( A + B + C - ) + P( A - B + C - ) Bell inequality: P( A - B + ) + P( A + C - )  P( B + C - )

Realism refuted Local realism  Bell inequality Aspect: Bell inequality is violated  No local realism in Qu Mechanics Properties found in a measurement do not exist before the measurement

Conceptual problems Paradoxes: Schroedinger cat etc. No reality previous to measurement Linear evolution c 1 |a 1  0  +c 2 |a 2  0   c 1 |a 1    +c 2 |a 2     reduction impossible

Everett interpretation Linear evolution c 1 |a 1  0  +c 2 |a 2  0   c 1 |a 1    +c 2 |a 2    Many classical realities (many worlds) Selection = consciousness

Quantum consciousness Qu world = many classical realities Consciousness = Selection Consciousness = selection of a class. reality Unconsciousness = all class. realities = qu world At the edge of consciousness (trance) Choice of reality (modification of probabilities) Contact with the quantum world (other realities)

Conclusion Physics of measurements: entanglement Open systems = continuously measured ones Entanglement  Quantum informatics Conceptual problems: no selection in QM Everett: Selection = consciousness Quantum consciousness: choice of reality etc.

Обзоры M.M., Квантовая механика и декогеренция, Москва, Физматлит, 2001 [translated from English (Quantum Measurements and Decoherence, Kluwer, Dordrecht etc., 2000)] M.M., Диссипация и декогеренция квантовых систем, УФН 173, 1199 (2003) [Physics-Uspekhi 46, 1163 (2003)] M.M., Понятие сознания в контексте квантовой механики, УФН 175, 413 (2005) [Physics-Uspekhi 175 (2005)

Reviews M.M., Quantum Measurements and Decoherence. Kluwer, Dordrecht etc., 2000 [Russian: translation: Москва, Физматлит, 2001] M.M., Dissipation and decoherence of quantum systems, УФН 173, 1199 (2003) [Physics-Uspekhi 46, 1163 (2003)] M.M., Conception of consciousness in the context of quantum mechanics, УФН 175, 413 (2005) [Physics-Uspekhi 175 (2005)]

Conceptual problems of QuantumMechanics M.M., Quantum mechanics: New experiments, new applications and new formulations of old questions, Physics-Uspekhi 43, (2000). [ Russian: М.М., УФН 170, 631 (2000)] М.М., Conception of consciousness in the context of quantum mechanics, Physics-Uspekhi 175, No.4 (2005)] [ Russian: М.М., 175, 413 (2005)]

Sections of the Talk Introduction Op en systems and continuous measurements Restricted Path Integrals (RPI) Non-minimally disturbing monitoring Realization by a series of soft observations Conclusion and reviews