Quantum measurements: status and problems

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Quantum measurements status and problems

• Michael B. Mensky
• P.N.Lebedev Physical Institute
• Moscow, Russia

MARKOV READINGSMoscow, May 12, 2005
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Quantum Gravity and Quantum Measurements

• M.A.Markov on Qu Meas
• Nature of physical knowledge (1947)
• Three interpretations of QM (1991)

M.A.Markov and Bryce DeWitt 3d Intern. Seminar on
Quantum Gravity Moscow, 1984
3
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
• Everetts interpretation and consciousness

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Plan of the talk

• Physics Entanglement and decoherence
• Continuous measurements
• open quantum systems and dissipation
• Quantum informatics
• Bells theorem
• Conceptual problems (M.A.Markov 1947)
• Everett interpretation (M.A.Markov 1991)

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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.
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Reduction postulate

• Von Neumann reduction postulate
• ??c1a1? c2a2 ? ? a1?, p1 c1 2
• ? a2?, p2 c2 2
• With projectors P1 a1? ? a1 , P2 a2? ?
  a2
• ?? ? P1 ?? , p1?? P1 ??
• ? P2 ?? , p2?? P2 ??

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Generalization of reduction postulate

• Many alternatives (? Pi 1) i
• ?? ? Pi ?? , pi ?? Pi ??
• Fuzzy measurement (?dx Rx Rx 1)
• x
• ?? ? Rx ?? , p(x) ?? Rx Rx ??

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Open systems and continuous measurements

• Decoherence and dissipation from interaction with
  environment

Measurement (phenomenology)
Environment
System
System

• Open quantum systems
• continuously measured ones

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Entanglement

• Measuring as an interaction evolution U
• a1??0? ? U a1??0? a1? ?1?
• a2??0? ? U a2??0? a2? ?2?
• Entanglement
• ???0? (c1a1?c2a2?)?0?
  c1a1??0?c2a2??0?
• ? U (c1a1??0?c2a2??0?) c1a1??1?c2a2??2
  ?

Entangled state
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Decoherence

• Entanglement
• ?0 ? ?? ?0? (c1a1? c2a2?) ?0?
• ? c1a1??1? c2a2??2? ??
• Decoherence
• ?0 ?? ?? (c1a1? c2a2 ?) (c1? a1 c2?
  a1)
• ? ? Tr? ?? ?? c12 a1? ? a1 c22
  a2? ? a2

Reduction interpretated
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Irreversible and reversible decoherence

• Macroscopic uncontrollable environment
• ? practically irreversible decoherence

Environment
Reservoir
Meter
System

• Microscopic or mesoscopic environment
• ? reversible
• decoherence

info
Meter
System
deco
Reversion U ? U-1
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Restricted Path Integrals (RPI)

• Continuous measurements
• presented by RPI
• Monitoring an observable
• ? decoherence
• Non-minimally disturbing monitoring
• ? dissipation

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Ideology of Feynman paths
q
q

• Feynman
• path integral over all paths

q
t

• Propagator Ut (q'',q') ?dq exp (i/?) Sq
• ? dp dq exp (i/?) ?0t (pdq-Hdt)
• Evolution ?t? Ut ?0?, ?t Ut ?0 Ut

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Restricting Feynman path integral
?
q

• Restricted Path Integral
• the paths, compatible
• with the readout

q
q
t

• Partial propagator Uta(q'',q')
• ?dpdq wa p,q exp(i/?) ?0t (p dq - H dt)

Weight functional
Evolution ?ta? Uta ?0?, ?t a Uta ?0
(Uta)
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Probabilities of measurement readouts
?
q

• Probability of the result
• PaTr ?ta Tr Uta?0(Uta)
• Non-selective description
• ?t ? d? ?ta
• ? d? Uta ?0 (Uta)
• Generalized unitarity
• ? d? (Uta) Uta 1

q
q
t
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Monitoring an observable
A
a

• Observable AA(p,q,t)
• Measurement readout
• a a(t) 0 ? t? ? t

A
A
t

• Gaussian weight functional
• wa p,q exp-? ?0t A(t) - a(t)2 dt
• Why Gaussian?
• Quantum Central Limit Theorem!

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Effective Schroedinger equation

• Restricted Path Integral for monitoring A
• Uta(q'',q')?dpdq exp(i/?)?0t (p dq - H
  dt)
• - ? ?0t
  A(t) - a(t)2 dt
• Effective Hamiltonian ?
• Ha (p,q,t) H(p,q,t) - i ? ? (A(p,q,t) -
  a(t))2
• Effective Schroedinger equation
• ??ta?/?t - (i/?) H - ? (A - a(t))2 ?ta?

Imaginary potential
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Density matrix and master equation

• Selective description
• ?ta? Uta?0?
• ? non-selective (total density matrix)
• ?t ? d a ?ta ? d a Uta ?0 (Uta)
• Density matrix ?t satisfies master equation
• ??t/?t - (i/?) H , ?t - (?/2) A , A , ?t
  
• ?
  decoherence !

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Non-minimally disturbing monitoring

• Imaginary terms in the exponent
• wa exp ? dt - ? (A-a(t))2 - (i/?) ? a(t)
  B
• Disturbed evolution
• conditioned by the observation of a(t)
• Uta ? dpdq exp ?0t (i/?) ( p dq-H(q,p)
  dt)
• - ? (A(q,p)-a(t))2 - (i/?) ? a(t) B(q,p)

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Master equation

• Calculate the selective density matrix
• ?ta Uta ?0(Uta)
• and the total density matrix
• ?t ? da ?ta
• The resulting ?t satisfies the master equation
• ? ? /? t - (i/?) H , ? - (i ? /2 ?) B ,
  A , ?
• - (?2/8 ? ?2) B , B , ? - (?/2) A ,
  A , ?

Dissipation
Decoherence
Correction to CL
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Lindblad form of the master equation

• Introduce the Lindblad operator
• L A- i(?/2 ? ?)B
• The equation takes then the Lindblad form
• ? ? /? t - (i/?) H - i(? ? /4) ( (L)2-
  L 2) , ?
• - (?/2) ( L L ? - 2 L ? L ? L
  L )
• Hamiltonian is shifted by the measurement

No positivity in CL

• Lindblad form ? positivity of ?

• Dissipation results from continuous measurements

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Dissipative harmonic oscillator

• Hamiltonian of an oscillator H P 2/2 ?2 Q2
  /2
• Momentum is monitored AP, B? Q
• ? ? /? t - (i/?) H , ? - (i ? ?/2 ? ) Q ,
  P, ?
• - (?2? 2/8 ? ? 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
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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

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Quantum informatics

• Qubits
• Quantum computer
• Quantum cryptography
• Quantum teleportation

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Qubits

• Two-level system
• 0?, 1?
• Superposition
• a 0? b 1?
• ? quantum parallelism (entangled states)
• (0? 1?)2 00? 01? 10? 11?
• (0? 1?)N ?02N -1 x?

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Quantum computer

• Quantum parallelism
• (0? 1?)N ?02N -1 x?
• Calculation time t?P(N) instead of t ?eN
• Quantum algorithms
• Factorization in prime numbers
• finding the period of a periodic function
• (digital Fourier decomposition)
• ?? Cryptography

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Quantum cryptography

• Quantum cloning ??A? ? ?? ?? A?
  impossible
• ?1?A? ? ?1? ?1? A1?, ?2?A? ? ?2? ?2?
  A2?
• 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)

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Quantum teleportation
Meas Result i
A
B
??A a 0? b 1?
?B
? Ui ?B ??B
Meas
?A?
Qu correlation (entanglement)

• Correlation takes no time (pre-arranged)
• Communication with light speed

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Quantum teleportation

• Arbitrary state ??A a 0? b 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 ? Ui ?B
• ? ??A teleported Ui ?B ? ?B a 0? b 1?

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Bells theorem

• EPR effect
• Local realism
• Bells inequality
• Aspects experiment

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EPR effect
S0
S1/2
S1/2

• Maximal entanglement
• ?? ?? - ?? ?? A?1 A-?2 - A-?1 A ?2
• anticorrelation of spin projections
• ? Correlation of projections on different axes

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Local realism

• Anticorrelation A?1 A-?2 - A-?1 A ?2
• Assumtion of local realism means
• If A-?2 , then really A?1
• If A ?2 , then really A-?1
• Then measurement is interpreted as
• Am?1 Bn ?2 ? Am?1 B-n ?1 (same particle)

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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-)

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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

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Conceptual problems

• Paradoxes Schroedinger cat etc.
• No reality previous to measurement
• Linear evolution
• c1a1??0?c2a2??0? ? c1a1??1?c2a2??2?
• ? reduction impossible

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Everett interpretation

• Linear evolution
• c1a1??0?c2a2??0? ? c1a1??1?c2a2??2?
• Many classical realities (many worlds)
• Selection consciousness

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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)

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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.

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??????

• M.M., ????????? ???????? ? ????????????, ??????,
  ?????????, 2001
• translated from English (Quantum
  Measurements and Decoherence, Kluwer, Dordrecht
  etc., 2000)
• M.M., ?????????? ? ???????????? ????????? ??????,
  
• ??? 173, 1199 (2003) Physics-Uspekhi 46, 1163
  (2003)
• M.M., ??????? ???????? ? ????????? ?????????
  ????????, ??? 175, 413 (2005) Physics-Uspekhi
  175 (2005)

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Reviews

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

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Conceptual problems of QuantumMechanics

• M.M., Quantum mechanics New experiments, new
  applications and new formulations of old
  questions,
• Physics-Uspekhi 43, 585-600 (2000).
• Russian ?.?., ??? 170, 631 (2000)
• ?.?., Conception of consciousness in the context
  of quantum mechanics,
• Physics-Uspekhi 175, No.4 (2005)
• Russian ?.?., 175, 413 (2005)

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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