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### Quantum measurements: status and problems

Michael B. Mensky

P.N.Lebedev Physical Institute

Moscow, Russia

MARKOV READINGSMoscow, May 12, 2005

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

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

|=c1|a1+ c2|a2 |a1, p1=| c1 |2

|a2, p2=| c2 |2

- With projectors P1 = |a1 a1| , P2 = |a2 a2|

| P1 | , p1=| P1 |

P2 | , p2=| P2 |

Generalization of reduction postulate

- Many alternatives( Pi = 1)i

| Pi| , pi=| Pi|

- Fuzzy measurement(dxRx†Rx = 1)

x

| Rx| , p(x) =| Rx†Rx|

Open systems andcontinuous measurements

- Decoherence and dissipation from interaction with environment

Measurement (phenomenology)

Environment

System

System

- Open quantum systems
- = continuously measured ones

Entanglement

- Measuring as an interaction: evolutionU

|a1|0 U|a1|0 = |a1 |1

|a2|0 U|a2|0 = |a2 |2

- Entanglement

||0 = (c1|a1+c2|a2)|0 = c1|a1|0+c2|a2|0

U(c1|a1|0+c2|a2|0) = c1|a1|1+c2|a2|2

Entangled state

Decoherence

- Entanglement

|0 = | |0 = (c1|a1+ c2|a2) |0

c1|a1|1 + c2|a2|2 = |

- Decoherence

0 = | | = (c1|a1+ c2|a2) (c1 a1|+ c2 a1|)

= Tr | | = |c1|2 |a1 a1| + |c2|2 |a2 a2|

Reduction interpretated

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

Restricted Path Integrals (RPI)

- Continuous measurements

presented by RPI

- Monitoring an observable

decoherence

- Non-minimally disturbing monitoring

dissipation

Restricted Path Integral:

the paths, compatible

with the readout

Partial propagator: Uta(q\'\',q\') =

=d[p]d[q] wa[p,q] exp{(i/ћ) 0t (p dq - H dt)}

Restricting Feynman path integral

q

q”

q’

t

Weight functional

Evolution: |ta= Uta |0, ta = Uta0(Uta)†

Effective Schroedinger equation

- Restricted Path Integral for monitoring A

Ut[a](q\'\',q\')=d[p]d[q] exp{(i/ћ)0t(p dq - H dt)

- 0t[A(t) - a(t)]2dt}

- 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

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

a|0+ b |1

- quantum parallelism (entangled states)

(|0+ |1)2 = |00+ |01 + |10 + |11

(|0+ |1)N = 02N-1|x

Quantum computer

- Quantum parallelism

(|0+ |1)N = 02N-1|x

- Calculation time tP(N) instead of teN
- 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 |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)

Quantum teleportation

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

Meas Result i

A

B

|A = a|0+ b |1

| B

Ui | B = |B

Meas

| A

Qu correlation (entanglement)

Bell’s theorem

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

EPR effect

S=0

- Maximal entanglement:

| | - | | =|A+1|A-2 - |A-1| A+2

anticorrelation of spin projections

- Correlation of projections on different axes

S=1/2

S=1/2

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

|Am1| Bn2 |Am1| B-n1(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

c1|a1|0+c2|a2|0 c1|a1|1+c2|a2|2

reduction impossible

Everett interpretation

- Linear evolution

c1|a1|0+c2|a2|0 c1|a1|1+c2|a2|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)

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: 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

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