Backward Evolving Quantum State Lev Vaidman 2 March 2006

3 March -- Feb Backward Evolving Workshop

We showed measurability of the basic concepts of the formalism such as the two-state vector and the nonlocal variable with eigenstates evolving in arbitrary directions of time. Conclusions The forward and backward evolving quantum states are similar in respect to ideal and demolition quantum measurements, no cloning theorem, and teleportation. They are different in respect to reversing the direction of time evolution with a “flip”. The two-state vector formalism of quantum mechanics allows to see peculiar properties of pre- and post-selected quantum systems. The concept of weak values explains in a natural way numerous bizarre quantum effects and suggests existence of novel amplification methods

Backward Evolving Quantum State The Quantum State Evolving Backward

Time symmetry and the two-state vector No cloning theorem for the backward evolving quantum state Teleportation of a backward evolving quantum state Measurements of nonlocal variables The Aharonov-Bergmann-Lebowitz formula Generalized two-state vector Three box paradox Weak values Outline

Time symmetry and the two-state vector

The standard (one-state vector) description of a quantum system at time t

We assume:

The standard (one-state vector) description of a quantum system at all times:

The time reversal of

The time reversed description of a quantum system

The backward (one-state vector) description of a quantum system at time t

?

? At time t:

?

?

Erasing the past

Alternative past for the same present Hint:

protection Erasing the past

Time symmetric description of a pre- and post-selected quantum system The two-state vector

The two-state vector is a complete description of a system at time t ? The two-state vector is what we can say now ( ) about the pre- and post-selected system at time t The two-state vector describes a single pre- and post-selected system, but to test predictions of the two-state vector we need a pre- and post-selected ensemble The probability of is not an issue

? ? ?? ? ? The pre- and post-selected ensemble

? ? ?? ? ?

A=? Are there any differences between what can be done to and ? Nondemolition (von Neumann) measurements ?

U U Unitary transformation Are there any differences between what can be done to and ? ?

? No cloning theorem? CLONER Are there any differences between what can be done to and ?

? No cloning theorem A=a U(a) Are there any differences between what can be done to and ?

Proof of no cloning theorem for backward evolving quantum state Cloner exists We can send signals to the past Cloner does not exist

Proof of no cloning theorem for backward evolving quantum state Cloner exists We can send signals to the past Cloner does not exist CLONER measurement or CLONER measurement mixture of

Teleportation? Teleportation of a backward evolving quantum state without sending classical information? Are there any differences between what can be done to and ?

The EPR State

The EPR- Bohm State David Bohm

Teleportation

Teleportation of a backward evolving quantum state without sending classical information?

No teleportation of a backward evolving quantum state without sending classical information because we cannot prepare backward evolving EPR pair.

YES, teleportation of a backward evolving quantum state without sending classical information when we add a “flip”.

YES, teleportation of a backward evolving quantum state without sending classical information when we add a “flip”.

NO teleportation of a forward evolving quantum state without sending classical information even if we add a “flip”.

Nonlocal variables

Measurement of nonlocal variables

Example:

BELL OPERATOR IS MEASURABLE

All nonlocal variables (for forward evolving states) are measurable in a destructive way Backward evolving states ? Mixed ? Two state vector ? Vaidman, PRL 90, (2003)

“OPPOSITE TIME DIRECTIONS” BELL OPERATOR IS MEASURABLE IN A NONDESTRUCTIVE WAY Completely correlated state

Destructive measurements of nonlocal variables with parts evolving in arbitrary directions of time L. Vaidman and I. Nevo, quant-ph/ The backward evolving state can be transformed to “flipped” state evolving forward in time. spin All forward evolving state nonlocal variables are measurable in a destructive way continuous

Destructive measurements of nonlocal variables with parts evolving in arbitrary directions of time L. Vaidman and I. Nevo, quant-ph/ The backward evolving state can be transformed to “flipped” state evolving forward in time. spin All forward evolving state nonlocal variables are measurable in a destructive way continuous local measurement

The two-state vector is a complete description of a system at time t ? The two-state vector is what we can say now ( ) about the pre- and post-selected system at time t So, what can we say?

The Aharonov-Bergmann-Lebowitz (ABL) formula: described by the two-state vector: Measurements performed on a pre- and post-selected system

described by the two-state vector: The Aharonov-Bergmann-Lebowitz (ABL) formula:

Measurements performed on a pre- and post-selected system described by the two-state vector: The Aharonov-Bergmann-Lebowitz (ABL) formula: At time t:

Measurements performed on a pre- and post-selected system described by the two-state vector: The Aharonov-Bergmann-Lebowitz (ABL) formula: Can we arrange at time t: ?

Generalized two-state vector protection

PRL 58, 1385 (1987)

Ascertaining the values of sigma(x), sigma(y), and sigma(z) of a polarization qubit Schulz O, Steinhubl R, Weber M, Englert BG, Kurtsiefer C, Weinfurter HSchulz OSteinhubl RWeber MEnglert BGKurtsiefer CWeinfurter H PRL 90, (2003) Abstract: In the 1987 spin-retrodiction puzzle of Vaidman, Aharonov, and Albert one is challenged to ascertain the values of sigma(x), sigma(y), and sigma(z) of a spin- 1/2 particle by utilizing entanglement. We report the experimental realization of a quantum-optical version in which the outcome of an intermediate polarization projection is inferred by exploiting single-photon two-qubit quantum gates. The experimental success probability is consistently above the 90.2% threshold of the optimal one-qubit strategy, with an average success probability of 95.6%.

The three box paradox Where is the ball? ?

The three box paradox It is in always !

The three box paradox It is always in

The three box paradox It is always in but if we open both, it might be in

A single photon sees two balls It scatters exactly as if there were two balls Y. Aharonov and L. Vaidman Phys. Rev. A 67, (2003)

Weak measurements

The outcomes of weak measurements are weak values Weak value of a variable C of a pre- and post-selected system described at time t by the two-state vector

Pointer probability distribution Weak measurements performed on a pre- and post-selected ensemble Weak Measurement of The particle pre-selected The particle post-selected

How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100 Y. Aharonov, D. Albert, and L. Vaidman PRL 60, 1351 (1988) Realization of a measurement of a ``weak value'' Realization of a measurement of a ``weak value'' N. W. M. Ritchie, J. G. Story, and R. G. Hulet Phys. Rev. Lett. 66, (1991) Measuring device is the particle itself, so it is easy to obtain the ensemble of pre- and post-selected particles

Properties of a quantum system during the time interval between two measurementsProperties of a quantum system during the time interval between two measurements Y. Aharonov and L. Vaidman PRA 41, 11 (1990 ) Another example: superposition of positive shifts yields negative shift A. Botero Superposition of Gaussians shifted by small values yields the Gaussian shifted by the large value

Two useful theorems: If with probability 1 then If then with probability 1 For dichotomic variables: The three box paradox

We showed measurability of the basic concepts of the formalism such as the two-state vector and the nonlocal variable with eigenstates evolving in arbitrary directions of time. Summary The forward and backward evolving quantum states are similar in respect to ideal and demolition quantum measurements, no cloning theorem, and teleportation. They are different in respect to reversing the direction of time evolution with a “flip”. The two-state vector formalism of quantum mechanics allows to see peculiar properties of pre- and post-selected quantum systems. The concept of weak values explains in a natural way numerous bizarre quantum effects and suggests existence of novel amplification methods.

A single pre- and post-selected system ? The two-state vector in the framework of the many-worlds interpretation of quantum mechanics The other world

A single pre- and post-selected system ? The two-state vector in the framework of the many-worlds interpretation of quantum mechanics This world