Quantum Measurements:some technical background

• “Measurement postulate”• “Projection postulate”• The two aspects of measurement• Density matrices, environments, et cetera• von Neumann measurements

(entanglement and decoherence)

Slides, and some other useful links, to be posted:http://www.physics.utoronto.ca/~steinberg/QMP.html

14 Oct 2003

(AKA: the boring lecture)

The measurement postulate

...measurement outcomes...

“Collapse of the wavefunction”

Future measurements of A will of course agree with this ai

Measurement State preparation

What are the effects of measurement?

Suppose we have two pawns, one black and one white, and I putone in each hand – we can write this state as something like

= +

Obviously, if I open my left hand and measure the colour ofits pawn, I find either black or white, not both – from that pointon, I describe the left pawn as one colour eigenstate, or .

Is the other pawn still in a state of uncertain colour?No – obviously, its state has also been “affected” by this measurement.

More refined version:the “projection postulate”

When ai is found, the state

Of course, this is not normalized, so the final state is actually

Finding the left pawn to be black leaves the system in astate where the right pawn is known to be white (unsurprisingly).

Two effects of measurement1. One thing happens, as opposed to all other possibilities

2. Interference between the different possibilities becomes impossible.

100%50% 50% What’s the state of the particles before the final beam-splitter?

If no bomb was present, half the particles are in path A and half are in path B.

If the bomb “measures” which path each is in... then still,half are in path A and half are in path B.

The measurement didn’t change the probabilistic descriptionof the state... but without the bomb, interference caused allparticles to interfere at the final beam splitter and go left;with the bomb, there is no such interference.

Measurement destroyed phase information, but left theprobabilities unchanged. |A + ei|B |A “OR” |B .

A B

det. 1det. 2

How does the bomb cause the other detector to fire?

The probability is given by the absolute square of this inner product,which is 1/4 + 1/4 = 1/2 (because the orthogonality of "peace" and

"BOOM" cause the cross-terms to vanish).

Sneaky fact...No one knows why one thing happens instead of manysimultaneous things... in fact, no one knows whetherthis is true (cf. “relative-state”, “many-worlds”, “many-minds”interpretations).

We can try (a) to understand what measurements do to coherence

and/or (b) to search for a real “collapse” process, supplementaryto quantum mechanics as we know it.

No “collapse” process has ever been observed – i.e., no casewhere we would make the wrong predictions if we didn’tassume collapse. Yet to make sense of probabilities, onetypically assumes that by the time you measure something,it’s one thing or another. (But how do you know that whenI measured it, I wasn’t still in a probabilistic state? “Wigner’s friend.”)

We need a formalism for this... Note that in that interferometer, |A |det. 1 and |det. 2;

|B |det. 1 and |det. 2; but |A + |B |det. 2 only

(because of interference).The state "|A OR |B" might be |A (and get to det. 1 half the time)... or itmight be |B (and get to det. 1 half the time). It's not |A + |B, |A – |B, etc.

Any QM wave function you write down which is half A andhalf B will exhibit some interference; no wave function candescribe the state after such a measurement.Technical example: there is no spin-1/2 state with <SZ> = <SX> = 0.

"Pure states" individual QM wave functions

"Mixed states" probabilistic mixtures of QM states.(e.g., results of measurements)"Density Matrices"

Intro to density matrices...

Interpretation of matrix elements

Diagonal elements = probabilities

Off-diagonal elements = "coherences"(provide info. about relative phase)

Connection to observables

And what about mixed states?

• The essential property of a statistical mixture is that all expectation values are just the weighted averages of those for the individual pure states.

• Our expression for expectation values is linear in the density matrix – i.e., we can keep using that expression with mixed states, if we define the mixed-state density matrix itself as a weighted average.

Density matrices for mixed states

Note: probabilities still 50/50, but no coherence.

What happens if you don't look at part of your system?

When you calculate expectation values, you trace over the system.If your operators depend only on a subsystem, then it makes nodifference whether you trace over other systems before or after:

Decoherence arises from throwing away information

Taking this trace over the environment retains only terms diagonalin the environment variables – i.e., no cross-terms (coherences) remainif they refer to different states of the environment.

(If there is any way – even in principle – to tell which of twopaths was followed, then no interference may occur.)

s when env is

s when env is

...

There is still coherence between and , but if theenvironment is not part of your interferometer, you mayas well consider it to have "collapsed" to or .This means there is no effective coherence if you lookonly at the system.

coherencelost

Decoherence: the party lineWhen a particle interacts with a measurement device, the two subsystems become entangled (no separable description).

Coherence is still present, but only in the entire system; if there is enough information in the measurement device to tell which pathyour subsystem followed, then it is impossible to observe interferencewithout looking at both parts of the system.The effective density matrix of your system (traced over states ofthe measuring apparatus) is that of a mixed state.

Coherence is never truly lost, as unitary evolution preserves the purity of states. In principle, this measurement interaction is reversible.

In practice, once the system interacts with the "environment", i.e., anythingwith too many degrees of freedom for us to handle, we cannot reverse it.Just as in classical statistical mechanics, it is the approximation of an opensystem which leads to effective irreversibility, and loss of information (increaseof entropy).

Loss of Information = Loss of Coherence

So, how does a system become "entangled" with a measuring device?

•First, recall: Bohr – we must treat measurement classically Wigner – why must we?•von Neumann:there are two processes in QM: Unitary and Reduction. He shows how all the effects of measurement we've described so far may be explained without any reduction, or macroscopic devices.

•[Of course, this gets us a diagonal density matrix – classical probabilities without coherence – but still can't tell us how those probabilities turn into one occurrence or another.]

To measure some observable A, let a "meter" interactwith it, so the bigger A is, the more the pointer on themeter moves.P is the generator of translations, so this just means weallow the system and meter to interact according to

Hint A P.

An aside (more intuitive?)Suppose instead of looking at the position of our pointer,we used its velocity to take a reading.

In other words, let the particle exert a force on the pointer,and have the force be proportional to A; then the pointer'sfinal velocity will be proportional to A too. F = g A

U(x) = g A XHint = g A X

This works with any pair of conjugate variables.In the standard case, Hint = g A Px , we can see

The pointer position evolves at a rateproportional to <A>.

A von Neumann measurement

Hint=gApx

System-pointercoupling

Initial State of Pointer

x

A

Initial State of System

x

A

Final state of both (entangled)

Back-ActionIn other words, the measurement does not simply cause thepointer position to evolve, while leaving the system alone.

The interaction entangles the two, and as we have seen, thisentanglement is the source of decoherence.

It is often also described as "back-action" of the measuringdevice on the measured system. Unless Px, the momentumof the pointer, is perfectly well-defined, then the interactionHamiltonian Hint = g A Px looks like an uncertain (noisy)potential for the particle.

A high-resolution measurement needs a well-defined pointer position X.This implies (by Heisenberg) that Px is not well-defined.The more accurate the measurement, the greater the back-action.Measuring A perturbs the variable conjugate to A "randomly"

(unless, that is, you pay attention to entanglement).

SummaryWe have no idea whether or not "collapse" really occurs.

Any time two systems interact and we discard information aboutone of them, this can be thought of as a measurement, whetheror not either is macroscopic, & whether or not there is collapse.

The von Neumann interaction shows how the two systems becomeentangled, and how this may look like random noise from the pointof view of the subsystem.

The "reduced density matrix" of an entangled subsystem appearsmixed, because the discarded parts of the system carry awayinformation. This is the origin of decoherence of the measuredsubsystem.