Measurement and Interference

Let us recall what we learned about projectivemeasurements. Any ideal physical measurement on asystem with a D-dimensional Hilbert spacecorresponds to an observable.

An observable is a D ×D Hermitian operator O = O†.Such operators are diagonalizable:

O =∑

j

λjPj , Pj = P†j , PjPk = Pjδjk,

with M ≤ D real distinct eigenvalues λj and Mcorresponding orthogonal projectors Pj.

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When a measurement is done, the outcome (i.e., themeasured value) is one of the eigenvalues λj.

After the measurement the system is left in an eigen-state of O corresponding to λj . This implies that if themeasurement is immediately repeated, the sameoutcome will occur. The probability pj of a particularoutcome λj is determined by Born’s rule.

Let’s see how this works for an observable that is non-degenerate, i.e., has M = D distinct real eigenvalues λj .

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Nondegenerate Observables

In this case, each of the projectors Pj = |φj〉〈φj| is one-dimensional. The eigenstates {|φj〉} are determineduniquely (up to a phase) and are an orthonormal basis.

We can write any state |ψ〉 in this basis:

|ψ〉 =D∑

j=1

αj|φj〉,

in which case the probability of outcome λj is pj = |αj|2and the system is left in the eigenstate |φj〉 after themeasurement.

Such a nondegenerate observable, with D distincteigenvalues and one-dimensional spectral projectors, iscalled complete.

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Degenerate Observables

Suppose that the observable is degenerate, i.e., at leastone of the projectors Pj corresponding to an eigenvalueλj is more than one-dimensional, Tr{Pj} ≡ dj > 1.

In this case, the basis of eigenvectors is not unique.What are the outcome probabilities? What state is thesystem left in?

Let us choose an eigenbasis |φjk〉, where states withthe label j correspond to λj , and

Pj =∑

k

|φjk〉〈φjk|.

There are many ways of choosing this eigenbasis, ingeneral, but it must be orthonormal.

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We can write the state in terms of this basis,

|ψ〉 =M∑

j=1

dj∑

k=1

αjk|φjk〉.

The probability of outcome j is then pj =∑

k |αjk|2, andthe system is left in the state

|ψj〉 =∑

k

(αjk/√pj)|φjk〉.

The important thing to note is that this outcome does notdepend on the particular choice of eigenbasis |φjk〉.Choosing any orthonormal eigenbasis will give thesame probabilities and the same final states.

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We can see this by rewriting the expressions in terms ofthe projector Pj, which is uniquely determined:

pj = 〈ψ|Pj |ψ〉, and |ψj〉 = Pj|ψ〉/√pj .

Each eigenvalue λj corresponds to a subspace of Hilbertspace; the probability of the outcome depends on thecomponent of |ψ〉 in that subspace, and the system’sstate is orthogonally projected into the subspace.

Note that these expressions include nondegenerateobservables as a special case, where Pj = |φj〉〈φj|.

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Interference

Interference occurs when amplitudes combine to augment orsuppress particular outcomes:

Suppose the system’s initial state is |0〉. How does the statechange as it passes through the two beam splitters?

|0〉 → 1√2|0〉+ 1√

2|1〉 → 1

2|0〉+ 1

2|1〉+ 1

2|0〉 − 1

2|1〉 = |0〉.

Interference between the two paths cancels the |1〉 termswhile enhancing the |0〉 terms, so detector 0 always clicks.

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Which-Way Information

What would happen if we did a projective measurement inbetween the two beam-splitters? That is, if we measuredwhich path the photon took? This would give outcomes of 0or 1 with equal probability.

|0〉 → 1√2|0〉+ 1√

2|1〉,

measurement → |0〉 measurement → |1〉OR

|0〉 → 1√2|0〉+ 1√

2|1〉, |1〉 → 1√

2|0〉 − 1√

2|1〉.

In both cases, there will be an equal probability of a click indetectors 0 or 1. The measurement destroys interference.

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The Meaning of Superposition States

One often sees superposition states like (|0〉+ |1〉)/√2

said to mean that “the system is either in state 0 orstate 1.” But this interpretation cannot be correct! If itwere, then measuring the system, to determine whichstate it really was in, should not change the outcome oflater measurements. But it does! Measuring which armof the interferometer held the photon lets detector 1click when it never would without the measurement.

This effect of measurement on interference is every-where in QM. Since most QIP protocols rely on inter-ference to work, including extra measurements candestroy their effectiveness. This includes interactionswith the environment, which can effectively “measure”the system. This effect is called decoherence, and it is theenemy of quantum information!

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Measurement and Entanglement

How does measurement affect entanglement? Considerthe maximally-entangled state |Φ+〉 = (|00〉+ |11〉) /

√2.

If we measure the first system in the Z basis, we getresults 0 and 1 with equal probability; after themeasurement, the system is left in the product state |00〉or |11〉. The same thing happens if we measure thesecond system in the Z basis.

What if we measure the first system in the X basis? Wecan re-write this state

|Φ+〉 =1

2√2

(

(|0〉+|1〉)⊗(|0〉+|1〉)+(|0〉−|1〉)⊗(|0〉−|1〉))

.

Once again, whatever the outcome, the final state willbe a product.

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After any complete local measurement (i.e., a completemeasurement of one of the subsystems) the system willbe left unentangled. So we see that, as with inter-ference, measurement can destroy entanglement.

However, joint measurements of both subsystems canalso create entanglement. For example, Bell statemeasurement. Even with an initial product state |ψ〉 ⊗ |φ〉,the system is left in one of the Bell states:

|Φ+〉 = (|00〉+ |11〉)/√2 |Φ−〉 = (|00〉 − |11〉)/

√2

|Ψ+〉 = (|01〉+ |10〉)/√2 |Ψ−〉 = (|01〉 − |10〉)/

√2.

In fact, entanglement is not an observable. There is nomeasurement which will reliably tell you if thepre-measurement state was entangled or not.

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We see both these effects at once in quantum teleportation.The initial state is |ψ〉 ⊗ |Ψ−〉, where Alice has the firsttwo q-bits and Bob has the third. There is no entangle-ment between Alice’s first q-bit and the other two; butthere is entanglement between her second q-bit andBob’s q-bit. Then Alice makes a Bell measurement onher two q-bits. Afterwards, there is entanglementbetween Alice’s two q-bits, but no longer betweenAlice’s q-bits and Bob’s.

The Bell state measurement in this case was acomplete local measurement with respect to thedivision between Alice and Bob’s systems, but anentangling joint measurement with respect to Alice’stwo subsystems.

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Compatible measurements

As stated before, if two observables A and B commute,

[A, B] ≡ AB − BA = 0,

then A and B are compatible, and can be measuredsimultaneously. Let us see why.

By the spectral theorem, both A and B can bedecomposed in terms of projectors:

A =

MA∑

j=1

ajPj , B =

MB∑

k=1

bkP ′k.

In order for [A, B] = 0 to be true, [Pj , P ′k] = 0 for all j, k.

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If two projectors Pj and P ′k commute, then their product

is also a projector:

(PjP ′k)

† = PjP ′k,

(PjP ′k)

2 = (Pj)2(P ′

k)2 = PjP ′

k.

This means that it is possible to write the spectraldecompositions of A and B in terms of more refinedprojectors:

A =

MA∑

j=1

MB∑

k=1

ajPjP ′k, B =

MA∑

j=1

MB∑

k=1

bkPjP ′k.

(Of course, many of the PjP ′k may actually be zero.)

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We have decomposed the two operators in terms of thesame set of projectors. We can find an orthonormalbasis of eigenvectors for both of these operators. Here’sa simple example:

A =

1 0 0

0 1 0

0 0 2

, B =

3 0 0

0 4 0

0 0 4

.

If two observables A and B are not compatible (i.e., donot commute), they are complementary: they cannot bemeasured simultaneously. This means that while astate |ψ〉 can be written in terms of the eigenbasis of Aor of B, it cannot be written in terms of a simultaneouseigenbasis of both.

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Our prime example of complementary observables arethe three Pauli operators X, Y and Z.

Each of these is an observable; they have eigenbases{(|0〉 ± |1〉)/

√2}, {(|0〉 ± i|1〉)/

√2}, {|0〉, |1〉}. If a spin is

in one of these eigenstates, it has a definite value of thecorresponding observable; but the other two arecompletely undetermined: any measurement of them has a50/50 chance of seing either possible result.

Complementarity is what makes BB84 work. Alicechooses definite values for one of two complementaryvariables to transmit her bits. Eve cannot measure onevariable without disturbing the other.

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Experimental measurements

The mathematical formulation of projectivemeasurement is extremely elegant. In this idealizeddescription, measurements disturb the system as littleas possible; they are repeatable, and always yielddefinite (though not deterministic) outcomes.

In a commonly used phrase, ideal projectivemeasurements are quantum nondemolition (QND)measurements.

In the real world, things are not so nice.

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Photodetection

A good example of this is the photodetector. While thereare various kinds of photodetectors, they for the mostpart work in roughly the same way. The photon isabsorbed by the detector; the energy liberates someelectric charges, which produce a measurable current(called the photocurrent) that is detected by a meter.

Unfortunately, this is very far from being a QNDmeasurement! It destroys the photon that is measured.

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Detecting, e.g., which arm of a Mach-Zenderinterferometer a photon is in without absorbing it isextremely difficult. Often the best that can be done isdetermining that the photon isn’t in one arm, by thefailure of a photodetector to click.

The spin-1/2 can be measured by a Stern-Gerlachdevice. Depending on the orientation of the spin, theatom it is attached to will be deflected up or down.However, we don’t know the outcome until the atomstrikes a photographic plate, at which point it is lost.

In some cases, we don’t need ideal measurements.Many QIP protocols only do measurements at the end.In principle, it is always possible to defer allmeasurements to the end of the protocol. This is theprinciple of deferred measurement. Let’s see how it works.

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Deferred Measurement

Suppose we wish to measure a q-bit in the Z basis, but westill need to use it after the measurement. How can we getaround this problem? Consider this quantum circuit:

If the first bit is in the state |0〉 the second bit will be left instate |0〉; if it is in state |1〉, the second bit will be flipped. Wecould then measure the second bit, and we will have,indirectly, measured the first—without destroying it.Moreover, we can wait to measure the second bit until theprocedure is over without altering the results.

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What if we want to measure in other than the standard Zbasis? Let U change from the basis we wish to measure tothe Z basis, and carry out this circuit:

The same trick works for measurements on more than onequantum bit. Here is a circuit for a Bell state measurement:

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Conditioning Actions

In some protocols, we may use the outcome of ameasurement to determine an action to perform. Forinstance, in the quantum teleportation protocol, theoutcome of Alice’s Bell state measurement determineswhich unitary transformation Bob must perform.

Even in this case, it is often possible to defer allmeasurements to the end of the procedure. We canreplace this conditioned operation by a controlledoperation.

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Suppose we wish to measure a q-bit in some basis, andif the result is a 1, do a unitary transformation V onanother q-bit.

The deferred measurement principle uses the fact thatwe can do measurements indirectly.

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Indirect Measurements

Such indirect measurements take the following form:

1. Prepare an extra system (e.g., another q-bit) in a knowninitial state (e.g., |0〉). This extra system is often referredto as an ancillary system or ancilla.

2. Have the system and the ancilla interact by carrying outsome circuit.

3. Measure the ancilla. This will give information about thesystem, thereby indirectly measuring it as well.

In practice, virtually all real measurements are indirect.Even our own eyesight works indirectly, by intercepting lightwhich has bounced off of the object we are viewing.

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Example of Indirect Measurement

Here is an example of an indirect measurement used inquantum information:

The system we wish to measure is the microwave mode inthe superconducting cavity. We probe it by preparing atomsin a particular state and sending them through the cavity;while inside, they interact with the microwaves, and aremeasured when they emerge.

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Generalized Measurements

In fact, the use of ancillas gives us more than just a wayof doing projective measurements. It enables us tocarry out a broad range of actions, of which projectivemeasurements are just a special case. These moregeneral procedures are called generalized measurements(or sometimes quantum operations).

Next time: Generalized measurements.

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