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Y. D. Chong PH4401: Quantum Mechanics III

Chapter 2: Quantum Entanglement

They don’t think it be like it is,

but it do.

Oscar Gamble

I. QUANTUM STATES OF MULTI-PARTICLE SYSTEMS

So far, we have studied quantum mechanical systems consisting of single particles. Thenext important step is to look at systems of more than one particle. We shall see thatthe postulates of quantum mechanics, when applied to multi-particle systems, give rise tointeresting and counterintuitive phenomena such as quantum entanglement.

Suppose we have two particles labeled A and B. If each individual particle is treated asa quantum system, the postulates of quantum mechanics require that its state be describedby a vector in a Hilbert space. Let HA and HB denote the respective single-particle Hilbertspaces. Then the Hilbert space for the combined system of two particles is

H = HA ⊗HB. (2.1)

The symbol ⊗ refers to a tensor product, a mathematical operation that combines twoHilbert spaces to form another Hilbert space. It is most easily understood in terms of explicitbasis vectors: let HA be spanned by a basis {|µ1〉, |µ2〉, |µ3〉, . . . }, and HB be spanned by{|ν1〉, |ν2〉, |ν3〉, . . . }. Then HA ⊗ HB is a space spanned by basis vectors consisting ofpairwise combinations of basis vectors drawn from the HA and HB bases:{

|µi〉 ⊗ |νj〉 for all |µi〉, |νj〉}. (2.2)

Thus, if HA has dimension dA and HB has dimension dB, then HA⊗HB has dimensiondAdB. Any two-particle state can be written as a superposition of these basis vectors:

|ψ〉 =∑ij

cij |µi〉 ⊗ |νj〉. (2.3)

The inner product between the tensor product basis states is defined as follows:(|µi〉 ⊗ |νj〉 , |µp〉 ⊗ |νq〉

)≡(〈µi| ⊗ 〈νj|

)(|µp〉 ⊗ |νq〉

)≡ 〈µi|µp〉〈νj|νq〉 = δipδjq. (2.4)

In other words, the inner product is performed “slot-by-slot”. We calculate the inner productfor A, calculate the inner product for B, and then multiply the two resulting numbers. Youcan check that this satisfies all the formal requirements for an inner product in linear algebra(see Exercise 1).

For example, suppose HA and HB are both 2D Hilbert spaces describing spin-1/2 degreesof freedom. Each space can be spanned by an orthonormal basis { |+z〉, |−z〉 }, representing“spin-up” and “spin-down”. Then the tensor product space H is a 4D space spanned by{

|+z〉 ⊗ |+z〉 , |+z〉 ⊗ |−z〉 , |− z〉 ⊗ |+z〉 , |−z〉 ⊗ |−z〉}. (2.5)

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We now make an important observation. If A is in state |µ〉 and B is in state |ν〉, thenthe state of the combined system is fully specified: |µ〉 ⊗ |ν〉 ∈ HA ⊗HB. But the reverseis not generally true! There exist states of the combined system that cannot be expressedin terms of definite states of the individual particles. For example, consider the followingquantum state of two spin-1/2 particles:

|ψ〉 =1√2

(|+z〉 ⊗ |−z〉 − |−z〉 ⊗ |+z〉

). (2.6)

This state is constructed from two of the four basis states in (2.5), and you can check thatthe factor of 1/

√2 ensures the normalization 〈ψ|ψ〉 = 1 with the inner product rule (2.4).

It is evident from looking at Eq. (2.6) that neither A nor B possesses a definite |+z〉 or |−z〉state. Moreover, we shall show (in Section VI) that there’s no choice of basis that allowsthis state to be expressed in terms of definite individual-particle states; i.e.,

|ψ〉 6= |ψA〉 ⊗ |ψB〉 for any |ψA〉 ∈HA, |ψB〉 ∈HB. (2.7)

In such a situation, the two particles are said to be entangled.

It is cumbersome to keep writing ⊗ symbols, so we will henceforth omit the ⊗ in caseswhere the tensor product is obvious. For instance,

1√2

(|+z〉 ⊗ |−z〉 − |−z〉 ⊗ |+z〉

)≡ 1√

2

(|+z〉|−z〉 − |−z〉|+z〉

). (2.8)

For systems of more than two particles, quantum states can be defined using multipletensor products. Suppose a quantum system contains N particles described by the individualHilbert spaces {H1,H2, . . . ,HN} having dimensionality {d1, . . . , dN}. Then the overallsystem is described by the Hilbert space

H = H1 ⊗H2 ⊗ · · · ⊗HN , (2.9)

which has dimensionality d = d1d2 · · · dN . The dimensionality scales exponentially with thenumber of particles! For instance, if each particle has a 2D Hilbert space, a 20-particle systemhas a Hilbert space with 220 = 1 048 576 dimensions. Thus, even in quantum systems witha modest number of particles, the quantum state can carry huge amounts of information.This is one of the motivations behind the active research field of quantum computing.

Finally, a proviso: although we refer to subsystems like A and B as “particles” for nar-rative convenience, they need not be actual particles. All this formalism applies to generalsubsystems—i.e., subsets of a large quantum system’s degrees of freedom. For instance, ifa quantum system has a position eigenbasis for 3D space, the x, y, and z coordinates aredistinct degrees of freedom, so each position eigenstate is really a tensor product:

| r = (x, y, z) 〉 ≡ |x〉 |y〉 |z〉.

Also, if the subsystems really are particles, we are going to assume for now that the particlesare distinguishable. There are other complications that arise if the particles are “identical”,which will be the subject of the next chapter. (If you’re unsure what this means, just readon.)

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II. PARTIAL MEASUREMENTS

Let us recall how measurements work in single-particle quantum theory. Each observableQ is described by some Hermitian operator Q, which has an eigenbasis {|qi〉} such that

Q|qi〉 = qi|qi〉. (2.10)

For simplicity, let the eigenvalues {qi} be non-degenerate. Suppose a particle initially has

quantum state |ψ〉. This can always be expanded in terms of the eigenbasis of Q:

|ψ〉 =∑i

ψi |qi〉, where and ψi = 〈qi|ψ〉. (2.11)

The measurement postulate of quantum mechanics states that if we measure Q, then(i) the probability of obtaining the measurement outcome qi is Pi = |ψi|2, the absolutesquare of the coefficient of |qi〉 in the basis expansion; and (ii) upon obtaining this outcome,the system instantly “collapses” into state |qi〉.

Mathematically, these two rules can be summarized using the projection operator

Π(qi) = |qi〉〈qi|. (2.12)

Applying this operator to |ψ〉 gives

|ψ′〉 = |qi〉〈qi|ψ〉, (2.13)

which is not normalized to unity. Then:

1. The probability of obtaining this outcome is 〈ψ′|ψ′〉 = |〈qi|ψ〉|2.

2. The post-collapse state is obtained by the re-normalization |ψ′〉 → |qi〉.

For multi-particle systems, there is a new complication: what if a measurement is per-formed on just one particle?

Consider a system of two particles A and B, with two-particle Hilbert space HA ⊗HB.We perform a measurement on particle A, corresponding to a Hermitian operator µ thatacts upon HA and has eigenvectors {|µ1〉, |µ2〉, . . . }. We can write any state |ψ〉 using theeigenbasis of µ for the HA part, and an arbitrary basis {|ν1〉, |ν2〉, . . . } for the HB part:

|ψ〉 =∑ij

ψij |µi〉|νj〉

=∑i

|µi〉|ϕi〉, where |ϕi〉 ≡∑j

ψij |νj〉.(2.14)

Unlike the single-particle case, the “coefficient” of |µi〉 in this basis expansion is not a com-plex number, but a vector |ϕi〉 ∈ HB. Proceeding by analogy, the probability of obtainingthe outcome µi ought to be the “absolute square” of this “coefficient”, 〈ϕi|ϕi〉. To make theanalogy a bit more rigorous, let us define the partial projector

Π(µi) = |µi〉〈µi| ⊗ I . (2.15)

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This contains a projector, |µi〉〈µi|, which only acts upon the HA part of the two-particlespace. Applying this to the above multi-particle state gives

|ψ′〉 = Π(µi) |ψ〉 = |µi〉|ϕi〉. (2.16)

Now we can follow the same measurement rules as before. The outcome probability is

Pi = 〈ψ′|ψ′〉 = 〈µi|µi〉〈ϕi|ϕi〉 =∑j

|ψij|2, (2.17)

and the post-measurement collapsed state is obtained by the re-normalization

|ψ′〉 → 1√∑j′ |ψij′ |2

∑j

ψij|µi〉|νj〉. (2.18)

Let’s work through an example. Consider a system of two spin-1/2 particles, with state

|ψ〉 =1√2

(|+z〉|−z〉 − |−z〉|+z〉

). (2.19)

For each particle, |+z 〉 and |−z 〉 denote eigenstates of the operator Sz, with eigenvalues+~/2 and −~/2 respectively. Suppose we measure Sz on particle A. There are two possibleoutcomes:

• First outcome: +~/2. The projection operator is |+z〉〈+z|⊗ I, and applying this to|ψ〉 yields |ψ′〉 = (1/

√2) |+z〉|−z〉. Hence, the outcome probability is P+ = 〈ψ′|ψ; 〉 = 1

2and the post-collapse state is |+z〉|−z〉.

• Second outcome: −~/2. The projection operator is |−z〉〈−z|⊗I, and applying this to|ψ〉 yields |ψ′〉 = (1/

√2) |−z〉|+z〉. Hence, the outcome probability is P− = 〈ψ′|ψ′〉 = 1

2and the post-collapse state is |−z〉|+z〉.

The two possible outcomes, +~/2 and −~/2, occur with equal probability. In either case,the two-particle state collapses so that A is in the observed spin eigenstate, and B has theopposite spin. After the collapse, the two-particle state is no longer entangled.

III. THE EINSTEIN-PODOLSKY-ROSEN “PARADOX”

In 1935, Einstein, Podolsky, and Rosen (EPR) used the counter-intuitive features ofquantum entanglement to formulate a thought experiment known as the EPR paradox.They tried to use this thought experiment to argue that quantum theory cannot serve asa fundamental description of reality. Subsequently, however, it was shown that the EPRparadox is not an actual paradox; physical systems really do have the strange behavior thatthe thought experiment highlighted.

The EPR paradox begins with an entangled state like this state of two spin-1/2 particles:

|ψ〉 =1√2

(|+z〉|−z〉 − |−z〉|+z〉

). (2.20)

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As before, let the two particles be labeled A and B. Measuring Sz on A collapses the systeminto a two-particle state that is unentangled, where each particle has a definite spin. If themeasurement outcome is +~/2, the new state is |+z〉|−z〉, whereas if the outcome is −~/2,the new state is |−z〉|+z〉.

The postulates of quantum theory seem to indicate that the state collapse happens in-stantaneously, regardless of the distance separating the particles. Imagine that we preparethe two-particle state in a laboratory, transport particle A to the Alpha Centauri star sys-tem, and transport particle B to the Betelgeuse system, separated by ∼ 640 light years. Inprinciple, this can be done carefully enough to avoid disturbing the two-particle quantumstate.

Once ready, an experimentalist at Alpha Centauri measures Sz on particle A, which in-duces an instantaneous collapse of the two-particle state. Immediately afterwards, an exper-imentalist at Betelgeuse measures Sz on particle B, and obtains—with 100% certainty—theopposite spin. During the time interval between these two measurements, no classical signalcould have traveled between the two star systems, not even at the speed of light. Yet thestate collapse induced by the measurement at Alpha Centauri has a definite effect on theresult of the measurement at Betelgeuse.

There are three noteworthy aspects of this phenomenon:

First, it dispels some commonsensical but mistaken “explanations” for quantum statecollapse in terms of perturbative effects. For instance, it is sometimes explained that if wewant to measure a particle’s position, we need to shine a light beam on it, or disturb itin some way, and this disturbance generates an uncertainty in the particle’s momentum.The EPR paradox shows that such stories don’t capture the full weirdness of quantum statecollapse, for we can collapse the state of a particle by doing a measurement on anotherparticle far away!

Second, the experimentalist has a certain amount of control over the state collapse, dueto the choice of what measurement to perform. So far, we have considered Sz measurementsperformed on particle A. But the experimentalist at Alpha Centauri can choose to measurethe spin of A along another axis, say Sx. In the basis of spin-up and spin-down states, theoperator Sx has matrix representation

Sx =~2

(0 11 0

). (2.21)

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The eigenvalues and eigenvectors are

sx =~2, |+x〉 =

1√2

(|+z〉+ |−z〉

)sx = −~

2, |−x〉 =

1√2

(|+z〉 − |−z〉

).

(2.22)

Conversely, we can write the Sz eigenstates in the {|+x〉, |−x〉} basis:

|+z〉 =1√2

(|+x〉+ |−x〉

)|−z〉 =

1√2

(|+x〉 − |−x〉

).

(2.23)

This allows us to write the two-particle entangled state in the Sx basis:

|ψ〉 =1√2

(|−x〉|+x〉 − |+x〉|−x〉

). (2.24)

The Alpha Centauri measurement still collapses the particles into definite spin states withopposite spins—but now spin states of Sx rather than Sz.

Third, this ability to choose the measurement axis does not allow for superluminal com-munication. The experimentalist at Alpha Centauri can choose whether to (i) measure Sz

or (ii) measure Sx, and this choice instantaneously affects the quantum state of particle B.If the Betelgeuse experimentalist can find a way to distinguish between the cases (i) and (ii),even statistically, this would serve as a method for instantaneous communication, violatingthe theory of relativity! Yet this turns out to be impossible. The key problem is that quan-tum states themselves cannot be measured; only observables can be measured. Suppose theAlpha Centauri measurement is Sz, which collapses B to either |+z〉 or |−z〉, each withprobability 1/2. The Betelgeuse experimentalist must now choose which measurement toperform. If Sz is measured, the outcome is +~/2 or −~/2 with equal probabilities. If Sx ismeasured, the probabilities are:

P (Sx = +~/2) =1

2

∣∣∣〈+x|+z〉∣∣∣2 +1

2

∣∣∣〈+x|−z〉∣∣∣2 =1

2

P (Sx = −~/2) =1

2

∣∣∣〈−x|+z〉∣∣∣2 +1

2

∣∣∣〈−x|−z〉∣∣∣2 =1

2.

(2.25)

The probabilities are still equal! Repeating this analysis for any other choice of spin axis,we find that the two possible outcomes always have equal probability. Thus, the Betelgeusemeasurement does not yield any information about the choice of measurement axis at AlphaCentauri.

Since quantum state collapse does not allow for superluminal communication, it is con-sistent in practice with the theory of relativity. However, state collapse is still nonlocal,in the sense that unobservable ingredients of the theory (quantum states) can change fasterthan light can travel between two points. For this reason, EPR argued that quantum theoryis philosophically inconsistent with relativity.

EPR suggested an alternative: maybe quantum mechanics is an approximation of somedeeper theory, whose details are currently unknown, but which is deterministic and local.

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Such a “hidden variable theory” may give the appearance of quantum state collapse inthe following way. Suppose each particle has a definite but “hidden” value of Sz, eitherSz = +~/2 or Sz = −~/2; let us denote these as [+] or [−]. We can hypothesize that thetwo-particle quantum state |ψ〉 is not an actual description of reality; rather, it correspondsto a statistical distribution of “hidden variable” states, denoted by [+;−] (i.e., Sz = +~/2for particle A and Sz = −~/2 for particle B), and [−; +] (the other way around).

When the Alpha Centauri experimentalist measures Sz, the value of the hidden variableis revealed. A result of +z implies [+;−], whereas −z implies [−; +]. When the Betelgeuseexperimentalist subsequently measures Sz, the result obtained is the opposite of the AlphaCentauri result. But those were simply the values all along—there is no instantaneousphysical influence traveling from Alpha Centauri to Betelguese.

Clearly, there are many missing details in this hypothetical description. Any actual hiddenvariable theory would also need to replicate the huge list of successful predictions made byquantum theory. Trying to come up with a suitable theory of this sort seems difficult, butwith enough hard work, one might imagine that it is doable.

IV. BELL’S THEOREM

In 1964, however, John S. Bell published a bombshell paper showing that the predictionsof quantum theory are inherently inconsistent with hidden variable theories. The amazingthing about this result, known as Bell’s theorem, is that it requires no knowledge aboutthe details of the hidden variable theory, just that it is deterministic and local. Here, wepresent a simplified version of Bell’s theorem, based on Mermin (1981).

We again consider spin-1/2 particle pairs, with particle A sent to Alpha Centauri andparticle B sent to Betelgeuse. At each location, an experimentalist can measure the particle’sspin along three distinct choices of spin axis. These spin observables are denoted by S1, S2,and S3. We will not specify the actual directions of these spin axes until later in the proof.For now, just note that the axes need not correspond to orthogonal spatial directions.

We now repeatedly prepare two-particle systems, and send the particles to Alpha Centauriand Betelgeuse. Each time, the prepared two-particle state is

|ψ〉 =1√2

(|+z〉|−z〉 − |−z〉|+z〉

). (2.26)

During each round of the experiment, each experimentalist randomly chooses one of thethree spin axes S1, S2, or S3, and performs that spin measurement. It doesn’t matter whichexperimentalist performs the measurement first; the experimentalists can’t influence each

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other, as there is not enough time for a light-speed signal to travel between the two locations.Many rounds of the experiment are conducted; for each round, both experimentalists’ choicesof spin axis are recorded, along with their measurement results.

At the end, the experimental records are brought together and examined. We assumethat the results are consistent with the predictions of quantum theory. Among other things,this means that whenever the experimentalists happen to choose the same measurementaxis, they always find opposite spins. (For example, this is the case during “Experiment 4”in the above figure, where both experimentalists happened to measure S3.)

Can a hidden variable theory reproduce the results predicted by quantum theory? In ahidden variable theory, each particle must have a definite value for each spin observable. Forexample, particle A might have S1 = +~/2, S2 = +~/2, S3 = −~/2. Let us denote this by[+ +−]. To be consistent with the predictions of quantum theory, the hidden spin variablesfor the two particles must have opposite values along each direction. This means that thereare 8 distinct possibilities, which we can denote as

[+ + +;−−−], [+ +−;−−+], [+−+;−+−], [+−−;−+ +],

[−+ +; +−−], [−+−; +−+], [−−+; + +−], [−−−; + + +].

For instance, [+ +−;−−+] indicates that for particle A, S1 = S2 = +~/2 and S3 = −~/2,while particle B has the opposite spin values, S1 = S2 = −~/2 and S3 = +~/2. So far,however, we don’t know anything about the relative probabilities of these 8 cases.

Let’s now focus on the subset of experiments in which the two experimentalists happenedto choose different spin axes (e.g., Alpha Centauri chose S1 and Betelgeuse chose S2). Withinthis subset, what is the probability for the two measurement results to have opposite signs(i.e., one + and one −)? To answer this question, we first look at the following 6 cases:

[+ +−;−−+], [+−+;−+−], [+−−;−+ +],

[−+ +; +−−], [−+−; +−+], [−−+; + +−].

These are the cases which do not have all + or all − for each particle. Consider one of these,say [+ +−;−−+]. The two experimentalists picked their measurement axes at randomeach time, and amongst the experiments where they picked different axes, there are twoways for the measurement results to have opposite signs: (S1, S2) or (S2, S1). There are fourways to get the same sign: (S1, S3), (S2, S3), (S3, S1) and (S3, S2). Thus, for this particular

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set of hidden variables, the probability for measurement results with opposite signs is 1/3.If we go through all 6 of the cases listed above, we find that in call cases, the probability foropposite signs is 1/3.

Now look at the remaining 2 cases:

[+ + +;−−−], [−−−; + + +].

For these, the experimentalists always obtain results that have opposite signs. Combiningthis with the findings from the previous paragraph, we obtain the following statement:

Given that the two experimentalists choose different spin axes, the probability that theirresults have opposite signs is P ≥ 1/3.

This is called Bell’s inequality. If we can arrange a situation where quantum theorypredicts a probability P < 1/3 (i.e., a violation of Bell’s inequality), that would mean thatquantum theory is inherently inconsistent with local deterministic hidden variables. Thisconclusion would hold regardless of the “inner workings” of the hidden variable theory. Inparticular, note that the above derivation made no assumptions about the relative proba-bilities of the hidden variable states.

To complete the proof, we must find a set {S1, S2, S3} such that the predictions of quantummechanics violate Bell’s inequality. One simple choice is to align S1 with the z axis, andalign S2 and S3 along the x-z plane at 120◦ (2π/3 radians) from S1, as shown below:

The corresponding spin operators can be written in the eigenbasis of Sz:

S1 =~2σ3

S2 =~2

[cos(2π/3)σ3 + sin(2π/3)σ1]

S3 =~2

[cos(2π/3)σ3 − sin(2π/3)σ1] .

(2.27)

Suppose the Alpha Centauri experimentalist chooses S1, and obtains +~/2. Particle Acollapses to state |+z〉, and particle B collapses to state |−z〉. The Betelgeuse experimentalistis assumed to choose a different spin axis. If the choice is S2, the expectation value is

〈−z |S2 | −z 〉 =~2

[cos(2π/3)〈−z |σ3| −z 〉+ sin(2π/3)〈−z |σ1| −z 〉

]=

~2· 1

2

(2.28)

If P+ and P− respectively denote the probability of measuring +~/2 and −~/2 in thismeasurement, the above equation implies that P+−P− = +1/2. Moreover, P+ +P− = 1 byprobability conservation. It follows that the probability of obtaining a negative value (the

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opposite sign from the Alpha Centauri measurement) is P− = 1/4. All the other possiblescenarios are worked out in a similar way.

The result is that if the two experimentalists choose different measurement axis, theyobtain results of opposite signs with probability 1/4. Therefore, Bell’s inequality is violated.

Last of all, we must consult Nature itself. Is it possible to observe, in an actual ex-periment, probabilities that violate Bell’s inequality? In the decades following Bell’s 1964paper, many experiments were performed to answer this question. These experiments areall substantially more complicated than the simple two-particle spin-1/2 model that we’vestudied, and they are subject to various uncertainties and “loopholes” that are beyond thescope of our discussion. But in the end, the experimental consensus appears to be a clearyes : Nature really does behave according to quantum mechanics, and in a manner that can-not be replicated by deterministic local hidden variables! A summary of the experimentalevidence can be found in Aspect (1999).

V. DENSITY OPERATORS

We will now introduce a theoretical construct called the density operator, which helpsto streamline many calculations in multi-particle quantum mechanics.

Consider a quantum system with a d-dimensional Hilbert space H . Given an arbitrarystate |ψ〉 ∈H , we can define an operator

ρ = |ψ〉〈ψ|. (2.29)

This is simply the projection operator associated with |ψ〉, but in this context we call it adensity operator. (Some authors call it a density matrix, based on the understanding thatquantum operators can be represented as matrices).

In the language of linear algebra, ρ is formed by the “matrix outer product” of |ψ〉 withits Hermitian conjugate. It has the following noteworthy features:

1. It is a Hermitian operator. Moreover, one eigenvalue is 1 (with eigenvector |ψ〉), andthe other d − 1 eigenvalues are all 0 (the corresponding eigenvectors consist of d − 1vectors spanning the subspace orthogonal to |ψ〉). Thus,

All eigenvalues of ρ are real numbers in [0, 1].

Sum of eigenvalues of ρ = Tr[ρ]

= 1.(2.30)

2. With the system in state |ψ〉, suppose we perform a measurement corresponding

to the Hermitian operator Q, which has eigenvalues {q1, q2, . . . } and eigenvectors{|q1〉, |q2〉, . . . }. The probability for outcome qi is |〈qi|ψ〉|2 = 〈qi|ψ〉〈ψ|qi〉; thus,

P (qi) = 〈qi| ρ |qi〉. (2.31)

3. The expectation value of the Q observable is 〈Q〉 =∑

i qiP (qi), and after a few linesof algebra we can show that

〈Q〉 = Tr[Q ρ

]. (2.32)

This is a nice expression, because the trace is a basis-independent operation, so 〈Q〉is explicitly shown to be basis-independent.

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Now, suppose again that the quantum system consists of two particles (subsystems) Aand B, with Hilbert spaces HA and HB, where H = HA⊗HB. Suppose we are interestedin the physical behavior of A—specifically, the expectation values of partial measurementsperformed on A, the corresponding outcome probabilities, etc. These quantities can, ofcourse, be calculated from the overall system state |ψ〉, but this might be inconvenient todeal with (e.g., if A is a relatively simple subsystem but B is large and complicated).

However, it turns out that there is a way to describe the behavior of A in its own terms.This is done by defining a reduced density operator for A:

ρA = TrB[ρ]. (2.33)

Here, TrB[· · · ] refers to a partial trace, which means tracing over the HB part of theHilbert space H = HA ⊗HB. The partial trace yields an operator acting on HA.

To better understand Eq. (2.33), let us work in an explicit basis. Let {|µ1〉, |µ2〉, · · · } bea basis for HA, and {|ν1〉, |ν2〉, · · · } be a basis for HB. Any given state of the combinedsystem can be written in the tensor product basis:

|ψ〉 =∑ij

ψij |µi〉|νj〉, (2.34)

for some complex coefficients ψij. We require∑

ij |ψij|2 = 1 so that |ψ〉 is normalized tounity. The density operator of the combined system is

ρ = |ψ〉〈ψ| =∑ii′jj′

(|µi〉|νj〉

)ψijψ

∗i′j′

(〈µi′ |〈νj′|

). (2.35)

The reduced density operator for subsystem A is

ρA =∑j

〈νj|ρ|νj〉

=∑ii′

|µi〉 ρAii′ 〈µi′ | where ρAii′ ≡∑j

ψijψ∗i′j.

(2.36)

From this, we see that ρA is Hermitian. Moreover, if we happen to take the {|µi〉} basis tobe the eigenbasis for ρA, the eigenvalues are ρAii =

∑j |ψij|2. We can thus prove that:

All eigenvalues of ρA are real numbers in [0, 1].

Sum of eigenvalues of ρA = Tr[ρA]

= 1.(2.37)

The reduced density operator contains all the information required to calculate outcomeprobabilities for any partial measurement performed on A. Suppose we perform a measure-ment on A corresponding to a Hermitian operator µ. Without loss of generality, we canchoose the basis {|µ1〉, |µ2〉, . . . } to be the eigenvectors of µ. We claim that the probabilityof obtaining outcome µi is

P (µi) = 〈µi|ρA|µi〉 (2.38)

= ρAii =∑j

|ψij|2. (2.39)

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To prove this, recall the rules for partial measurements from Section II. The projectionoperator for measurement outcome µi is

Π(µi) = |µi〉〈µi| ⊗ I . (2.40)

Therefore,

P (µi) = 〈ψ|Π(µi)Π(µi)|ψ〉= 〈ψ|Π(µi)|ψ〉

= 〈ψ|(|µi〉〈µi| ⊗ I

)|ψ〉

=∑j

〈ψ|(|µi〉〈µi| ⊗ |νj〉〈νj|

)|ψ〉

=∑j

(〈µi|〈νj|

)|ψ〉〈ψ|

(|µi〉|νj〉

)= 〈µi| TrB[ρ] |µi〉 (Q.E.D.)

(2.41)

It also follows that the expectation value for this measurement is

〈µ〉 = TrA[µ ρA

]. (2.42)

The reduced density operator’s properties (2.37), (2.38), and (2.42) are precisely the sameas the original density operator’s properties (2.30), (2.31), (2.32). Hence, we deduce thatthese are the general properties of any density operator, reduced or not. The special classof density operators that can be written in the form ρ = |ψ〉〈ψ| (i.e., in terms of an explicitquantum state, like the one we started this discussion with) are said to describe pure states.Density operators that cannot be written in such a form are said to describe mixed states.

A mixed state can also be interpreted as a probability distribution of quantum states. Tosee this, let us return to Eq. (2.36):

ρA =∑ii′j

|µi〉ψijψ∗i′j〈µi′ |

=∑j

|ψ′j〉〈ψ′j| where |ψ′j〉 ≡∑i

ψij|µi〉.(2.43)

However, the |ψ′〉 vectors inside the sum are not normalized. Re-normalizing them yieldsthe expression

ρA =∑j

pj |Ψj〉〈Ψj| where

pj = 〈ψ′j|ψ′j〉 =

∑i

|ψij|2

|Ψj〉 =1√pj|ψ′j〉, so 〈Ψj|Ψj〉 = 1.

(2.44)

This is a weighted sum, where each term involves a pure state |Ψj〉〈Ψj| and a weight pj.The pj’s are real numbers between 0 and 1, and sum to unity, so they can be regardedas probabilities. We can therefore interpret ρA as describing an “ensemble” of differentquantum systems, weighted by a set of probabilities pj.

12


Although this interpretation is highly suggestive, one should not take it too seriously.For one thing, the decomposition is not unique: it is possible to express a given densityoperator using different weighted sums, each having a different set of probabilities. For anexample, see Exercise 3. Ultimately, the information about the mixed state is encoded inthe mathematical object ρA itself, which cannot really be simplified any further.

VI. ENTANGLEMENT ENTROPY

Previously, we said that a multi-particle system is entangled if the individual particleslack definite quantum states. It would be nice to make this statement more precise, and infact physicists have come up with several different quantitive measures of entanglement. Inthis section, we will describe the most common measure, entanglement entropy, whichis closely related to the entropy concept from thermodynamics, statistical mechanics, andinformation theory.

We have seen from the previous section that if a subsystem A is (possibly) entangledwith some other subsystem B, the information required to calculate all partial measurementoutcomes on A is stored within a reduced density operator ρA. We can use this to define aquantity called the entanglement entropy of A:

SA = −kb TrA

{ρA ln

[ρA]}. (2.45)

In this formula, ln[· · · ] denotes the logarithm of an operator, which is the inverse of the

exponential: ln(P ) = Q ⇒ exp(Q) = P . The prefactor kb is Boltzmann’s constant, andensures that SA has the same units as thermodynamic entropy.

The definition of the entanglement entropy is based on an analogy with the entropy con-cept from classical thermodynamics, statistical mechanics and information theory. In thoseclassical contexts, entropy is a quantitative measure of uncertainty (i.e, lack of information)about a system’s underlying microscopic state, or “microstate”. Suppose a system has Wpossible microstates that occur with probabilities {p1, p2, . . . , pW}, satisfying

∑i pi = 1.

Then we define the classical entropy

Scl. = −kbW∑i=1

pi ln(pi). (2.46)

In a situation of complete certainty where the system is known to be in a specific microstatek (pi = δik), the formula gives Scl. = 0. (Note that x ln(x) → 0 as x → 0). In a situationof complete uncertainty where all microstates are equally probable (pi = 1/W ), we getScl. = kb lnW , the entropy of a microcanonical ensemble in statistical mechanics. For anyother distribution of probabilities, it can be shown that the entropy lies between these twoextremes: 0 ≤ Scl. ≤ kb lnW . For a review of the properties of entropy, see Appendix C.

The concept of entanglement entropy aims to quantify the uncertainty arising from aquantum (sub)system’s lack of a definite quantum state, due to it being possibly entangledwith another (sub)system. When formulating it, the key issue we need to be careful about ishow to extend classical notions of probability to quantum systems. We have seen that whenperforming a measurement on A whose possible outcomes are {µ1, µ2, . . . }, the probability

13


of getting µi is pi = 〈µi|ρA|µi〉. However, it is problematic to directly substitute theseprobabilities {pi} into the classical entropy formula, since they are basis-dependent (i.e., theset of probabilities is dependent on the choice of measurement). Eq. (2.45) bypasses thisproblem by using the trace, which is basis-independent.

In the special case where {|µi〉} is the eigenbasis for ρA, the connection is easier to see.From (2.37), the eigenvalues {pi} are all real numbers between 0 and 1, and summing tounity, so they can be regarded as probabilities. Then the entanglement entropy is

SA = −kb∑i

〈µi|ρA ln(ρA)|µi〉

= −kb∑i

pi ln(pi).(2.47)

Therefore, in this particular basis the expression for the entanglement entropy is consistentwith the classical definition of entropy, with the eigenvalues of ρA serving as the relevantprobabilities.

By analogy with the classical entropy formula (see Appendix C), the entanglement entropyhas the following bounds:

0 ≤ SA ≤ kb ln(dA), (2.48)

where dA is the dimension of HA.

For an unentangled system—i.e., a pure state—we can write ρ = |ψ〉〈ψ| for some state |ψ〉.In this case, |ψ〉 is itself an eigenstate of ρ with eigenvalue 1, so SA = 0. This corresponds tothe lower bound of (2.48). Conversely, if we find that a system has SA 6= 0, that implies thatρA cannot be written as a pure state |ψ〉〈ψ| for any |ψ〉; hence, it must be entangled—i.e.,ρA must describe a mixed state.

A system is said to be maximally entangled if it saturates the upper bound of (2.48),SA = kb ln(dA). This occurs if and only if the eigenvalues of the density operator are allequal: i.e., pi = 1/dA for all i = 1, . . . , dA.

As an example, consider the following state of two spin-1/2 particles:

|ψ〉 =1√2

(|+z〉|−z〉 − |−z〉|+z〉

). (2.49)

The density operator for the two-particle system is

ρ(ψ) =1

2

(|+z〉|−z〉 − |−z〉|+z〉

)(〈+z|〈−z| − 〈−z|〈+z|

). (2.50)

Tracing over subsystem B gives the reduced density operator

ρA(ψ) =1

2

(|+z〉〈+z| + |−z〉〈−z|

). (2.51)

This can be expressed as a matrix in the {|+ z〉, |− z〉} basis:

ρA(ψ) =

(12

00 1

2

). (2.52)

Next, we can use ρA to compute the entropy of subsystem A:

SA = −kbTr {ρA ln(ρA)} = −kbTr

(12

ln(12

)0

0 12

ln(12

)) = kb ln(2). (2.53)

Hence, subsystems A and B are maximally entangled.

14


VII. THE MANY WORLDS INTERPRETATION

We conclude this chapter by discussing a set of compelling but controversial ideas arisingfrom the phenomenon of quantum entanglement: the Many Worlds Interpretation asformulated by Hugh Everett (1956).

So far, when describing the phenomenon of state collapse, we have relied on the mea-surement postulate (see Section II), which is part of the Copenhagen Interpretation ofquantum mechanics. This is how quantum mechanics is typically taught, and how physiciststhink about the theory when doing practical, everyday calculations.

However, the measurement postulate has two bad features:

1. It stands apart from the other postulates of quantum mechanics, for it is the onlyplace where randomness (or “indeterminism”) creeps into quantum theory. The otherpostulates do not refer to probabilities. In particular, the Schrodinger equation

i~∂

∂t|ψ(t)〉 = H(t)|ψ(t)〉 (2.54)

is completely deterministic. If you know H(t) and are given the state |ψ(t0)〉 at sometime t0, you can in principle determine |ψ(t)〉 for all t. This time-evolution consistsof a smooth, non-random rotation of the state vector within its Hilbert space. Ameasurement process, however, has a completely different character: it causes thestate vector to jump discontinuously to a randomly-selected value. It is strange thatquantum theory contains two completely different ways for a state to change.

2. The measurement postulate is silent on what constitutes a measurement. Does mea-surement require a conscious observer? Surely not: as Einstein once exasperatedlyasked, are we really expected to believe that the Moon exists only when we look atit? But if a given device interacts with a particle, what determines whether it acts viathe Schrodinger equation, or performs a measurement?

The Many Worlds Interpretation seeks to resolve these problems by positing that themeasurement postulate is not a fundamental postulate of quantum mechanics. Rather, whatwe call “measurement”, including state collapse and the apparent randomness of measure-ment results, is an emergent phenomenon that can be derived from the behavior of complexmany-particle quantum systems obeying the Schrodinger equation. The key idea is that ameasurement process can be described by applying the Schrodinger equation to a quantumsystem containing both the thing being measured and the measurement apparatus itself.

We can study this using a toy model formulated by Albrecht (1993). Consider a spin-1/2particle, and an apparatus designed to measure Sz. Let HS be the spin-1/2 Hilbert space(which is 2D), and HA be the Hilbert space of the apparatus (which has dimension d). Wewill assume that d is very large, as actual experimental apparatuses are macroscopic objectscontaining 1023 or more atoms! The Hilbert space of the combined system is

H = HS ⊗HA, (2.55)

and is 2d-dimensional. Let us suppose the system is prepared in an initial state

|ψ(0)〉 =(a+|+ z〉+ a−|− z〉

)⊗ |Ψ〉, (2.56)

15


where a± ∈ C are the quantum amplitudes for the particle to be initially spin-up or spin-down, and |Ψ〉 ∈HA is the initial state of the apparatus.

The combined system now evolves via the Schrodinger equation. We aim to show that ifthe Hamiltonian has the form

H = Sz ⊗ V , (2.57)

where Sz is the operator corresponding to the observable Sz, then time evolution has aneffect equivalent to the measurement of Sz.

It turns out that we can show this without making any special choices for V or |Ψ〉. We

only need d� 2, and for both V and |Ψ〉 to be “sufficiently complicated”. We choose |Ψ〉 to

be a random state vector, and choose random matrix components for the operator V . Theprecise generation procedures will be elaborated on later. Once we decide on |Ψ〉 and V , wecan evolve the system by solving the Schodinger equation

|ψ(t)〉 = U(t)|ψ(0)〉, where U(t) = exp

[− i~Ht

]. (2.58)

Because the part of H acting on the HS subspace is Sz, the result necessarily has thefollowing form:

|ψ(t)〉 = U(t)(a+|+z〉+ a−|−z〉

)⊗ |Ψ〉

= a+|+z〉 ⊗ |Ψ+(t)〉 + a−|−z〉 ⊗ |Ψ−(t)〉.(2.59)

Here, |Ψ+(t)〉 and |Ψ−(t)〉 are apparatus states that are “paired up” with the |+z〉 and |−z〉states of the spin-1/2 subsystem. At t = 0, both |Ψ+(t)〉 and |Ψ−(t)〉 are equal to |Ψ〉; fort > 0, they rotate into different parts of the state space HA. If the dimensionality of HA

is sufficiently large, and both V and |Ψ〉 are sufficiently complicated, we can guess (and wewill verify numerically) that the two state vectors rotate into completely different parts ofthe state space, so that

〈Ψ+(t)|Ψ−(t)〉 ≈ 0 for sufficiently large t. (2.60)

Once this is the case, the two terms in the above expression for |ψ(t)〉 can be interpretedas two decoupled “worlds”. In one world, the spin has a definite value +~/2, and theapparatus is in a state |Ψ+〉 (which might describe, for instance, a macroscopically-sizedphysical pointer that is pointing to a “Sz = +~/2” reading). In the other world, the spinhas a definite value −~/2, and the apparatus has a different state |Ψ−〉 (which might describea physical pointer pointing to a “Sz = −~/2” reading). Importantly, the |Ψ+〉 and |Ψ−〉states are orthogonal, so they can be rigorously distinguished from each other. The twoworlds are “weighted” by |a+|2 and |a−|2, which correspond to the probabilities of the twopossible measurement results.

The above description can be tested numerically. Let us use an arbitrary basis for theapparatus space HA; in that basis, let the d components of the initial apparatus state vector|Ψ〉 be random complex numbers:

|Ψ〉 =1√N

Ψ0

Ψ1...

Ψd−1

, where Re(Ψj), Im(Ψj) ∼ N(0, 1). (2.61)

16


In other words, the real and imaginary parts of each complex number Ψj are independentlydrawn from the the standard normal (Gaussian) distribution, denoted by N(0, 1). Thenormalization constant N is defined so that 〈Ψ|Ψ〉 = 1.

Likewise, we generate the matrix elements of V according to the following random scheme:

Aij ∼ uij + ivij, where uij, vij ∼ N(0, 1)

V =1

2√d

(A+ A†

).

(2.62)

This scheme produces a d× d matrix with random components, subject to the requirementthat the overall matrix be Hermitian. The normalization factor 1/2

√d is relatively unimpor-

tant; it ensures that the eigenvalues of V lie in a fixed range, [−2, 2], instead of scaling withd. If you are interested in learning more about the properties of such “random matrices”,see Edelman and Rao (2005).

The Schrodinger equation can now be solved numerically. The results are shown below:

In the inital state, we let a+ = 0.7, so a− =√

1− 0.72 = 0.71414 . . . The upper panelplots the overlap between the two apparatus states, |〈Ψ+|Ψ−〉|2, versus t. In accordancewith the preceding discussion, the overlap is unity at t = 0, but subsequently decreases tonearly zero. For comparison, the lower panel plots the entanglement entropy between thetwo subsystems, SA = −kbTrA {ρA ln ρA}, where ρA is the reduced density matrix obtainedby tracing over the spin subspace. We find that SA = 0 at t = 0, due to the fact that thespin and apparatus subsystems start out with definite quantum states in |ψ(0)〉. As thesystem evolves, the subsystems become increasingly entangled, and SA increases up to

SmaxA /kb = −

(|a+|2 ln |a+|2 + |a−|2 ln |a−|2

)≈ 0.693 (2.63)

This value is indicated in the figure by a horizontal dashed line, and corresponds to theresult of the classical entropy formula for probabilities {|a+|2, |a−|2}. Moreover, we seethat the entropy reaches Smax

A at around the same time that |〈Ψ+|Ψ−〉|2 reaches zero. Thisdemonstrates the close relationship between “measurement” and “entanglement”.

17


For details about the numerical linear algebra methods used to perform the above calcu-lation, refer to Appendix D.

The “many worlds” concept can be generalized from the above toy model to the universeas a whole. In the viewpoint of the Many Worlds Interpretation of quantum mechanics, theentire universe can be described by a mind-bogglingly complicated quantum state, evolvingdeterministically according to the Schrodinger equation. This evolution involves repeated“branchings” of the universal quantum state, which continuously produces more and moreworlds. The classical world that we appear to inhabit is just one of a vast multitude. It isup to you to decide whether this conception of reality seems reasonable. It is essentially amatter of preference, because the Copenhangen Interpretation and the Many Worlds Inter-pretation have identical physical consequences, which is why they are referred to as different“interpretations” of quantum mechanics, rather than different theories.

Exercises

1. Let HA and HB denote single-particle Hilbert spaces with well-defined inner products.That is to say, for all vectors |µ〉, |µ′〉, |µ′′〉 ∈ HA, that Hilbert space’s inner productsatisfies the inner product axioms

(a) 〈µ|µ′〉 = 〈µ′|µ〉∗

(b) 〈µ|µ〉 ∈ R+0 , and 〈µ|µ〉 = 0 if and only if |µ〉 = 0.

(c) 〈µ|(|µ′〉+ |µ′′〉

)= 〈µ|µ′〉+ 〈µ|µ′′〉

(d) 〈µ|(c|µ′〉

)= c〈µ|µ′〉 for all c ∈ C,

and likewise for vectors from HB with that Hilbert space’s inner product.

In Section I, we defined a tensor product space HA ⊗HB as the space spanned bythe basis vectors {|µi〉 ⊗ |νj〉}, where |µi〉 are basis vectors for HA and |νj〉 are basisvectors for HB. Prove that we can define an inner product using(

〈µi| ⊗ 〈νj|)(|µp〉 ⊗ |νq〉

)≡ 〈µi|µp〉〈νj|νq〉 = δipδjq (2.64)

which satisfies the inner product axioms.

2. Consider the density operator

ρ =1

2|+z〉〈+z| +

1

2|+x〉〈+x| (2.65)

where |+x〉 = 1√2

(|+z〉+ |−z〉). This can be viewed as an equal-probability sum of

two different pure states. However, the density matrix can also be written as

ρ = p1 |ψ1〉〈ψ1| + p2 |ψ2〉〈ψ2| (2.66)

where |ψ1,2〉 are the eigenvectors of ρ. Show that p1 and p2 are not 1/2.

18


3. Consider two distinguishable particles, A and B. The 2D Hilbert space of A is spannedby {|µ〉, |ν〉}, and the 3D Hilbert space of B is spanned by {|p〉, |q〉, |r〉}. The two-particle state is

|ψ〉 =1

3|µ〉|p〉+

1√6|µ〉|q〉+

1√18|µ〉|r〉+

√2

3|ν〉|p〉+

1√3|ν〉|q〉+

1

3|ν〉|r〉. (2.67)

Find the entanglement entropy.

Further Reading

[1] Bransden & Joachain, §14.1—14.4, §17.1–17.5

[2] Sakurai, §3.9

[3] A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Phys-ical Reality Be Considered Complete?, Physical Review 47, 777 (1935). [link]

[4] J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics 1, 195 (1964). [link]

[5] N. D. Mermin, Bringing home the atomic world: Quantum mysteries for anybody, Amer-ican Journal of Physics 49, 940 (1981). [link]

[6] A. Aspect, Bell’s inequality test: more ideal than ever, Nature (News and Views) 398,189 (1999). [link]

[7] H. Everett, III, The Theory of the Universal Wave Function (PhD thesis), PrincetonUniversity (1956) [link]

[8] A. Albrecht, Following a “collapsing” wave function, Physical Review D 48, 3768 (1993).[link]

[9] A. Edelman and N. R. Rao, Random matrix theory, Acta Numerica 14, 233 (2005). [link]

19

https://journals.aps.org/pr/abstract/10.1103/PhysRev.47.777

http://inspirehep.net/record/31657/files/

http://aapt.scitation.org/doi/abs/10.1119/1.12594

https://www.nature.com/articles/18296

http://ucispace.lib.uci.edu/handle/10575/1302

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.48.3768

https://doi.org/10.1017/S0962492904000236

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