The Mathematics of Entanglement - Summer 2013 27 May, 2013

Quantum States

Lecturer: Fernando G.S.L. Brandao Lecture 1

Entanglement is a quantum mechanical form of correlation, which appears in many areas, suchas condensed matter physics, quantum chemistry, and other areas of physics. This week we willdiscuss a perspective from quantum information, which means we will abstract away the underlyingphysics, and make statements about entanglement that apply independent of the underlying physi-cal system. This will also allow us to discuss information-processing applications, such as quantumcryptography.

1 Probability theory and Tensor products

Before discussing quantum states, we explain some aspects of probability theory, which turns outto have many similar features.

Suppose we have a system with d possible states, for some integer d, which we label by 1, . . . , d.Thus a deterministic state is simply an element of the set {1, . . . , d}. The probabilistic states areprobability distributions over this set, i.e. vectors in Rd

+ whose entries sum to 1. The notationRd+ means that the entries are nonnegative. Thus, a probability distribution p = (p(1), . . . , p(d))

satisfies∑d

x=1 p(x) = 1 and p(x) ≥ 0 for each x. Note that we can think of a deterministic statex ∈ {1, . . . , d} as the probability distribution where p(x) = 1 and all other probabilities are zero.

Composition. Suppose we are given p ∈ Rm+ and q ∈ Rn

+ which correspond to independentprobability distributions. Their joint distribution is given by the vector

p⊗ q :=

p(1)q(1)p(1)q(2)

...p(1)q(n)

...p(m)q(n)

.

We have introduced the notation ⊗ to denote the tensor product, which in general maps a pair ofvectors with dimensions m,n to a single vector with dimension mn. Later we will also considerthe tensor product of matrices. If Mn denotes n× n matrices, and we have A ∈Mn, B ∈Mm thenA⊗ B ∈ Mnm is the matrix whose entries are all possible products of an entry of A and an entry

of B. For example, if n = 2 and A =

(a11 a12a21 a22

)then A⊗B is the block matrix(

a11B a12Ba21B a22B

).

One useful fact about tensor products, which simplifies many calculations, is that

(A⊗B)(C ⊗D) = AC ⊗BD.

We also define the tensor product of two vector space V ⊗W to be the span of all v ⊗ w forv ∈ V and w ∈W . In particular, observe that Cm ⊗ Cn = Cmn.

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2 Quantum Mechanics

We will use Dirac notation in which a “ket” |ψ〉 denote a column vector in a complex vector space,i.e.

|ψ〉 =

ψ1

ψ2...ψd

∈ Cd.

The “bra” 〈ψ| denotes the conjugate transpose, i.e.

〈ψ| =(ψ∗1 ψ∗2 · · · ψ∗d

).

Combining a bra and a ket gives a “bra[c]ket”, meaning an inner product

〈ϕ|ψ〉 =

d∑i=1

ϕ∗iψi.

In this notation the norm is

‖ψ‖2 =√〈ψ|ψ〉 =

√√√√ d∑i=1

|ψi|2.

Now we can define a quantum state. The quantum analogue of a system with d states is thed-dimensional Hilbert space Cd. For example, a quantum system with d = 2 is called a qubit. Unitvectors |ψ〉 ∈ Cd, where 〈ψ|ψ〉 = 1, are called pure states. They are the analogue of deterministicstates in classical probability theory. For example, we might define the following pure states of aqubit:

|0〉 =

(10

), |1〉 =

(01

), |+〉 =

1√2

(|0〉+ |1〉), |−〉 =1√2

(|0〉+ |1〉).

Note that both pairs |0〉 , |1〉 and |+〉 , |−〉 form orthonormal bases of a qubit.

2.1 Measurements

A projective measurement is a collection of projectors {Pk} such that Pk ∈Md for each k, P †k = Pk,PkPk′ = δk,k′Pk and

∑k Pk = I. For example, we might measure in the computational basis, which

consists of the unit vectors |k〉 with a one in the kth position and zeros elsewhere. Thus define

Pk = |k〉 〈k| =

0 0. . .

01

0. . .

0 0

,

which is the projector onto the one-dimensional subspace spanned by |k〉.

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Born’s rule states that Pr[k], the probability of measurement outcome k, is given by

Pr[k] = 〈ψ|Pk |ψ〉 .

As an exercise, verify that this is equal to tr(Pk |ψ〉 〈ψ|). In our example, this is simply |ψk|2.Example. If we perform the measurement {|0〉 〈0| , |1〉 〈1|} on |+〉, then Pr[0] = Pr[1] = 1/2. If

we perform the measurement {|+〉 〈+| , |−〉 〈−|}, then Pr[+] = 1 and Pr[−] = 0.

3 Mixed states

Mixed states are a common generalization of probability theory and pure quantum mechanics. Ingeneral, if we have an ensemble of pure quantum states |ψx〉 with probabilities p(x), then definethe density matrix to be

ρ =∑x

p(x) |ψx〉 〈ψx| .

The vectors |ψx〉 do not have to be orthogonal.Note that ρ is always Hermitian, meaning ρ = ρ†. Here † denotes the conjugate transpose, so

that (A†)i,j = Aj,i. In fact, ρ is positive semi-definite (“PSD”). This is also denoted ρ ≥ 0. Twoequivalent definitions (assuming that ρ = ρ†) are:

1. For all |ψ〉, 〈ψ| ρ |ψ〉 ≥ 0.

2. All the eigenvalues of ρ are nonnegative. That is,

ρ =∑i

λi |ϕi〉 〈ϕi| (1)

for an orthonormal basis {|ϕ1〉 , . . . , |ϕd〉} with each λi ≥ 0.

Exercise: Prove that these definitions are equivalent.A density matrix should also have trace one, since tr ρ =

∑x p(x) 〈ψx|ψx〉 =

∑x p(x) = 1.

Conversely, any PSD matrix with trace one can be written in the form∑

x p(x) |ψx〉 〈ψx| forsome probability distribution p and some unit vectors {|ψx〉}, and hence is a valid density matrix.This is just based on the eigenvalue decomposition: we can always take p(x) = λx in (1).

Note that this decomposition is not unique in general. For example, consider the maximally

mixed state ρ = I/2 =

(1/2 00 1/2

). This can be decomposed either as 1

2 |0〉 〈0| +12 |1〉 〈1| or as

12 |+〉 〈+|+

12 |−〉 〈−|, or indeed as 1

2 |u〉 〈u|+12 |v〉 〈v| for any orthonormal basis {|u〉 , |v〉}.

Born’s rule. For pure states if we measured {Pk} then the probability of outcome k would betr(Pk |ψ〉 〈ψ|). For mixed states ρ this becomes Pr[k] = tr(ρPk) by linearity.

4 Composite systems and Entanglement

We will work always with distinguishable particles. A pure state of two quantum systems is givenby a unit vector |η〉 in the tensor product Hilbert space Cn ⊗ Cm ∼= Cnm.

For example, if particle A is in the pure state |ψ〉A and particle B is in the pure state |ϕ〉Bthen their joint state is |η〉AB = |ψ〉A ⊗ |ϕ〉B. If |ψ〉A ∈ Cn and |ϕ〉B ∈ Cm, then we will have|η〉AB ∈ Cmn.

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This should have the property that if we measure one system, say A, then we should obtainthe same result in this new formalism that we would have had if we treated the states separately.If we perform the projective measurement {Pk} on system A then this is equivalent to performingthe measurement {Pk ⊗ I} on the joint system. We can then calculate

Pr[k] = 〈η|Pk ⊗ I |η〉=A 〈ψ|B |ϕ〉 (Pk ⊗ I) |ψ〉A |ϕ〉B= 〈ψ|Pk |ψ〉 〈ϕ|ϕ〉= 〈ψ|Pk |ψ〉

Just as there are joint distributions for which two random variables are not independent (e.g.,...), there are quantum states which are cannot be written as a tensor product |ψ〉 ⊗ |ϕ〉 for anychoice of |ψ〉 , |ϕ〉. In quantum mechanics, this situation can even occur for pure states of thesystem. For example, consider the EPR pair (we will see the reason for the name later):

∣∣Φ+⟩

=|0〉 ⊗ |0〉+ |1〉 ⊗ |1〉√

2.

We say that a pure state |η〉 is entangled if, for any |ψ〉 , |ϕ〉, we have |η〉 6= |ψ〉 ⊗ |ϕ〉.Entangled states have many counterintuitive properties. For example, suppose we measure the

state |Φ+〉 using the projectors {Pj,k = |j〉 〈j| ⊗ |k〉 〈k|}. Then we can calculate

Pr[(0, 0)] =⟨Φ+∣∣P0,0

∣∣Φ+⟩

=⟨Φ+∣∣ |0〉 〈0| ⊗ |0〉 〈0| ∣∣Φ+

⟩=

1

2

Pr[(1, 1)] =1

2Pr[(0, 1)] = 0

Pr[(1, 0)] = 0

The outcomes are perfectly correlated.However, observe that if we measure in a different basis, we will also get perfect correlation.

Consider the measurement with outcomes

{|++〉 〈++| , |+−〉 〈+−| , |−+〉 〈−+| , |−−〉 〈−−| ,

where we have used the shorthand |++〉 := |+〉 ⊗ |+〉, and similarly for the other three. Then onecan calculate (and doing so is a good exercise) that, given the state |Φ+〉, we have

Pr[(+,+)] = Pr[(−,−)] =1

2,

meaning again there is perfect correlation.

4.1 Partial trace

Suppose we have ρAB ∈ D(Cn⊗Cm). We would like a quantum analogue of the notion of a marginaldistribution in probability theory.

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Define the reduced state of A to be

ρA := trB ρAB

trB(ρAB) =∑k

(I ⊗ 〈k|)ρAB(I ⊗ ketk),

where {|k〉} is any orthonormal basis on B. The operation trB is called a partial trace.We observe that if we perform a measurement {Pj} on A, then we have

Pr[j] = tr(Pj ⊗ I)ρAB = tr(Pk trB(ρAB)) = tr(PkρA).

Thus the reduced state ρA perfectly reproduces the statistics of any measurement on the system A.

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