PHY305: Notes on Entanglement and the Density Matrix

Here follows a short summary of the deﬁnitions of qubits, EPR states, entanglement, the density matrix, pure states, mixed states, measurement, and decoherence.

Deﬁnition of a qubit A qubit is the simplest quantum mechanical system that one can consider: it only has two states. The most familiar example is to think of a qubit as the spin state of a spin 1/2 particle. Consider a qubit, called A. A general state of state of A be written as a linear sum of a spin up and a spin down state

|ψiA = a0| ↑ iA + a1| ↓ iA (1)

Here a0 and a1 are two arbitrary complex numbers, that satisfy

2 2 |a0| + |a1| = 1 . (2)

The real dimension of the Hilbert space of states of a qubit is therefore

dim Hqubit = 3 (3)

Instead of using spin up or spin down, we can also write the two states as

|ψiA = a0 | 0 i + a1 | 1 i, (4) and think of | 0 i and | 1 i as the two possible states of a classical bit – the bit that is used in binary computations. The diﬀerence between a qubit and a classical bit is that the state of a qubit is a linear superposition of states of a classical bit. We can measure the state of a qubit, by measuring the eigenvalue of some hermitian operator O, that acts on the Hilbert space of the qubit. These observables O are 2 × 2 hermitian matrices. Any such observable can be expanded as 3 X O = v01 + vjσj (5) j=1 where σj, j = 1, .., 3 denote the three Pauli matrices. Special cases of observables are the spin operators σj themselves, which measure (up to a prefactor ~/2) the component of the spin in the direction j. Measuring σ3 yields either spin up or spin down. Doing this particular measurement is called reading the qubit. The resulting outcome is a classical bit. 2 2 The numbers |a0| and |a1| are the probabilities that a measurement yields as outcome the corresponding classical bit.

1 Two qubit states Now let us look at a system consisting of two qubits, called A and B. Now things get more interesting. A possible state of the two qubits is

| Ψproduct i = |ψiA |φiB (6)

with |ψiA as in eqn (1) and

|ψiB = b0| ↑ iB + b1| ↓ iB . (7)

Now while | ψ iA and | φ iB are general states for each individual qubit, the state (5) is not the most general state of the two qubit system. State (5) is called a product state, because it factorizes as the product of a state of qubit A times a state of qubit B. A more interesting two qubit state is the EPR state 1 |ΨEPRi = √ (| ↑ i | ↓ i − | ↓ i | ↑ i ) (8) 2 A B A B This state has the special property that, if we would read out the state of qubit B (measure its spin along the z-axis), then thet state of qubit A depends on the outcome of the measurement on qubit B. We say that the EPR state is an entangled state. We will give a more general deﬁnition of an entanglement in a moment.

N qubit states Let us take a brief look at the most general state of N qubits. As a quick warm-up, the state of N classical bits is characterized by N numbers

x = {s1, s2, s3, . . . , sN } (9)

where si = 0, 1 is the i-th classical bit. The number of possible states x of N classical bits is 2N . The most general quantum mechanical state of N qubits is a linear combination of all possible classical states X |Ψi = cx | x i (10) N qubits 0≤ x <2N where x is short-hand notation of the state of N classical bits. So the general N qubit states N P 2 is speciﬁed by 2 complex amplitudes cX , subject to one single constraint x |cx| = 1 . Thus the dimension of the N qubit Hilbert space is

N dim HN qbits = 2 · 2 − 1 (11)

For say N = 100, this is already very large number! This large number arises because the general state of N qubits is not the product of N single qubit states, but is highly entangled. And for N = 100, this entanglement contains a lot of information! (More on this later.)

2 Entanglement with environment The notion of entanglement is important in any realistic description of a quantum system A that is in contact with some other quantum system B. This other system B is in fact always present, since any ﬁnite quantum system A is always in contact with some environment, which for example includes the measuring aparatus by which we plan to measure the quantum system A. So let us just call system B the environment of system A. Up to now we have considered situations where the state of the quantum system and its environment factorizes into a product state

| Ψproduct i = |ψiA |φiB . (12)

In this product situation, we can simply ignore the environment, and just describe the quantum system A by the state |ψiA . Indeed, if we consider an operator OA that just acts on the quantum system A (and not on the quantum system b), we have

hΨproduct|OA |Ψproducti = A hψ| OA | ψiA (13)

We say that the quantum system A is in a pure state. Now let | i i label some basis of the Hilbert space of system A. Similarly, let | j i denote the basis of quantum states of the environment. Then we can decompose X X | ψ iA = ai | i iA , | φ iB = bj | j iB , (14) i j X | Ψproduct i = aibj | i iA | j iB (15) i,j for some set of amplitudes ai and bj. Product states of this form are special, however, and they are hard to realize experimentally. A general quantum state of systems A and B looks as follows X | Ψ i = cij | i iA | j iB (16) i,j where cij are arbitrary complex amplitudes. We have

cij = aibj for a product state (17) but in general, cij can not be factorized in this way. If cij can not be factorized in the above form, we say that the quantum system A and its environment B are entangled. In this case, the state of the quantum system A is called a mixed state. We would like to ﬁnd a way to describe such a mixed state, without having to include the environment in our description.

3 Deﬁnition of a density matrix Let | i i label some basis of the Hilbert space of system A. The density matrix is a matrix with entries ρij. One often write the density matrix as an operator that acts on the Hilbert space, as follows X ρ = | i iρijh j |. (18) ij P (Note that this expression indeed acts on the Hilbert space via: ρ| φ i = i,j ρijh j |φi| i i.) The three properties that we will require of the density matrix ρij are

∗ X 1) ρij = ρji ; 2) ρii = 1 ; 3) ρii ≥ 0. (19) i The ﬁrst two conditions say that the density matrix is hermitian and has trace equal to 1:

1) ρ = ρ† ; 2) Tr(ρ) = 1 . (20)

The main use of the density matrix is to deﬁne the expectation values of operators O that act on the Hilbert space. This deﬁnition is as follows X h O i = ρij h j | O | i i = Tr(ρ O) (21) i,j

Diagonal form of the density matrix For a given density matrix ρ it is possible to ﬁnd a special basis | n i such that the density matrix ρ is diagonal: X ρ = | n iρnnh n | (22) n

0 ≤ ρnn ≡ pn ≤ 1. (23)

Here pn is the probability that the system is in the state | n i. A density matrix thus describes statistical mixture of quantum states. The requirement that Tr(ρ) = 1 amounts to the condition that the probabilities add up to 1: X pn = 1 . (24)

In this diagonal basis, the formula for the expectation value of the operator O takes the form of a statistical average X h O i = pn h n |O| n i (25) n

4 Density matrix of a pure state. A quantum system in a pure state is described by a wave function X |ψi = ai| i i (26) i The expectation value of an operator O then reads

X ∗ h O i = h ψ | O| ψ i = aiaj h j | O| i i (27) i,j From the deﬁnition (21) we read oﬀ that the density matrix associated to the state (26) is

X ∗ ρ = | i i aiaj h j | = |ψihψ| (28) i,j Comparing with eqn (22), we see that the density matrix of a pure state | ψ i, when diago- nalized, has one single eigenvalue pn = 1 (for | n i = | ψ i) while pm = 0 for all other m 6= n. This implies that the density matrix of a pure state has the special property that ρ2 = ρ. In general, we deduce the following criterion for a density matrix: Tr(ρ2) = 1 for a pure state

Tr(ρ2) < 1 for a mixed state (29)

Mixed state from an entangled state Let us return to the general entangled state (16) for the two quantum systems A and B. We want to compute the density matrix that describes the mixed state of system A.

Consider an operator OA that acts within the Hilbert space of A only. Its expectation value can be evaluated as h O i = h Ψ | O| Ψ i

X ∗ = cij ckl Bh l | j iBAh k | O | i iA i,j,k,l

X X ∗ = cij ckj Ah k | O | i iA (30) i,k j So we read oﬀ that the density matrix for the quantum system A is

A X ∗ ρik = cij ckj (31) j ∗ For the product state (17), this is of the form ρik = aiak, which describes a pure state.

However, for general cij, the density matrix (31) can not be written is this pure state form, and in this case the quantum system A is in a mixed state.

5 Measurement and Decoherence The notion of pure states and mixed states plays an important role in the description of a quantum mechanical measurement. Again, consider a quantum system A. To do a measurement on A, we must put it in contact with a measuring device B. Let us assume that, initially, system A is in a pure state (26). The measuring apparatus is in some other

state | φ iB that initially is uncorrelated with the state of A. So before the measurement starts, the combined system is in a product state of the form (12), or when written out in some eigenbasis, as in eqn (15). The process of doing the measurement then proceeds as follows. The measuring device has a basis of states that correlates with the basis of states of system A. This correlation is such that, when system A is in the eigenstate | i iA , then the measuring apparatus will go in a measurement state that says: “I have just measured that system A is in the state | i iA .”

Let us denote this state of the measuring apparatus by | i iB . (In reality, for every state

| i iA , there are in fact many such measuring states | i iB , but for simplicity let us assume that there is only one.) The measurement process is described by the Schr¨odinger equation of system A together with the measuring device B. The total Hamiltonian will have an interaction term that acts on the combined Hilbert space of A and B. For a well designed measuring apparatus, this Hamiltonian is such that after a while, the combined state looks as folllows X |Ψafter i = ai | i iA | i iB . (32) i This is a maximally entangled state: there is a one-to-one correlation between the state of the quantum system A and that of the measuring device B. For an ideal measurement, the amplitudes ai of |Ψafteri are identical to the amplitudes ai of the initial pure state (16) of system A. After the measurement, however, system A is no longer in a pure state, but in a mixed state with a diagonal density matrix

A X 2 ρafter = | i iA pi Ah i | , pi = |ai| . (33) i So we see that, in complete accord with the familiar description of quantum mechanical 2 measurement, the probability that system A ends up in state | i i is pi = |ai| . What has happened, however, is that after the interaction with the measurement device, the different states | i i and amplitudes ai are no longer added up coherently as in the pure state (26), but they are added as in a statistical ensemble with classical probabilities pi. The time-evolution from the initial product state (12) to the final entangled state (32), or equivalenly, from the initial pure state (26) to the final mixed state (33), is known as decoherence.