Elements of quantum probability

In this section we shall introduce some basic principles of quantum probability and show how it compares to its classical counterpart. For a detailed account of the subject see[61, Ch. 1]or[81, Ch. 10.2]for an easy introduction. We start with a motivating example. SupposeX is a random variable on a certain probability space(Ω,F,P)which

takesnfinite real values(x1, ...,xn)with probabilities (p1, ...,pn),

P

ipi =1. Then the

expectation ofX is given byE[X] =Pn_i=1xipi, which can be rewritten as

E_P[X] =tr       p1 0 ... 0 p_n             x1 0 ... 0 x_n       ,

where tr(A)is trace ofA, i.e. the sum of its diagonal entires. This gives us a motivation to think of a random variable as a diagonal n_×n matrix and of its distribution as a diagonaln×nmatrix with trace 1.

LetY be another random variable with denumerable state space defined on the same probability space (Ω,_F,_P). In classical probability theory X Y and Y X are the same random variables, that is X and Y commute, as do their associated matrices; in particular, they have the same distribution. Quantum probability, at the other hand, can deal withnon-commutingrandom variables. In fact, we will see that classical probability is a special (commuting) case of quantum probability.

We start with some preliminary notation. LetH be a complex Hilbert space with inner product_〈·,_·〉, s.t. for anyu,v∈H

〈u,v〉=u∗v=X

i

u∗_ivi and kuk=〈u,u〉1/2 .

Let_O(H)denote a collection of all self-adjoint operators fromHto itself. Recall that an operatorAonH is calledHermitianorself-adjointif

〈Au,v〉=〈u,Av〉, _∀u,v∈H .

In case of a finite-dimensional operatorAthis is equivalent toAbeing its own conjugate transpose:A=A∗=AT.

In quantum probability an equivalent of a random variable is a Hermitian oper- ator, which is called anobservable. IfH has dimensionn<∞, then each such operator A∈ O(H)has the following spectral decomposition

A=α1P1+...+αnPn , (3.23)

where (α₁, ...,α_n) are n (not necessarily distinct) real eigenvalues of Aand each P_i, fori ∈ {1, ...,n}, is an orthogonal projection on the eigenspace corresponding to the eigenvalueα_i, denoted byV_i, withP_iP_j=0 for alli₆= j.

By an orthogonal projection on a subspaceW ⊆ H we mean an operatorPW ∈

O(H) with P_W = P_W∗ = P_W2. If W is finite-dimensional with an orthonormal basis

{w1,w2, ...}, then the matrix of the operator is given by

PW =

X

i

wiw∗i . (3.24)

Thus, if the eigenspaceVi, for anyi∈ {1, ...,n}, is one-dimensional, one hasPi =vivi∗,

wherev_i is the normalised (i.e.kvk=1) eigenvector corresponding toα_i. We denote a set of all orthogonal projections onHby_P(H).

pointed out by Parthasarasy in[61], one can compare (3.23) with a decomposition X(ω) =x₁1_{E1}(ω) +...+x_n1_{E

n}(ω), ω∈Ω,

of a random variableX on probability space(Ω,_F,P)taking values(x1, ...,xn), where

1_{E

i},i∈ {1, ...,n}, is an indicator random variable of the setEi={ω∈Ω:X(ω) =xi}.

A Hermitian operator ρ in _P(H) is positive if 〈ρu,u_{〉 ≥} 0, _∀u _∈ H. We call a positive operator in _P(H) with unit trace adensity matrix. A triplet (H,_P(H),ρ) is called aquantum probability space. If H is finite-dimensional, then(H,_P(H),ρ) is called afinite- dimensional quantum probability space.

Letρ_{∈ P}(H)be a density matrix and letλandube one of its eigenvalues and the corresponding eigenvector respectively. Then〈ρu,u〉 = λkuk> 0, which implies thatρhas positive eigenvalues. Then we can writeρas a sumP_ip_iu_iu∗_i, wherep_i>0, i≥1, with P

ipi =1, are eigenvalues ofρ andui, i≥1, are the corresponding nor-

malised eigenvectors. Ifρis a one-dimensional projection, then we can writeψψ∗=ρ for someψ_∈H withkψk=1. Vectorψis calleda pure state. We writeρ_ψ.

We are now ready to define probabilities on possible measurements of an ob- servable A∈ O(H). If the system is measured with respect to a density matrix ρ =

P

ipiuiu∗i, then

P(Ais measured asαi with respect toρ) =Pρ(Ais measured asαi):=tr(ρPi),

wherePi is the orthogonal projection on the eigenspace associated toαi. In particular,

for a pure stateψ, such thatψψ∗=ρ, we havePρ(Ais measured asαi) =tr(ψψ∗Pi) =

kP_iψk2. We can now calculate expectation of an observableAin stateρ.

E_ρ[A] = n X i=1 αiPρ(Ais measured asαi) = n X i=1 αit r(ρPi) =t r( n X i=1 αiρPi) =t r(ρ n X i=1 αiPi) =t r(ρA),

Quantum conditioning. Just like in the classic probability theory, there is a notion of conditional probability in its quantum counterpart.

Definition 3.9. (Quantum conditioning) Let P be an orthogonal projection, i.e. a quan- tum event in stateρ, and such thattr(PρP)6=0. A quantum probability conditioned on P is given by the state

PρP tr(PρP) .

LetAandBbe two observables defined on the same quantum probability space with the stateρ, and let P_αand P_β be some events associated toAandB respectively. Then, using the above definition,

Pρ(Bis measured asβ|Ais measured asα) =

tr(P_αρP_αP_β) tr(P_αρP_α) .

One notices that this expression is reminiscent of the classical formula for condi- tional probability_P(B=β|A=α) =P(A=α,B=β)/P(A=α). Indeed, sinceP_α2=Pα, tr(P_αρP_α) =tr(ρP_α2) =tr(ρP_α) =Pρ(Ais measured asα). Thus, rearranging the above expression gives

tr(P_αρP_αP_β) =Pρ(B=β|A=α)Pρ(A=α)

and, similarly,

tr(P_βρP_βP_α) =Pρ(A=α|B=β)Pρ(B=β)

So, unlike in classical probability, the right-hand sides of the above two equa- tions are not in general equal. However, if AB = BA, and so P_αP_β = P_βP_α, by using P_α2=P_αandP_β2=P_β, it follows that

tr(P_αρP_αP_β) =tr(ρP_αP_βP_α) =tr(ρP_βP_α)

=tr(ρP_αP_β) =tr(ρP_βP_αP_β) =tr(P_βρP_βP_α),

i.e.

that is

Pρ(Ais measured asα,then B is measured asβ)

=P_ρ(Bis measured asβ,then Ais measured asα).

In other words, if observablesAandB commute, then it doesn’t matter in what order we measure them, something that is not in general true for non-commuting quan- tum random variables. It is precisely when observables commute, that they have a joint distribution in the sense of the classical probability.