This note is intended to make a brief introduction to density matrix\textbf{density matrix} and quantum stochastic process\textbf{quantum stochastic process}, also as a lecture note for seminar of non-equilibrium statistic.

Density matrix

we will review the properties of the density matrix and utilize them to rewrite the axioms of quantum mechanics.
The density matrix is widely applied in quantum information and quantum computation, condensed matter and other fields.

Definition and Properties

In some cases that a quantum system is in the state ∣ϕi⟩|\phi_i\rangle with pip_i, either in pure state which probability is corresponded to ∣ci∣2\lvert c_i\rvert^2or in mixed state which probability only has the statistical meaning, thus we could represent the system with the density operator:

ρ^≡∑ipi∣ϕi⟩⟨ϕi∣\hat{\rho}\equiv\sum_i p_i |\phi_i\rangle\langle\phi_i|

where ∣ϕj⟩|\phi_j\rangle satisfies the condition of normalization and not needs to satisfies the condition of orthogonality. Later we will not not distinguish density operator and density matrix.
The mean value of operator/observable is given by

⟨⟨A⟩⟩=∑i,jpiaj∣⟨ϕj∣aj⟩∣2=∑ipi⟨ϕi∣A∣ϕi⟩\langle\langle A\rangle\rangle=\sum_{i,j} p_i a_j |\langle \phi_j|a_j\rangle |^2=\sum_i p_i\langle\phi_i|A|\phi_i\rangle

the second step we use the von Neumann theorem that a self-adjoint operator can written as sum of projective operator, i.e., A=∑iai∣ai⟩⟨ai∣A=\sum_i a_i | a_i\rangle \langle a_i|, where ∣ai⟩{|a_i\rangle} is a set of orthonormal basis.
And the last formula could be written as

⟨⟨A⟩⟩=Tr(ρA) \langle\langle A \rangle\rangle=\mathrm{Tr}(\rho A)

Its properties are listed as below:

  1. positive semi-definite, i.e., $\langle \phi|\rho|\phi\rangle \geq 0 $ for any $|\phi\rangle $ in Hilbert space, self-adjoint or Hermitary ρ†=ρ\rho^{\dagger}=\rho and tr ρ=1\rho=1
  2. A system is in pure state if and only if ρ2=ρ\rho^2=\rho or trρ2=1\rho^2=1, meanwhile in mixed state, trρ2<1\rho^2 <1

QM in density matrix

  1. A quantum system could be represented by density matrix in Hilbert space, and if quantum system is in ρ\rho state with probability pip_i, thus

    ρ=∑ipiρi \rho=\sum_i p_i\rho_i

  2. A closed system is evolved with a unitary operator U(t)U(t), $|\phi(0)\rangle \rightarrow U(t)|\phi(0)\rangle $, meanwhile

    ρ(0)→∑ipiU(t)^∣ϕi(0)⟩⟨ϕi(0)∣U†(t)^=ρ′(t)=Uρ(0)U† \rho(0) \rightarrow \sum_i p_i \hat{U(t)}|\phi_i(0)\rangle \langle\phi_i(0)|\hat{U^{\dagger}(t)}=\rho^{\prime}(t)=U \rho(0) U^{\dagger}

    specifically, U=Texp(−iℏ∫0tHdt)U=Texp(-\frac{i}{\hbar}\int_0^t H\mathrm{d}t), which could be saw in property 3.

  3. In the Schr\"odinger picture, the state of system is governed by the Liouville—von Neumann equation\textbf{Liouville—von Neumann equation} iℏddtρ=[H,ρ]i\hbar \frac{d}{dt}\rho=[H,\rho], where HH is the system Hamiltonian.

  4. Measurement operator, there are such ways of measurement, projection measurement and POVM measurement, the POVM can be understood as the mixed state corresponding to pure state when compared to PVM. A POVM is a set of positive semi-definite matrices Fm{F_m} on a Hilbert space H{\mathcal {H}} that sum to the identity matrix.

∑mFm=I \sum_m F_m =\mathrm{I}

And the POVM can be decomposed to a product:

Fm=Mm†MmF_m=M_m^{\dagger} M_m

If the system is in the state of ρ\rho, thus the probability of mm is

p(m)=tr(ρFi)=tr(Mm†Mmρ)p(m)=\mathrm{tr}(\rho F_i)=\mathrm{tr}(M_m^{\dagger} M_m \rho)

and the state of system after measurement is

MmρMm†tr(Mm†Mmρ)\frac{M_m \rho M_m^{\dagger}}{\mathrm{tr}(M_m^{\dagger} M_m \rho)}

especially the POVM itself is a PVM, thus the MmM_m can be the projective operator.

  1. The joint state of the system is ρ1⨂ρ2⨂⋯⨂ρn\rho_1 \bigotimes \rho_2 \bigotimes \cdots \bigotimes \rho_n

Quantum stochastic process

In this section the classical stochastic process is generalized to quantum stochastic process, from which we find the classical-quantum correspondence clearly.Although quantum mechanics is an intrinsically probabilistic theory, the application of probabilistic concepts to quantum mechanics is quite different from that of the classical theory.

From classical to quantum

In the classical stochastic process, the mean value of an observable/random variable is

E(A(t))=⟨A(t)⟩=∑ipiai\mathrm{E(A(t))}=\langle A(t) \rangle=\sum_i p_i a_i

which the aia_i is the value of observable that we measure from the system observed and pip_i is the probability of variable to take the value, or statistically the proportion of ensembles we observe either classical or quantum.

And note equ(10) could be written as

⟨A(t)⟩=tr((p1p2⋱pn)(a1a2⋱an))=tr(ρA) \langle A(t) \rangle=\mathrm{tr}\left( \begin{pmatrix} p_1 \\ & p_2 \\ & & \ddots \\ & & & p_n \end{pmatrix} \begin{pmatrix} a_1 \\ & a_2 \\ & & \ddots \\ & & & a_n \end{pmatrix} \right)=\mathrm{tr(\rho A)}

where ρ\rho could be seen as classical density matrix, by which we generalize the classical stochastic to its quantum.

Quantum version

we could easily see the classical-quantum correspondence, the classical density matrix is diagonal, and in contrast the quantum version usually non-diagonal, i.e., non-commutative.

Actually, we could recall what we have learned in quantum physics. what does it mean when the Hamiltonian contains off-diagonal terms. In a classical system, the Hamiltonian would only contain diagonal terms corresponding to the energy of each individual particle. Even with two-body and more body interaction terms, it is only contribute to the diagonal terms. However, in a quantum system, the Hamiltonian must also take into account transitions between different quantum states, which can lead to non-diagonal terms. These non-diagonal terms represent quantum fluctuations that cannot be explained classically, and they play a crucial role in determining the properties of the system. In a way, the presence of non-diagonal terms in the Hamiltonian is a hallmark of quantum behavior, since it indicates the possibility of entanglement and superposition between particles.

We also know that the relationship and similarity between SM and QM, and the wick rotation could make them connected.

ρ=e−βH\rho=e^{-\beta H}

This also means that the non-diagonal Hamiltonian could result in non-diagonal density matrix.

ρclassical=(p1p2⋱pn)⟺ρquantum=∑ipi∣ϕi⟩⟨ϕi∣ \rho_{classical}=\begin{pmatrix} p_1 \\ & p_2 \\ & & \ddots \\ & & & p_n \end{pmatrix} \Longleftrightarrow \rho_{quantum}= \sum_i p_i |\phi_i\rangle \langle\phi_i|

As long as the basis is not orthogonal, thus the density matrix is not diagonal. It is worth mentioning that the density always could be diagonalized because the operator is unitary.

Contrasted to discrete condition, the continuous one that we need to consider the probability distribution function on Hilbert space, also to define the probability measure on Hilbert space.

As we’ve learned that fundamental concepts of traditional probability theory are three basic ingredients, Ω,σ,P\Omega, \sigma, P, which means $$\textbf{the sample space, the σ\sigma-algebra of events and the probability measure on the σ\sigma-algebra.} $$

So we need to find the so-called event satisfying σ\sigma-algebra as classical ones, to form the probability space on Hilbert space. And the so-called measure satisfies Kolmogorov axioms.

We could surprisingly find that the deep connection between probability theory and quantum mechanics. Specifically, A pure state with such components of observable corresponding to their probability

∣ψ⟩=∑iai∣ai⟩|\psi \rangle=\sum_i a_i |a_i\rangle

expanded as a set of complete basis, here we take the observable A’s eigenstate, where the coefficients {∣ai∣2}\{|a_i|^2\} just are reckoned as the probability distribution of random variable A, its Sample Space is the Hilbert Space of this system, comprising a set of complete basis {∣ai⟩}\{|a_i \rangle\}, in quantum computation we usually denotes as {∣0⟩,∣1⟩}\{|0 \rangle, |1 \rangle \}, the Measuring is just a Trial.

The probability density functional P[ψ]P[\psi], where ψ\psi is any normalized wave functions and not be differ by a phase factor, satisfying

∫HilbertspaceDψDψ∗P[ψ]=1 \int_{Hilbert space} D\psi D\psi^*P[\psi]=1

is exactly what we need.

And an amazing view to treat quantum entanglement is the joint probability distribution(JPD). And it can easily introduce the Bell inequality. Supposing we have two subsystems A, B, which both has two eigenstate {∣0⟩,∣1⟩}\{|0 \rangle, |1 \rangle \} spanning the subspace of total Hilbert space. The Bell state is an entangled state: ∣ψ⟩=12(∣00⟩+∣11⟩)|\psi \rangle=\frac{1}{\sqrt{2}}(|00 \rangle+|11 \rangle), thus its joint probability distribution is:

B\A PA=0P_{A=0} PA=1P_{A=1}
PB=0P_{B=0} $\frac{1}{2} $ 0
PB=1P_{B=1} 0 $\frac{1}{2} $

we could see the two dimensional probability distribution could not be written as two marginal probability distribution part’s product, i.e. the random variable A,B aren’t independent:

P((A,B))≠P(A)P(B)P((A,B)) \neq P(A)P(B)

The Bell inequality is just use this kind of way about JPD to test whether our theory about Quantum mechanics is true or not.

Let’s see the local hidden variable theory, whose basic hypothesis is that the JDP of two subsystem A,B is independent, guiding by a hidden law of hidden variable. We write the probability distribution of observables A,B measured, let’s say, a,ba, b :

p(a)=∑ξp(a∣ξ)p(ξ)p(b)=∑ξp(b∣ξ)p(ξ)p(a)=\sum_{\xi}p(a|\xi)p(\xi) \\ p(b)=\sum_{\xi}p(b|\xi)p(\xi)

The local Hidden variable hypothesis assumes:

p(a,b)=∑ξp(a∣ξ)p(b∣ξ)p(ξ)=∑ξp(a,b∣ξ)p(ξ)p(a,b)=\sum_{\xi}p(a|\xi)p(b|\xi)p(\xi)=\sum_{\xi}p(a,b|\xi)p(\xi)

Its physical meaning is obvious: Observables of two space-like seperated experimenters should not influence each other.

CHSH version Bell inequality is:

∣⟨A1⨂B1⟩+⟨A1⨂B2⟩+⟨A2⨂B1⟩−⟨A2⨂B2⟩∣≤2\left|\langle A_1 \bigotimes B_1 \rangle+\langle A_1 \bigotimes B_2\rangle +\langle A_2 \bigotimes B_1 \rangle - \langle A_2 \bigotimes B_2 \rangle \right| \leq 2

stochastic process

Now it is worth considering what if the probability density functional is time-dependent P[ψ,t]P[\psi,t], it is the stochastic process part of this note.

It leads to a stochastic process ψ(t)\psi(t) in the system’s Hilbert space which reproduces the density matrix through its covariance matrix, that is through the
expectation value ρS(t)=E(∣ψ(t)⟩⟨ψ(t)∣)\rho_S(t)=E(|\psi(t)\rangle\langle\psi(t)|), what is the transfer matrix of coefficients {∣ai∣2}\{|a_i|^2\}? we will state in next section.