Density Matrices

Introduction

We can represent a system as an ensemble of pure states {pi,ψi}\{p_i, \psi_i\}{pi​,ψi​}﻿. Now, if you have exact knowledge of the system then it is for sure in a pure state, i.e., ρ=∣ψ⟩⟨ψ∣\rho = |\psi\rangle\langle\psi|ρ=∣ψ⟩⟨ψ∣﻿. However, if we have classical uncertainty amongst the possible states. The system can be represented as a mixed state ρ=∑ipiψi\rho = \sum_{i} p_i\psi_iρ=∑i​pi​ψi​﻿ where the probabilities pip_ipi​﻿ are classical in nature.

This formulation helps us a lot in dealing with quantum information, noisy systems and helps us represent measurements better, as well. Why? Because it provides a convenient means for describing quantum systems whose state is not completely known.

Postulates in Density Matrices formulation

1.

Isolated physical system is given by its density matrix operating on a certain Hilbert space.

2.

Evolution of a closed quantum system is given by a unitary transformation as ρundefinedUUρU†\rho \xrightarrow{U} U\rho U^\daggerρU​UρU†﻿.

3.

The state space of a composite physical system is the tensor product of the state spaces of the component systems.

\rho = \rho_1\otimes...\otimes\rho_n

4.

Quantum measurements are described by a collection {Mm}\{M_m\}{Mm​}﻿ of measurement operators acting on the state space of the system. 

Probability that upon measurement the outcome is m=p(m)=tr(Mm†Mmρ)m = p(m) = tr(M_m^\dagger M_m\rho)m=p(m)=tr(Mm†​Mm​ρ)﻿ and the state of the system becomes as follows.

\rho \xrightarrow{\text{on measuring}} \frac{M_m\rho M_m^\dagger}{p(m)}

Measurement operators also follow the completeness equation, ∑mMm†Mm=I\sum_m M_m^\dagger M_m = I∑m​Mm†​Mm​=I﻿.

Properties

Theorem 1: An operator ρ\rhoρ﻿ is a density operator if and only if it is both positive (ρ=ρ†)(\rho = \rho^\dagger)(ρ=ρ†)﻿ and tr(ρ)=1tr(\rho) = 1tr(ρ)=1﻿.

Converse is easy to prove. If an operator is both positive and has trace as one, then it shall have a spectral decomposition of the form ∑iλi∣i⟩⟨i∣\sum_i\lambda_i|i\rangle\langle i|∑i​λi​∣i⟩⟨i∣﻿.

For the direct proof, let us consider ρ=∑ipitr(∣ψi⟩⟨ψi∣)=1\rho = \sum_i p_i tr(|\psi_i\rangle\langle\psi_i|) = 1ρ=∑i​pi​tr(∣ψi​⟩⟨ψi​∣)=1﻿.

Theorem 2: The sets ∣ψ~⟩|\tilde\psi\rangle∣ψ~​⟩﻿ and ∣ϕ~⟩|\tilde\phi\rangle∣ϕ~​⟩﻿ generate the same density matrix if and only if,

|\tilde\psi\rangle = \sum_j u_{ij} |\tilde\phi_j\rangle

where uiju_{ij}uij​﻿ is a unitary matrix of complex numbers, with indices iii﻿ and jjj﻿, and we 'pad' whichever set of vectors ∣ψ~i⟩|\tilde\psi_i\rangle∣ψ~​i​⟩﻿ or ∣ϕ~j⟩|\tilde\phi_j\rangle∣ϕ~​j​⟩﻿ is smaller with additional vectors 000﻿ so that the two sets have the same number of elements.

Theorem 3: If ρ\rhoρ﻿ is a density operator, then ρ\rhoρ﻿ is a pure state if and only if tr(ρ2)=1tr(\rho^2) = 1tr(ρ2)=1﻿ and mixed state if and only if tr(ρ2)<1tr(\rho^2) < 1tr(ρ2)<1﻿.

Theorem 4: Observable MMM﻿ has expectation ∑x⟨ψx∣M∣ψx⟩=tr(Mρ)\sum_x \langle \psi_x|M|\psi_x\rangle = tr(M\rho)∑x​⟨ψx​∣M∣ψx​⟩=tr(Mρ)﻿.

Reduced Density Operator

This is the single-most important application of density operator formulation is the existence of reduced density operator. It is defined as follows.

\rho_A = tr_B(\rho_{AB})

This allows us to talk about sub-systems of a composite system.

tr_B(|a_1\rangle\langle b_1| \otimes |b_1\rangle\langle b_2|) = |a_1\rangle\langle a_2| tr(|b_1\rangle\langle b_2|)

Schmidt Decomposition

Suppose ∣ψ⟩|\psi\rangle∣ψ⟩﻿ is a pure state of a composite system, ABABAB﻿. Then, there exists orthonormal states ∣iA⟩|i_A\rangle∣iA​⟩﻿ for system AAA﻿, and orthonormal states ∣iB⟩|i_B\rangle∣iB​⟩﻿ of system BBB﻿ such that

|\psi\rangle = \sum_i \lambda_i |i_A\rangle |i_B\rangle

where λi\lambda_iλi​﻿ are non-negative real numbers satisfying ∑iλi2=1\sum_i \lambda_i^2 = 1∑i​λi2​=1﻿ known as Schmidt co-efficients.

Purification

Suppose we have a mixed state ρA\rho_AρA​﻿ for a system AAA﻿. Then, we can introduce another system RRR﻿ such that ARARAR﻿ forms a pure state ∣AR⟩|AR\rangle∣AR⟩﻿ and ρA=trR(∣AR⟩⟨AR∣)\rho_A = tr_R(|AR\rangle\langle AR|)ρA​=trR​(∣AR⟩⟨AR∣)﻿.

Given  ρA=∑ipi∣iA⟩⟨iA∣\rho_A = \sum_i p_i |i_A\rangle\langle i_A|ρA​=∑i​pi​∣iA​⟩⟨iA​∣﻿, we shall have the following where ∣iR⟩ |i_R\rangle∣iR​⟩﻿ are orthonormal basis states.

|AR\rangle = \sum_i \sqrt{p_i}|i_A\rangle|i_R\rangle