The Exercise

A system $A$ is prepared in one of $n$ pure states $|\psi_1\rangle, \dots, |\psi_n\rangle$, the state $|\psi_i\rangle$ with probability $p_i$, and $\rho$ is the density operator of this mixture. A second system $R$, the reference, has dimension $n$ and an orthonormal basis $|1\rangle, \dots, |n\rangle$. Pair each $|\psi_i\rangle$ with $|i\rangle$, weight the pair by $\sqrt{p_i}$ and add: the sum $|AR\rangle$ is a pure state of $A$ and $R$ together.

The first part of the exercise is that tracing $R$ out of $|AR\rangle$ gives back $\rho$, so that $|AR\rangle$ is a purification of $\rho$.^✦ The second is that measuring $R$ in the basis $|i\rangle$ gives outcome $i$ with probability $p_i$ and leaves $A$ in the state $|\psi_i\rangle$. The third is that any pure state of $A$ and $R$ whose reduced state on $A$ is $\rho$ does the same, once the basis of $R$ is chosen to suit it.^✦

A Quantum System

A quantum system $S$ is a physical system whose state space $\mathcal{H}_S$ is a finite dimensional complex Hilbert space.^✦ A state $\sigma$ of $S$ is a positive operator of unit trace on $\mathcal{H}_S$.^✦ A pure state $|\psi\rangle$ of $S$ is a unit vector of $\mathcal{H}_S$, and it determines the state $|\psi\rangle\langle\psi|$ of $S$, the projector onto that vector.

The composite system $S \otimes T$ of two systems $S$ and $T$ is the system whose state space is $\mathcal{H}_S \otimes \mathcal{H}_T$.^✦ The reduced state of a state $\sigma$ of $S \otimes T$ is the state $\mathrm{Tr}_T\,\sigma$ of $S$ given by the partial trace over $T$, taken in an orthonormal basis $|k\rangle$ of $\mathcal{H}_T$: $$\mathrm{Tr}_T\,\sigma = \sum_k \big(I_S \otimes \langle k|\big)\,\sigma\,\big(I_S \otimes |k\rangle\big).$$(1)^✦

An Ensemble and Its Purification

An ensemble $\{p_i, |\psi_i\rangle\}$ of pure states of a system $A$ consists of $n$ pure states $|\psi_1\rangle, \dots, |\psi_n\rangle$ of $A$ and weights $p_1, \dots, p_n$ that form a probability distribution: every $p_i$ is nonnegative and $\sum_i p_i = 1$.^✦ The density operator of the ensemble is the state of $A$ $$\rho = \sum_{i=1}^n p_i\, |\psi_i\rangle\langle\psi_i|.$$(2)^✦

The reference system $R$ of the ensemble is the quantum system whose state space is $\mathbb{C}^n$ itself, with its standard basis $|1\rangle, \dots, |n\rangle$.^✦ The purification vector of the ensemble is the vector of $\mathcal{H}_A \otimes \mathbb{C}^n$ $$\sum_{i=1}^n \sqrt{p_i}\; |\psi_i\rangle \otimes |i\rangle.$$(3)^✦ The purification $|AR\rangle$ of the ensemble is the pure state of $A \otimes R$ whose vector is (3), a vector of unit norm: $$\langle AR|AR\rangle = \sum_{i,j} \sqrt{p_i p_j}\,\langle\psi_i|\psi_j\rangle\langle i|j\rangle = \sum_i p_i = 1.$$(4)^✦

Measurements

A measurement on a system $S$ with a finite set of outcomes is a family of measurement operators $M_i$ on $\mathcal{H}_S$, one for each outcome $i$, satisfying the completeness relation $\sum_i M_i^\dagger M_i = I$.^✦ The positive operator valued measure (POVM) of the measurement has the effects $E_i = M_i^\dagger M_i$, positive operators that sum to $I$.^✦ Performed on a state $\sigma$ of $S$, the measurement gives outcome $i$ with the probability $p(i)$ that its POVM assigns, and when $p(i) \neq 0$ the state after outcome $i$ is $\sigma_i$: $$p(i) = \mathrm{Tr}(E_i\,\sigma), \qquad \sigma_i = \frac{M_i\,\sigma\,M_i^\dagger}{p(i)}.$$(5)

The projective measurement in an orthonormal basis $\{|b_i\rangle\}$ of $\mathcal{H}_S$ is the measurement with operators $M_i = |b_i\rangle\langle b_i|$.^✦ For a measurement on $T$ with operators $M_i$ and a further system $S$, the measurement on the second factor of $S \otimes T$ is the measurement with the same outcomes and operators $I_S \otimes M_i$.^✦ The reference measurement of an ensemble is the projective measurement of $R$ in its standard basis, made on the second factor of $A \otimes R$: $$M_i = I_A \otimes |i\rangle\langle i|, \qquad i = 1, \dots, n.$$(6)^✦

The Claim

What Is Checked

The theorems are stated over the density operator (2) of an ensemble, its purification (4) and its reference measurement (6), the reduced state (1), and the probability of an outcome and the state after it (5). The third theorem ranges besides over every pure state of $A \otimes R$ that purifies $\rho$ and over the projective measurements of $R$ in the orthonormal bases of $\mathbb{C}^n$. The density operator (2) of an ensemble is a state of $A$, and the purification vector (3) has unit norm, so that $|AR\rangle$ is a pure state of $A \otimes R$. The reduced state (1) of a state of $A \otimes R$ is a state of $A$, and the state after an outcome of nonzero probability in (5) is again a state.