Lean, informalized

Nielsen & Chuang · Chapter 2, Introduction to quantum mechanics · p. 111

Exercise 2.81 - Pure states of a composite system with equal reduced states on the first system are related by a unitary on the second

Two purifications differ by a unitary on the reference.

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The Exercise

Let ρ\rho be a state of a quantum system SS. A purification of ρ\rho is a pure state |ψ⟩|\psi\rangle of a composite system S⊗RS \otimes R whose reduced state on SS is ρ\rho; the second system RR is the reference system. An observer who holds SS alone sees only ρ\rho, and cannot tell two purifications of ρ\rho on the same reference system apart. The exercise is to show that any two such purifications differ by a unitary acting on RR alone, with SS left untouched.✦

A Quantum System

A quantum system SS is described by its state space, a finite dimensional complex Hilbert space.✦ The composite system S⊗RS \otimes R of two systems SS and RR is the quantum system whose state space is the tensor product of the state spaces of SS and RR.✦

A pure state |ψ⟩|\psi\rangle of SS is a unit vector in the state space of SS, ⟨ψ|ψ⟩=1\langle \psi | \psi \rangle = 1.✦ A state ρ\rho of SS is a positive operator of unit trace on the state space of SS.✦ A pure state |ψ⟩|\psi\rangle gives the state |ψ⟩⟨ψ||\psi\rangle\langle\psi| of SS, the projector onto |ψ⟩|\psi\rangle.✦

The reduced state of a state σ\sigma of S⊗RS \otimes R is the state Tr⁡R(σ)\operatorname{Tr}_R(\sigma) of SS obtained by tracing out RR in an orthonormal basis {|j⟩}\{|j\rangle\} of the state space of RR, Tr⁡R(σ)=∑j(I⊗⟨j|)σ(I⊗|j⟩),\operatorname{Tr}_R(\sigma) = \sum_j \, (I \otimes \langle j|)\, \sigma \, (I \otimes |j\rangle),(1)✦ where II is the identity.

Unitary Evolution

A unitary UU of a quantum system RR is an evolution of RR as a closed system: an operator UU on the state space of RR with U†U=UU†=I.U^\dagger U = U U^\dagger = I.(2)✦ Applied to a pure state |χ⟩|\chi\rangle of RR, a unitary UU of RR gives the pure state U|χ⟩U|\chi\rangle.✦ The identity evolution of SS is the unitary of SS whose operator is the identity II.✦

For a unitary VV of SS and a unitary UU of RR, the product unitary V⊗UV \otimes U is the unitary of S⊗RS \otimes R whose operator is the tensor product of the two operators, acting on product vectors as (V⊗U)(|a⟩⊗|b⟩)=V|a⟩⊗U|b⟩(V \otimes U)(|a\rangle \otimes |b\rangle) = V|a\rangle \otimes U|b\rangle. With the identity evolution of SS in the first place, I⊗UI \otimes U is the evolution of S⊗RS \otimes R in which UU acts on RR and nothing acts on SS.

The Claim

Two pure states of S⊗RS \otimes R purify one and the same state of SS exactly when their reduced states on SS are equal.

Theorem 1. Let SS and RR be quantum systems, and let |ψ⟩|\psi\rangle and |φ⟩|\varphi\rangle be pure states of S⊗RS \otimes R with the same reduced state on SS, Tr⁡R(|ψ⟩⟨ψ|)=Tr⁡R(|φ⟩⟨φ|).\operatorname{Tr}_R\bigl(|\psi\rangle\langle\psi|\bigr) = \operatorname{Tr}_R\bigl(|\varphi\rangle\langle\varphi|\bigr).(3) Then there is a unitary UU of RR such that |φ⟩=(I⊗U)|ψ⟩.|\varphi\rangle = (I \otimes U)\,|\psi\rangle.(4)✦

What Is Checked

The claim is stated over the composite system S⊗RS \otimes R and its pure states, the reduced state (1) of the projector |ψ⟩⟨ψ||\psi\rangle\langle\psi|, and the product unitary I⊗UI \otimes U of the identity evolution of SS with a unitary of RR, applied to |ψ⟩|\psi\rangle.

About these constructions the library shows that the projector onto a unit vector is a state, and that so is the reduced state of a state; both sides of (3) are therefore states of SS. It shows as well that a unitary preserves norms and that the tensor product of two unitaries is unitary, so that (I⊗U)|ψ⟩(I \otimes U)|\psi\rangle is again a pure state of S⊗RS \otimes R.