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Positive-probability instrument branches are density operators

Statement

Let $ρ$ be an input density operator for a finite instrument $(I_{a})_{a \in Ω}$ , and let $p_{a} = tr (I_{a} (ρ))$ . If $p_{a} > 0$ , then the conditional operator $ρ_{a} = I_{a} (ρ) / p_{a}$ is a density operator on the instrument's output space.

Facts & Assumptions

Given: A finite instrument, an input density operator $ρ$ , and an outcome $a$ with $p_{a} > 0$ .

[F1]

Finite instruments and conditional states: each branch map is completely positive, $p_{a} = tr (I_{a} (ρ))$ , and the conditional operator is $I_{a} (ρ) / p_{a}$ when $p_{a} > 0$ .

[F2]

Completely positive maps: at ancilla dimension $r = 1$ , a completely positive map preserves nonnegative operators.

[F5]

Density operator: on a nonzero finite-dimensional complex inner-product space, a density operator is nonnegative and has trace one.

[F3]

Instrument outcome weights normalize: the branch weights satisfy $p_{a} \geq 0$ and sum to one.

[F4]

Trace is a linear functional on $M_{n} (F)$ : trace commutes with scalar multiplication.

[F6]

Non-negative and positive operators: a nonnegative operator is self-adjoint and has nonnegative quadratic form on every vector.

Proof

1.1givenF3

The outcome-law theorem [F3] shows that $p_{a}$ is the probability of branch $a$ ; the hypothesis $p_{a} > 0$ selects a positive-probability branch.

2.1givenF1F2F5F6step 1.1

Put $τ_{a} = I_{a} (ρ)$ . By [F1]–[F2] and [F5]–[F6], $τ_{a}$ is nonnegative. Since $p_{a} > 0$ is real and $τ_{a}$ is self-adjoint, for all $u, v$ one has $⟨ (τ_{a} / p_{a}) u, v ⟩ = p_{a}^{- 1} ⟨ τ_{a} u, v ⟩ = p_{a}^{- 1} ⟨ u, τ_{a} v ⟩ = ⟨ u, (τ_{a} / p_{a}) v ⟩$ , so $τ_{a} / p_{a}$ is self-adjoint. Division by $p_{a} > 0$ also preserves nonnegativity of every quadratic form. Thus $ρ_{a} = τ_{a} / p_{a}$ is nonnegative by [F6].

3.1givenF1F4F5step 2.1∎

By the definition in [F1], $tr (τ_{a}) = p_{a}$ . Linearity [F4] gives $tr (ρ_{a}) = p_{a}^{- 1} tr (τ_{a}) = 1$ . Thus $ρ_{a}$ is a density operator by [F5]. The hypothesis $p_{a} > 0$ also forces the output space to be nonzero, since its zero space has only trace-zero endomorphisms.

Depends on

Used by

One Simon Fourier-sampling sample Algorithm
Finite authenticated key-distribution protocol Definition
Heralded success and conditional verification contract Definition
Normalized output law for finite quantum query branches Definition
Simon query and output measurement produce a coset pair Theorem

Cited to discharge well-definedness by Finite instruments and conditional states.

Dependency tree · two levels

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Sources

John Watrous, The Theory of Quantum Information (2018), §2.3.2 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 3: Foundations II: Measurement and Evolution, §3.2.4 (standard reference, not scraped)