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The nonselective output of an instrument is a density operator

Statement

For an input density operator $ρ$ and a finite instrument $(I_{a})_{a \in Ω}$ , define its nonselective output by $\overset{ˉ}{ρ} : = \sum_{a \in Ω} I_{a} (ρ)$ . Then $\overset{ˉ}{ρ}$ is a density operator on the output space. Moreover,

$\overset{ˉ}{ρ} = \sum_{a : p_{a} > 0} p_{a} ρ_{a},$

where $p_{a} = tr (I_{a} (ρ))$ and $ρ_{a}$ is the positive-probability conditional state.

Facts & Assumptions

Given: An input density operator $ρ$ and a finite instrument with outcome set $Ω$ .

[F1]

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

[F6]

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

[F2]

Instrument outcome weights normalize: $p_{a} \geq 0$ and $\sum_{a} p_{a} = 1$ .

[F3]

A zero-probability instrument branch has no conditioned state: if $p_{a} = 0$ , then $I_{a} (ρ) = 0$ .

[F4]

Completely positive maps: a completely positive branch map sends nonnegative inputs to nonnegative outputs at ancilla dimension one.

[F5]

Trace is a linear functional on $M_{n} (F)$ : trace preserves finite sums.

[F7]

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

Proof

1.1givenF1F2F4F5F6F7

For each $a$ , $τ_{a} = I_{a} (ρ)$ is nonnegative by [F1], [F4], and [F6]–[F7]. For all $x, y$ , self-adjointness of each $τ_{a}$ gives $⟨ \overset{ˉ}{ρ} x, y ⟩ = \sum_{a} ⟨ τ_{a} x, y ⟩ = \sum_{a} ⟨ x, τ_{a} y ⟩ = ⟨ x, \overset{ˉ}{ρ} y ⟩$ , so the finite sum $\overset{ˉ}{ρ}$ is self-adjoint. For every $x$ , its quadratic form is $⟨ \overset{ˉ}{ρ} x, x ⟩ = \sum_{a} ⟨ τ_{a} x, x ⟩ \geq 0$ . Hence $\overset{ˉ}{ρ}$ is nonnegative by [F7]. By [F5] and [F2], $tr (\overset{ˉ}{ρ}) = \sum_{a} tr (τ_{a}) = \sum_{a} p_{a} = 1$ . Hence $\overset{ˉ}{ρ}$ is a density operator; its trace one also shows the output space is nonzero.

2.1givenF1F3step 1.1∎

If $p_{a} > 0$ , the instrument definition [F1] gives $p_{a} ρ_{a} = τ_{a}$ . If $p_{a} = 0$ , [F3] gives $τ_{a} = 0$ . Summing these equalities over the finite outcome set yields $\overset{ˉ}{ρ} = \sum_{a : p_{a} > 0} p_{a} ρ_{a}$ .

Depends on

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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, §§3.2.4–3.2.5 (standard reference, not scraped)