TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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POVM outcome weights are probabilities

Statement

Let $ρ$ be a density operator on a nonzero finite-dimensional complex inner-product space $H$ , and let $E = (E_{a})_{a \in Ω}$ be a finite POVM on $H$ . Then

$p_{ρ} (a) = tr (ρ E_{a}) \in [0, 1] for each a \in Ω, \sum_{a \in Ω} p_{ρ} (a) = 1.$

Facts & Assumptions

Given: $H$ , the density operator $ρ$ , the finite outcome set $Ω$ , and the POVM $E$ stated above.

[F1]

Finite POVMs and their outcome law: $E_{a}$ is nonnegative, $\sum_{a} E_{a} = I_{H}$ , and $p_{ρ} (a) = tr (ρ E_{a})$ .

[F7]

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

[F8]

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

[F2]

A non-negative operator has a unique non-negative square root: every nonnegative operator on a finite-dimensional real or complex inner-product space has a unique nonnegative square root $R$ with $R^{2} = T$ .

[F3]

A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S: a nonnegative operator has an orthonormal basis in which its matrix is diagonal with nonnegative real diagonal entries.

[F4]

For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ : for compatible rectangular matrices $A, B$ , $tr (A B) = tr (B A)$ , including zero-sized cases.

[F5]

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

[F6]

The basis-independent trace of an endomorphism of a finite-dimensional vector space: endomorphism trace is the matrix trace in any basis and is basis-independent.

Proof

1.1givenF1F2F3F4F6F7F8

Fix $a \in Ω$ . Let $M_{a} = E_{a}^{1 / 2}$ be the nonnegative square root supplied by [F2]. Since $M_{a}$ and $ρ$ are self-adjoint by [F2] and [F7]–[F8], for every $x, y \in H$ one has $⟨ M_{a} ρ M_{a} x, y ⟩ = ⟨ ρ M_{a} x, M_{a} y ⟩ = ⟨ M_{a} x, ρ M_{a} y ⟩ = ⟨ x, M_{a} ρ M_{a} y ⟩$ , so $M_{a} ρ M_{a}$ is self-adjoint. For every $x \in H$ , $⟨ M_{a} ρ M_{a} x, x ⟩ = ⟨ ρ M_{a} x, M_{a} x ⟩ \geq 0$ by [F7]–[F8], so $M_{a} ρ M_{a}$ is nonnegative. By [F3] it is diagonal with nonnegative entries in an orthonormal basis, hence its trace is nonnegative by [F6]. By [F4] and $M_{a}^{2} = E_{a}$ , $p_{ρ} (a) = tr (ρ E_{a}) = tr (M_{a} ρ M_{a}) \geq 0$ .

1.2givenF1F5F6F7

Linearity of trace [F5] and the completeness relation in [F1] give $\sum_{a \in Ω} p_{ρ} (a) = tr (ρ \sum_{a} E_{a}) = tr (ρ I_{H}) = tr (ρ) = 1$ by [F7] and [F6].

2.1step 1.1algebra∎

For fixed $a$ , $1 - p_{ρ} (a) = \sum_{b \in Ω, b \neq a} p_{ρ} (b) \geq 0$ by step 1.1; if $Ω = {a}$ this is the empty sum $0$ . Thus $p_{ρ} (a) \leq 1$ , and together with step 1.1 each weight lies in $[0, 1]$ .

Depends on

Used by

Finite adiabatic path model Definition
Finite cq block codes, average error, and capacity Definition
An orthogonal ensemble attains its Holevo information Example
Controlled Global Phase Changes an X-Outcome Example
A finite verifier induces an effect on its witness Lemma
Global Phase Under Control Lemma
A Controlled Global Phase Becomes a Relative Control Phase Theorem
Nonnegativity and equality for finite quantum relative entropy Theorem
Parallel QMA amplification remains sound for entangled witnesses Theorem
Standalone Global Phase Leaves Pure-State POVM Probabilities Unchanged Theorem
The finite Holevo measurement bound Theorem
The finite POVM law is a probability kernel Theorem

Cited to discharge well-definedness by Finite POVMs and their outcome law.

Dependency tree · two levels

29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Watrous, The Theory of Quantum Information (2018), §2.3.1 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 3: Foundations II: Measurement and Evolution, §3.1.2 (standard reference, not scraped)