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Projective measurement outcome weights are probabilities

Statement

Let $ρ$ be a density operator on a nonzero finite-dimensional complex inner-product space $H$ , and let $(P_{a})_{a \in Ω}$ be a finite projective measurement on $H$ . Then $p_{ρ} (a) = tr (ρ P_{a})$ lies in $[0, 1]$ for every $a$ , and $\sum_{a \in Ω} p_{ρ} (a) = 1$ .

Facts & Assumptions

Given: The density operator $ρ$ and the finite projective measurement $(P_{a})_{a \in Ω}$ stated above.

[F1]

Finite projective measurements: $P_{a}^{*} = P_{a} = P_{a}^{2}$ , $P_{a} P_{b} = 0$ for $a \neq b$ , and $\sum_{a} P_{a} = I_{H}$ ; zero projections are allowed.

[F6]

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

[F7]

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

[F2]

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.

[F3]

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

[F4]

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

[F5]

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

1.1givenF1F2F3F5F6F7

Fix $a$ . Set $τ_{a} = P_{a} ρ P_{a}$ . Self-adjointness of $P_{a}$ and $ρ$ gives $⟨ τ_{a} x, y ⟩ = ⟨ ρ P_{a} x, P_{a} y ⟩ = ⟨ x, τ_{a} y ⟩$ for all $x, y$ , so $τ_{a}$ is self-adjoint. It is nonnegative, since $⟨ τ_{a} x, x ⟩ = ⟨ ρ P_{a} x, P_{a} x ⟩ \geq 0$ for every $x$ by [F6]–[F7]. By [F2], $τ_{a}$ has nonnegative diagonal entries in an orthonormal basis, so [F5] gives $tr (τ_{a}) \geq 0$ . By [F3] and $P_{a}^{2} = P_{a}$ , $p_{ρ} (a) = tr (ρ P_{a}) = tr (P_{a} ρ P_{a}) = tr (τ_{a}) \geq 0$ .

1.2givenF1F4F5F6

By [F4], the resolution of the identity in [F1], and [F6], $\sum_{a} p_{ρ} (a) = tr (ρ \sum_{a} P_{a}) = tr (ρ I_{H}) = tr (ρ) = 1$ .

2.1step 1.1algebra∎

For each $a$ , $1 - p_{ρ} (a) = \sum_{b \neq a} p_{ρ} (b) \geq 0$ by step 1.1, with the empty sum interpreted as $0$ when $Ω = {a}$ . Therefore every weight is at most one and, by step 1.1, belongs to $[0, 1]$ .

Depends on

Used by

Amplitude amplification with a fixed integer iteration count Algorithm
Amplitude amplification with independently chosen phase rotations Algorithm
Exact success amplification by attenuating the good branch Algorithm
Known-amplitude amplification with integer rounding Algorithm
Power-of-two dihedral hidden-subgroup sieve Algorithm
Quantum phase estimation Algorithm
Szegedy-walk marked-set detection Algorithm
Unknown-amplitude randomized search Algorithm
Coherent variable-time algorithm model Definition
Finite adiabatic path model Definition
Prepared unitary and success subspace for amplitude amplification Definition
QAOA circuit and expected combinatorial objective Definition
Symmetric and antisymmetric swap-test measurement Definition
A projective measurement branch with probability zero Example
One-bit feed-forward and deferred measurement Example
VQE and MaxCut-QAOA objective calculation Example
Amplitude-amplification invariant plane Lemma
Dihedral hidden-state pair combination Lemma
Finite amplification preserves the good state with a deterministic call cap Lemma
Fourier sample from a generalized-dihedral reflection coset state Lemma
Phase-estimation finite outcome law Lemma
Conditional state for a projective measurement Theorem
CSS syndrome correction Theorem
Dense-coding correctness and resources Theorem
Element-distinctness gate cost in the QRAM model Theorem
Exact success for known positive amplitude with explicit dilution Theorem
Exact success probability after amplitude amplification Theorem
Fingerprint swap-test error and communication cost Theorem
Finite Naimark isometric compression Theorem
Finite reversible-walk finding cost Theorem
Fusion measurement branches are normalized exactly when nonzero Theorem
Phase estimation on a superposition of eigenspaces Theorem
QAOA objective is an expected feasible score only Theorem
QPE output law and conditional state for a spectral input Theorem
Success bound under an explicit amplitude-amplification phase mismatch Theorem
Swap-test measurement and pure-state outcome law Theorem
Teleportation outcomes are equiprobable Theorem
Teleportation preserves a finite reference Theorem
The measurement law for a finite continuous-time walk Theorem
Unknown-amplitude search has no false positives and halts almost surely Theorem

…and 1 more result.

Cited to discharge well-definedness by Finite projective measurements.

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Sources

John Watrous, The Theory of Quantum Information (2018), §§2.3.1–2.3.2 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 3: Foundations II: Measurement and Evolution, §3.1.1 (standard reference, not scraped)