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Conditional state for a projective measurement

Statement

Let $ρ$ be a density operator and $(P_{a})_{a \in Ω}$ a finite projective measurement on a nonzero finite-dimensional complex inner-product space $H$ . Put $p_{a} = tr (ρ P_{a})$ . If $p_{a} > 0$ , then

$ρ_{a} : = \frac{P_{a} ρ P_{a}}{p_{a}}$

is a density operator whose image $im (ρ_{a}) : = {ρ_{a} x : x \in H}$ is contained in $im (P_{a})$ (Kernel and image of a linear map). This statement assigns a conditional state only on positive-probability branches.

Facts & Assumptions

Given: The density operator $ρ$ , the projective measurement $(P_{a})_{a \in Ω}$ , and an outcome $a$ with $p_{a} > 0$ .

[F1]

Finite projective measurements: $P_{a}^{*} = P_{a} = P_{a}^{2}$ and distinct outcome projections are orthogonal.

[F2]

Projective measurement outcome weights are probabilities: $p_{a} = tr (ρ P_{a})$ is the probability of outcome $a$ .

[F5]

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

[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 scalar multiples.

[F6]

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

[F7]

Kernel and image of a linear map: the image of a linear map $T$ is $im T = {T (x) : x is in the domain}$ .

Proof

1.1givenF1F5F6

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. For every $x \in H$ , $⟨ τ_{a} x, x ⟩ = ⟨ ρ P_{a} x, P_{a} x ⟩ \geq 0$ by [F5] and [F6], so $τ_{a}$ is nonnegative. Since $p_{a} > 0$ is real, scaling the same pairing identity by $1 / p_{a}$ makes $ρ_{a} = τ_{a} / p_{a}$ self-adjoint; its quadratic form is nonnegative as well, so $ρ_{a}$ is nonnegative.

1.2givenF1F2F3F4F5

By [F3] and $P_{a}^{2} = P_{a}$ , $tr (τ_{a}) = tr (P_{a} ρ P_{a}) = tr (ρ P_{a}^{2}) = p_{a}$ by [F2]. Hence [F4] gives $tr (ρ_{a}) = p_{a} / p_{a} = 1$ , so $ρ_{a}$ is a density operator by [F5].

2.1givenF1F7step 1.1step 1.2∎

For every $x \in H$ , $ρ_{a} x = P_{a} (ρ P_{a} x) / p_{a}$ is of the form $P_{a} y$ with $y = ρ P_{a} x / p_{a}$ . By [F7], every such vector lies in $im (P_{a})$ , so $im (ρ_{a}) \subseteq im (P_{a})$ .

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
Szegedy-walk marked-set detection Algorithm
Unknown-amplitude randomized search Algorithm
Coherent variable-time algorithm model Definition
Prepared unitary and success subspace for amplitude amplification Definition
One-bit feed-forward and deferred measurement Example
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
CSS syndrome correction Theorem
Exact success for known positive amplitude with explicit dilution Theorem
Finite reversible-walk finding cost Theorem
Fusion measurement branches are normalized exactly when nonzero Theorem
Kuperberg dihedral hidden-subgroup scope and resource claim Theorem
Teleportation preserves a finite reference Theorem
Unknown-amplitude search has no false positives and halts almost surely Theorem
Walk detection correctness and cost Theorem

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, §3.1.1 (standard reference, not scraped)