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Fusion measurement branches are normalized exactly when nonzero

Statement

Let $(P_{a})_{a \in A}$ be the finite fusion-channel projectors of Fusion-channel measurement access on a finite-dimensional fusion space $H$ , and let $ψ \in H$ be a unit vector. The channel probabilities $p_{a} = ∥ P_{a} ψ ∥^{2}$ are nonnegative and sum to one. If $p_{a} > 0$ , the conditional branch is the unit vector $P_{a} ψ / \sqrt{p_{a}}$ . If $p_{a} = 0$ , then $P_{a} ψ = 0$ and no normalized conditional state is defined.

Facts & Assumptions

Given: A finite orthogonal resolution $(P_{a})$ of the identity and a unit input vector $ψ$ .

[F1]

The channel family is a finite projective measurement: its projections are pairwise orthogonal and sum to identity. Fusion-channel measurement access

[F2]

The projective measurement outcome law assigns probability $p_{a} = tr (∣ ψ ⟩ ⟨ ψ ∣ P_{a})$ to outcome $a$ . Projective measurement outcome weights are probabilities

[F3]

A unit vector defines a pure-state ray whose rank-one projector is a density operator. Pure state and ray Rank-one density operators are pure-state projectors

[F4]

On an outcome of positive probability, the conditional density operator is $P_{a} ρ P_{a} / p_{a}$ . Conditional state for a projective measurement

[F5]

The Dirac outer product acts by $∣ x ⟩ ⟨ y ∣ z = ⟨ z, y ⟩_{r o o t} x$ ; trace is the sum of diagonal entries, and $∥ x ∥^{2} = ⟨ x, x ⟩_{r o o t}$ . Dirac Notation Conversion The basis-independent trace of an endomorphism of a finite-dimensional vector space The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product

[F6]

In orthonormal coordinates, $∣ x ⟩ ⟨ y ∣$ has matrix $[x] [y]^{†}$ , so its trace is $⟨ x, y ⟩_{r o o t}$ . Dirac Notation Coordinate Identity

Proof

1.1F1F2F3F5F6givenalgebra

Set $ρ = ∣ ψ ⟩ ⟨ ψ ∣$ , a density operator by [F3]. The Dirac rule and $P_{a} = P_{a}^{*} = P_{a}^{2}$ give $∣ ψ ⟩ ⟨ ψ ∣ P_{a} = ∣ ψ ⟩ ⟨ P_{a} ψ ∣$ , whose trace is $⟨ ψ, P_{a} ψ ⟩_{r o o t}$ by [F6]. By self-adjointness and idempotence this equals $⟨ P_{a} ψ, P_{a} ψ ⟩_{r o o t} = ∥ P_{a} ψ ∥^{2}$ by [F5]. Thus [F2] identifies $p_{a} = ∥ P_{a} ψ ∥^{2} \geq 0$ as the outcome probability. The resolution identity gives $\sum_{a} p_{a} = ⟨ ψ, (\sum_{a} P_{a}) ψ ⟩_{r o o t} = ∥ ψ ∥^{2} = 1$ .

2.1F1F4F5step 1.1algebra∎

If $p_{a} > 0$ , then $ρ_{a} = P_{a} ρ P_{a} / p_{a}$ is the conditional density operator by [F4]. The Dirac rule and self-adjointness give $P_{a} ρ P_{a} = ∣ P_{a} ψ ⟩ ⟨ P_{a} ψ ∣$ ; using step 1.1, $ρ_{a}$ is the rank-one projector of the unit vector $P_{a} ψ / \sqrt{p_{a}}$ . If $p_{a} = 0$ , step 1.1 gives $∥ P_{a} ψ ∥ = 0$ , hence $P_{a} ψ = 0$ . The zero-probability convention is that this branch is omitted and no normalized conditional vector or state is assigned; division by $\sqrt{p_{a}} = 0$ is undefined.

Depends on

Used by

Fibonacci fusion measurement branches and zero outcomes Example

Dependency tree · two levels

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Sources

John Preskill, Lecture Notes for Physics 219: Quantum Computation, Chapter 9: Topological Quantum Computation (standard reference, not scraped)
Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)