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Measurement Variation At Most Trace Distance

Statement

Let $ρ, σ$ be density operators on a finite-dimensional complex Hilbert space and let ${E_{a}}_{a \in A}$ be any finite POVM. If $p_{a} = tr (ρ E_{a})$ and $q_{a} = tr (σ E_{a})$ , then $\frac{1}{2} \sum_{a \in A} ∣ p_{a} - q_{a} ∣ \leq D (ρ, σ) .$ The two-effect projective measurement from the Holevo–Helstrom theorem attains equality.

Facts & Assumptions

Given: Density operators $ρ, σ$ and a finite POVM ${E_{a}}$ .

[F1]

A finite POVM has nonnegative effects $E_{a}$ , $\sum_{a} E_{a} = I$ , and probabilities $p_{a} = tr (ρ E_{a})$ (Finite POVMs and their outcome law). The current QC04 definition body is present as a draft.

[F2]

$D (ρ, σ) = \frac{1}{2} ∥ Δ ∥_{1}$ for $Δ = ρ - σ$ (Trace Distance).

[L1]

For Hermitian trace-zero $Δ = Δ_{+} - Δ_{-}$ , $tr Δ_{+} = tr Δ_{-} = \frac{1}{2} ∥ Δ ∥_{1}$ (Hermitian Trace Norm Positive Negative Parts).

[L2]

The positive spectral projection of $Δ$ and its complement form an attaining binary projective measurement (Helstrom Binary Discrimination).

[F3]

If $A, B$ are nonnegative finite-dimensional operators, then $tr (A B) = tr (B^{1 / 2} A B^{1 / 2}) \geq 0$ , since cyclicity identifies the traces and $B^{1 / 2} A B^{1 / 2}$ is nonnegative (For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ , A non-negative operator has a unique non-negative square root, 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).

Proof

1.1givenF1algebra

Set $δ_{a} = p_{a} - q_{a} = tr (E_{a} Δ)$ . Since $\sum_{a} E_{a} = I$ and $tr Δ = tr ρ - tr σ = 0$ , one has $\sum_{a} δ_{a} = 0$ . Therefore the sum of the positive $δ_{a}$ equals $\frac{1}{2} \sum_{a} ∣ δ_{a} ∣$ .

2.1step 1.1F1F2F3L1algebra

Let $J = {a : δ_{a} > 0}$ and $E_{J} = \sum_{a \in J} E_{a}$ . Positivity and completeness in [F1] imply $0 \leq E_{J} \leq I$ . By [L1], $Δ = Δ_{+} - Δ_{-}$ with both parts non-negative. Then $tr (E_{J} Δ_{-}) \geq 0$ and $tr ((I - E_{J}) Δ_{+}) \geq 0$ by [F3], so $tr (E_{J} Δ) \leq tr Δ_{+}$ . Consequently $\frac{1}{2} \sum_{a} ∣ δ_{a} ∣ = \sum_{a \in J} tr (E_{a} Δ) = tr (E_{J} Δ) \leq tr Δ_{+} = D (ρ, σ)$ by [L1] and [F2].

3.1L1L2step 2.1∎

For the positive spectral projection $E_{+}$ of $Δ$ , the binary measurement $(E_{+}, I - E_{+})$ has signed outcome differences $tr Δ_{+}$ and $- tr Δ_{+}$ ; hence its total variation is $tr Δ_{+} = D$ . It attains the bound in step 2.1.

Depends on

Used by

One uniform phase query gives small trace-distance advantage Counterexample
Finite braid compilation and adiabatic error budget Lemma
Concatenated Threshold Under Local-Stochastic Correlated Noise Theorem
Conditional Concatenated Threshold for Proper Codes Theorem
Entanglement-free, randomness-free quantum fingerprints require Ω(log n) qubits Theorem
Fuchs Van De Graaf Theorem
Hybrid bound for a composed channel circuit Theorem
Operator error bounds measured output variation Theorem
Tensor-product state errors add under trace distance Theorem
Trace Distance Channel Contractivity Theorem

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Sources

John Watrous, The Theory of Quantum Information (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 2: Foundations I: States and Ensembles (updated July 2015) (standard reference, not scraped)