Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

Robustness of ancilla-free testers

Abstract

Ancilla-free testers with a common normalization state have the robustness of their canonical POVMs.

Definition 1.1 (Quantum testers).

Formalization. D5/S3/Quantum/QuantumChannels/AncillaFreeTesterRobustnessEquality.IsTester (✓ std3).

Citation. Michal Sedlák; Daniel Reitzner; Giulio Chiribella; Mário Ziman (2016). Incompatible measurements on quantum causal networks. DOI: 10.1103/PhysRevA.93.052323. URL: https://arxiv.org/abs/1511.00976v2.

Commentary.

A tester is a finite family t of positive semidefinite effects on the output-input tensor product. Its sum is the output identity tensored with a density matrix σ on the input. Here IsDensity means positive semidefinite with complex trace equal to one. This is the tester definition in ti_pra_07.tex, lines 247–259.

Definition 1.2 (Compatibility after adding tester noise).

Formalization. D5/S3/Quantum/QuantumChannels/AncillaFreeTesterRobustnessEquality.TesterCompatibleAt (✓ std3).

Citation. Michal Sedlák; Daniel Reitzner; Giulio Chiribella; Mário Ziman (2016). Incompatible measurements on quantum causal networks. DOI: 10.1103/PhysRevA.93.052323. URL: https://arxiv.org/abs/1511.00976v2.

Commentary.

The noise testers e and f have the outcome types of a and b. A joint tester c has outcomes j × k, and its marginals are the two mixtures at the same real weight lam. The three testers may have different normalization states, and the noise need not be ancilla-free. The predicate itself allows every real lam; robustness restricts lam to the unit interval. These are the marginal and noise conditions in ti_pra_07.tex, lines 451–459 and 567–570.

Definition 1.3 (Tester robustness).

Formalization. D5/S3/Quantum/QuantumChannels/AncillaFreeTesterRobustnessEquality.testerRobustness (✓ std3).

Citation. Michal Sedlák; Daniel Reitzner; Giulio Chiribella; Mário Ziman (2016). Incompatible measurements on quantum causal networks. DOI: 10.1103/PhysRevA.93.052323. URL: https://arxiv.org/abs/1511.00976v2.

Commentary.

Tester robustness is the infimum of the real weights in the closed unit interval for which separate tester noises give compatible mixtures. The source uses a minimum in ti_pra_07.tex, lines 574–577; this infimum formulation makes no assertion of attainment.

Definition 1.4 (Compatibility after adding measurement noise).

Formalization. D5/S3/Quantum/QuantumChannels/AncillaFreeTesterRobustnessEquality.POVMCompatibleAt (✓ std3).

Citation. Michal Sedlák; Daniel Reitzner; Giulio Chiribella; Mário Ziman (2016). Incompatible measurements on quantum causal networks. DOI: 10.1103/PhysRevA.93.052323. URL: https://arxiv.org/abs/1511.00976v2.

Commentary.

The families u, v and g are POVMs: every effect is positive semidefinite and each whole family sums to the identity. The two marginals of g are the mixtures of p and q with the separate noises u and v. All scalar actions in these formulas use real scalars on complex matrices. The measurement noise condition is stated in ti_pra_07.tex, lines 761–765.

Definition 1.5 (Measurement robustness).

Formalization. D5/S3/Quantum/QuantumChannels/AncillaFreeTesterRobustnessEquality.povmRobustness (✓ std3).

Citation. Michal Sedlák; Daniel Reitzner; Giulio Chiribella; Mário Ziman (2016). Incompatible measurements on quantum causal networks. DOI: 10.1103/PhysRevA.93.052323. URL: https://arxiv.org/abs/1511.00976v2.

Commentary.

Measurement robustness is the infimum of the weights in the closed unit interval that permit a joint POVM after adding separate noise POVMs. It expresses the robustness of ti_pra_07.tex, lines 767–770, using an infimum without asserting attainment.

Definition 1.6 (Equality for a common ancilla-free state).

Formalization. D5/S3/Quantum/QuantumChannels/AncillaFreeTesterRobustnessEquality.claim (✓ std3).

Source. Repository-derived.

Acknowledgement. Michal Sedlák; Daniel Reitzner; Giulio Chiribella; Mário Ziman (2016). Incompatible measurements on quantum causal networks. DOI: 10.1103/PhysRevA.93.052323. URL: https://arxiv.org/abs/1511.00976v2.

Commentary.

For arbitrary finite output and input types y and x and outcome types j and k, take a common density matrix ρ and output POVMs m and n. Their ancilla-free testers have effects m(a) ⊗ ρ and n(b) ⊗ ρ. The associated canonical POVMs have effects m(a) ⊗ I and n(b) ⊗ I on y × Fin(rank ρ), using coordinates on the support of ρ. The equality concerns precisely these two robustness values. The product characterization is Proposition 3, ti_pra_07.tex, lines 333–338; the canonical POVM is defined at lines 281–303. The common-state setting appears at line 772 and the ancilla-free question at line 807.

Theorem 1.7 (Trace one implies positive rank).

Proof. Machine-checked in Lean as D5/S3/Quantum/QuantumChannels/AncillaFreeTesterRobustnessEquality.rank_pos_of_trace_one (✓ std3). ∎

Source. Repository-derived.

Commentary.

For any finite index type a, a complex square matrix A with trace one has positive rank. If its rank were zero, the range of its multiplication map would be zero, so the map and hence the matrix would vanish, contradicting the trace equation. No positivity assumption on A is needed.

Theorem 1.8 (Equality of tester and canonical measurement robustness).

Proof. Machine-checked in Lean as D5/S3/Quantum/QuantumChannels/AncillaFreeTesterRobustnessEquality.result (✓ std3). ∎

Resolves. Problems/sedlak-reitzner-chiribella-ziman-2016-ancilla-free-tester-robustness (proved) by D5/S3/Quantum/QuantumChannels/AncillaFreeTesterRobustnessEquality.result.

Source. Repository-derived.

Acknowledgement. Michal Sedlák; Daniel Reitzner; Giulio Chiribella; Mário Ziman (2016). Incompatible measurements on quantum causal networks. DOI: 10.1103/PhysRevA.93.052323. URL: https://arxiv.org/abs/1511.00976v2.

Commentary.

First, tensoring each effect of a POVM with ρ gives a tester: Kronecker products preserve positive semidefiniteness, and the effects sum to I ⊗ ρ. Conversely, take the partial trace over x of each tester effect. This preserves positive semidefiniteness and finite sums. A tester normalized by any density matrix σ is sent to a POVM because the partial trace of I ⊗ σ is (tr σ)I = I.

For a fixed lam, suppose u, v and g are noise and joint POVMs for m and n. Tensor all three families with ρ. Distributing the Kronecker product over the marginal sums, addition and real scalar multiplication gives noise and joint testers for m ⊗ ρ and n ⊗ ρ. In the reverse direction, start with arbitrary noise testers e and f and a joint tester c. Apply the partial trace over x to all three families and their marginal equations. They become POVMs, and tr ρ = 1 sends m(a) ⊗ ρ to m(a) and n(b) ⊗ ρ to n(b). Thus compatibility at lam is equivalent for the product testers and the output POVMs, even when e, f and c have different normalization states.

Next tensor noise and joint POVMs with the identity on Fin r. Positivity, unit normalization and the marginal equations are preserved. For the reverse direction assume 0 < r and compress every effect to the principal submatrix indexed by pairs (b,0). Principal submatrices preserve positivity. The inclusion b ↦ (b,0) is injective, so compression preserves the identity, and it commutes with finite sums, addition and real scalar multiplication. The compressed entries of t ⊗ I are exactly the entries of t. Consequently compression turns noise and joint POVMs on y × Fin r into noise and joint POVMs on y with the required marginals.

The trace equation tr ρ = 1 implies ρ ≠ 0, hence some column has a nonzero entry. The span of the columns therefore has positive dimension, giving 0 < rank ρ. The first pair of implications identifies the feasible weights in [0,1] for the product testers with those for m and n. The tensor and compression implications identify the latter set with the feasible weights for the canonical POVMs when r = rank ρ. Composing these set equalities and taking infima proves the displayed equality. No minimizer or numerical robustness value is needed.

References