bibkey: parzygnat2021conditionaldistributions
authors: Arthur J. Parzygnat
year: 2021
title: "Conditional Distributions for Quantum Systems"
doi: 10.4204/EPTCS.343.1
url: https://arxiv.org/abs/2102.01529v3
claim: "For a density matrix ν on M_m ⊗ M_n with invertible marginals ρ and σ, the conditional domains B_ν = {B : [ρ, tr_B(ν(1 ⊗ B))] = 0} and A_ν = {A : [σ, tr_A((A ⊗ 1)ν)] = 0} are operator systems (Definition 5.5, Lemma 5.8); they need not be C*-subalgebras for m = n = 3, and the author suspects that they always are for 4 × 4 density matrices (footnote 13)."
strata_touched:
  - D5/S3/Quantum/Information/ConditionalDomainSubalgebraRefutation
license: citation-only
triage: anchor

Parzygnat, conditional distributions for quantum systems

A. J. Parzygnat, Conditional Distributions for Quantum Systems, in M. Backens and C. Heunen (eds.), Proceedings 18th International Conference on Quantum Physics and Logic (QPL 2021), Electronic Proceedings in Theoretical Computer Science 343 (2021) 1–13, DOI 10.4204/EPTCS.343.1; arXiv:2102.01529 [quant-ph] (v1 2 February 2021, v2 30 May 2021, v3 13 September 2021).

Verified locator

DOI: 10.4204/EPTCS.343.1

Source: https://arxiv.org/abs/2102.01529v3; the text was read from the PDF. Versions v1 and v2 were compared for the statements below.

• page 6: Corollary 4.5 (the conditionals F and G for invertible marginals);
• page 9: Lemma 5.1 (∗-preservation criteria);
• page 10: Definition 5.5 (the commutants and the conditional domains), Lemma 5.8 (operator systems), Example 5.10, the remark on the 9 × 9 counterexample, and footnote 13.

Footnote 13 has the same wording in all three versions; it is footnote 12 in v1.

Source statements

Corollary 4.5: “Set A := M_m(C) and B := M_n(C). Let ζ ≡ tr(ν · ) be a state on A ⊗ B (with density matrix ν) whose marginals on A and B are given by ζ ∘ ι_A =: ω ≡ tr(ρ · ) and ζ ∘ ι_B =: ξ ≡ tr(σ · ), respectively. Suppose that ρ and σ are invertible.”

Definition 5.5: “Set A^ρ_c := {A ∈ A : [ρ, A] = 0} and B^σ_c := {B ∈ B : [σ, B] = 0} to be the commutants of {ρ} and {σ} inside A and B, respectively. Set B_ν := {B ∈ B : tr_B(ν(1_m ⊗ B)) ∈ A^ρ_c} and A_ν := {A ∈ A : tr_A((A ⊗ 1_n)ν) ∈ B^σ_c} to be the conditional domains of ν inside B and A, respectively.”

Lemma 5.8: “In the notation of Definition 5.5, B_ν and A_ν are operator systems.”

After Example 5.10: “Example 5.10 suggests that B_ν and A_ν are not only operator systems, but they might even be C∗-subalgebras. Is this always the case? The answer to this question is no, though the simplest counterexample I could currently find involves a 9×9 rank 2 density matrix with A = M3(C) and B = M3(C). Its expression is not particularly enlightening, so I have chosen to not record it here.”

Footnote 13: “Also, I could not find a 4 × 4 density matrix ν for which the conditional domains are not C∗-subalgebras, and I suspect that this may always be the case. I hope to resolve this in future work.”

Reading used by the repository

A 4 × 4 density matrix with both factors nontrivial is a two-qubit state, m = n = 2, with invertible marginals as in Corollary 4.5. An operator system in M₂(ℂ) containing 1 is a C*-subalgebra exactly when it is closed under multiplication, since a finite-dimensional *-subalgebra is norm-closed. The repository states the suspicion as: for every two-qubit density matrix with invertible marginals, both conditional domains are carriers of unital star subalgebras of M₂(ℂ).