bibkey: bunthpitriktitkosvirosztek2024wasserstein authors: G. Bunth; J. Pitrik; T. Titkos; D. Virosztek year: 2024 title: “On the metric property of quantum Wasserstein divergences” doi: 10.1103/PhysRevA.110.022211 url: https://arxiv.org/abs/2402.13150v4 claim: “Section 4.1 Proposition 3, equation (35), states D²(ρ,ρ)=4(1−√(1−|bρ|²)), corresponding for eigenvalues (1±r)/2 to 8−4(tr√ρ)²; Section 5.1 defines the Wasserstein complexity and states that its convexity is open.” strata_touched:

• D5/S3/Quantum/QuantumChannels/QubitWassersteinCouplings
• D5/S3/Quantum/QuantumChannels/WassersteinComplexityNonconvex license: citation-only triage: anchor

Quantum Wasserstein divergences and channel complexity

Verified locator

DOI: https://doi.org/10.1103/PhysRevA.110.022211

Source: https://arxiv.org/abs/2402.13150v4

Source statements

The paper defines couplings by prescribed partial traces, the quadratic quantum Wasserstein distance in equation (eq:2-Wass-quant-def), and the divergence $d_{\mathcal{A}}$ in equation (eq:mod-qw-dist-def). Section 5.1 (PDF p. 17) defines the Wasserstein complexity by

Given a finite collections of observables $\mathcal{A}=\{A_1,\dots ,A_k\}$, let us define the Wasserstein complexity of a channel $\Phi :\mathcal{S}(\mathcal{H})\to \mathcal{S}(\mathcal{H})$ by $C_W(\Phi ):={\max }_{\rho \in \mathcal{S}(\mathcal{H})}{d}_{\mathcal{A}}(\rho ,\Phi (\rho ))$.

The same section (PDF p. 18) states:

We note that whether the Wasserstein complexity (37) is convex or not is an open question.

The self-cost identity used by the qubit settlement is literature-attested by Section 4.1, Proposition 3, equation (35): $D^2(\rho ,\rho )=4(1-\sqrt{1-|b_{\rho }{|}^2})$, which becomes $8-4(\mathrm{tr}\sqrt{\rho }{)}^2$ for eigenvalues $(1\pm r)/2$.

The Lean modules use the preregistered finite-dimensional qubit conventions and refute the convexity question for the Pauli triple by an explicit midpoint.

Definitions and encoding

Section 1.2 defines the coupling set by

$$\mathcal{C}(\rho ,\omega )=\{\Pi \in \mathcal{S}(\mathcal{H}\otimes {\mathcal{H}}^{\ast })\mid {\mathrm{tr}}_{{\mathcal{H}}^{\ast }}\Pi =\omega ,{\mathrm{tr}}_{\mathcal{H}}\Pi ={\rho }^T\}.$$

Equation (eq:2-Wass-quant-def) is

$$D_{\mathcal{A}}^2(\rho ,\omega )=\underset{\Pi \in \mathcal{C}(\rho ,\omega )}{\inf }\sum _{j=1}^k\mathrm{tr}[(A_j\otimes I-I\otimes A_j^T)\Pi (A_j\otimes I-I\otimes A_j^T)].$$

Equation (eq:mod-qw-dist-def) is

$$d_{\mathcal{A}}(\rho ,\omega )=\sqrt{D_{\mathcal{A}}^2(\rho ,\omega )-\frac{1}{2}(D_{\mathcal{A}}^2(\rho ,\rho )+D_{\mathcal{A}}^2(\omega ,\omega ))}.$$

The finite-dimensional encoding uses complex matrices on Fin d; the coupling matrix is indexed by Fin d × Fin d. Its positive semidefinite, trace-one condition is stated on that product index type. This is a Lean elaboration repair preserving the source state condition on the tensor product. The dual factor uses Matrix.transpose. The frozen partialTraceRight and partialTraceLeft are the literal explicit sums, and the frozen IsDensity is the literal positive semidefinite, trace-one condition. CW uses the supremum of the source maximum’s value set. Its boundedness and the existence of admissible couplings are proved in the supporting module. The universal claim retains the literal zero-dimensional case.

Scope of the refutation

The repository construction uses the Pauli triple and dephasing parameter $t=1/4$. The unitary endpoint channels are mixed with weight $s=1/2$. The midpoint complexity is at least $\sqrt{1+\sqrt{3}}$, while both endpoints have complexity at most $\sqrt{2}$. The supporting module proves the exact self-cost formula for every qubit density state. No assumption of the source’s previously proved metric statements is changed by this refutation. The same midpoint exceeds both endpoint complexities, so quasi-convexity also fails. All three channels are unital, so restriction to unital channels does not restore convexity. The exact value of the complexity for the dephasing family remains open.

Counterexample family

The following extension is proved in prose; the Lean settlement specializes it to $t=1/4$. For $0<t<1/2$, set $U_{\pm }=\sqrt{1-t}I\pm i\sqrt{t}Z$ and ${\Phi }_{\pm }=\mathrm{Ad}(U_{\pm })$. Their midpoint is the unital channel ${\Psi }_t=(1-t)\mathrm{id}+t\mathrm{Ad}(Z)$.

For a qubit state $\rho$, put $S=\sqrt{\rho }$. The vectorized $U_{\pm }S$ coupling has cross cost $8-4|\mathrm{tr}(U_{\pm }S){|}^2$. Since $S$ is positive, its diagonal entries are real, so $|\mathrm{tr}(U_{\pm }S){|}^2\ge (1-t)(\mathrm{tr}S{)}^2$. Unitary covariance preserves the self cost, giving $d_{\mathcal{A}}(\rho ,{\Phi }_{\pm }(\rho ){)}^2\le 4t(\mathrm{tr}S{)}^2\le 8t$.

At $\rho =|+⟩⟨+|$, the pure marginal fixes the midpoint cross cost at $4+4t$. The input self cost is $4$ and the output self cost is $4-8\sqrt{t(1-t)}$. Consequently,

$$C_W({\Psi }_t{)}^2\ge 4t+4\sqrt{t(1-t)}>8t\ge C_W({\Phi }_{\pm }{)}^2.$$

The strict inequality follows from $t(1-t)>t^2$. Thus subtracting the output self cost causes the convexity failure: channel mixing lowers that self cost while raising the cross cost. The same family refutes quasi-convexity among unital channels.

The pinned numerical check, rerun by the implementation seat, gives: python3 check.py, exit 0, bad= 0. The experiment entry is docs/reports/bunth-pitrik-titkos-virosztek-2024-wasserstein-complexity-convexity/check.py in the-omega-institute/trureturing-experiments at commit 10a541e9aacff47302154c721e10ff180fda60aa, SHA-256 04787a6acc2c5bccda908bcb90ffd00d928e418e14cebc28bfa5f2d90f9c68db.