Pith Number

pith:ZONV7FJ3

pith:2026:ZONV7FJ36J6QDH7EQX6EWMYPIG

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Relations between different definitions of the quantum Wasserstein distance for qubits

G\'eza T\'oth, J\'ozsef Pitrik

Two quantum Wasserstein distances coincide for qubits when the cost function uses only one operator, so the self-distance equals the Wigner-Yanase skew information.

arxiv:2605.03027 v1 · 2026-05-04 · quant-ph · math-ph · math.MP

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Record completeness

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Claims

C1strongest claim

The quantum Wasserstein distances defined by Golse, Mouhot, Paul, and Caglioti and by De Palma and Trevisan coincide for qubits when a single operator appears in the cost function. As a consequence, the self-distance equals the Wigner-Yanase skew information in this case.

C2weakest assumption

The assumption that the cost function contains only a single operator and that the systems are strictly two-level qubits; the equivalence may fail for higher-dimensional systems or multi-operator costs.

C3one line summary

Two quantum Wasserstein distance definitions coincide for qubits with single-operator cost functions, implying the self-distance equals the Wigner-Yanase skew information.

References

46 extracted · 10 resolved · 3 Pith anchors

[1] K. ˙Zyczkowski and W. Slomczynski, The Monge distance between quantum states, J. Phys. A: Math. Gen.31, 9095 (1998) 1998

[2] K. ˙Zyczkowski and W. Slomczynski, The Monge metric on the sphere and geometry of quantum states, J. Phys. A: Math. Gen.34, 6689 (2001) 2001

[3] I. Bengtsson and K. ˙Zyczkowski,Geometry of Quan- tum States: An Introduction to Quantum Entanglement (Cambridge University Press, 2006) 2006

[4] P. Biane and D. Voiculescu, A free probability ana- logue of the Wasserstein metric on the trace-state space, GAFA, Geom. Funct. Anal.11, 1125 (2001) 2001

[5] E. Carlen and J. Maas, An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy, Commun. Math. Phys.331, 8 2014

Formal links

3 machine-checked theorem links

Receipt and verification

First computed	2026-05-21T01:05:19.764386Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

cb9b5f953bf27d019fe485fc4b330f41b827c6db0f29d5ae701533b29e4836ed

Aliases

arxiv: 2605.03027 · arxiv_version: 2605.03027v1 · doi: 10.48550/arxiv.2605.03027 · pith_short_12: ZONV7FJ36J6Q · pith_short_16: ZONV7FJ36J6QDH7E · pith_short_8: ZONV7FJ3

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ZONV7FJ36J6QDH7EQX6EWMYPIG \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: cb9b5f953bf27d019fe485fc4b330f41b827c6db0f29d5ae701533b29e4836ed

Canonical record JSON

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    "cross_cats_sorted": [
      "math-ph",
      "math.MP"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "quant-ph",
    "submitted_at": "2026-05-04T18:01:04Z",
    "title_canon_sha256": "8f3f1bce53bb62d7d74799f5856747fec52d9e2ed78d3d61871d16a374948da9"
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  "source": {
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