Formalize Open Quantum Problem #45. Catalytic entropy conjecture

[MarioKrenn6240]

What is the conjecture

This is problem #45 on the Open Quantum Problems site. It was proposed in P. Boes, J. Eisert, R. Gallego, M. P. Müller, and H. Wilming (2019).

Status: this problem is listed as solved on the Solved Quantum Problems page (exact conjecture solved by Wilming, 2022; an approximate variant was solved earlier in 2021; see below).

Problem (Open Quantum Problem #45: “Catalytic entropy conjecture”).
Consider a density matrix ρ on a finite-dimensional system S. Let H(ω) = -tr(ω log ω) be the von Neumann entropy.
Prove or disprove that for any density matrix ρ' on the same system S such that H(ρ') > H(ρ) and rank(ρ') ≥ rank(ρ), there exists a finite-dimensional system C (“catalyst”) with state σ_C and a unitary operator U on S ⊗ C such that the following holds:
tr_C[U (ρ ⊗ σ_C) U†] = ρ'
tr_S[U (ρ ⊗ σ_C) U†] = σ_C.

Concretely, if such σ_C and U exist, we say that the exact catalytic transition ρ → ρ' is possible. The final joint state on S ⊗ C need not factor as ρ' ⊗ σ_C: correlations between system and catalyst are allowed, as long as the system marginal is ρ' and the catalyst marginal is returned exactly unchanged.

The conjecture asks whether, aside from the trivial case where ρ and ρ' are already unitarily equivalent, exact catalytic transitions are completely characterized by two simple state parameters:

• strict increase of von Neumann entropy, and
• non-decrease of rank/support size.

A positive solution gives a single-shot operational characterization of von Neumann entropy, without appealing to an i.i.d. asymptotic limit or to external randomness.

Solved statement to formalize (Wilming, 2022):
For finite-dimensional density matrices ρ, ρ' on the same system that are not unitarily equivalent, the following are equivalent:

• there exist a finite-dimensional catalyst state σ_C and a unitary U on S ⊗ C such that
  tr_C[U (ρ ⊗ σ_C) U†] = ρ'
  and
  tr_S[U (ρ ⊗ σ_C) U†] = σ_C;
• H(ρ') > H(ρ) and rank(ρ') ≥ rank(ρ).

Wilming’s 2022 paper also proves an equivalent majorization-style formulation: for some finite n, the tensor power ρ^{⊗ n} majorizes a state Ω on S^{⊗ n} whose one-body marginals are all exactly ρ', if and only if the same entropy/rank conditions hold. This provides the route from correlated many-copy constructions to the exact catalytic transition.

Earlier approximate result (Wilming, 2021):
The OQP page records that Wilming first solved an approximate variant of the conjecture: for arbitrary ε > 0, the exact requirement tr_C[U (ρ ⊗ σ_C) U†] = ρ' is relaxed to
|| tr_C[U (ρ ⊗ σ_C) U†] - ρ' ||_1 ≤ ε,
while the catalyst marginal is still returned exactly.

Important caveat (also on the OQP page):
The boundary case H(ρ') = H(ρ) remains open. It lies outside the original conjecture: the OQP page notes that such a transformation cannot be implemented with a finite-dimensional catalyst, though a suitable infinite-dimensional catalyst may still exist.

The 2022 exact solution also proves the analogous classical statement for finite probability vectors, with permutations replacing unitaries and Shannon entropy replacing von Neumann entropy.

Where to find the details / references

Primary sources:

• Open Quantum Problems site (Problem #45)
• Solved Quantum Problems list (shows #45 solved)

Original problem / motivation:

• P. Boes, J. Eisert, R. Gallego, M. P. Müller, and H. Wilming, “Von Neumann Entropy from Unitarity”, Phys. Rev. Lett. 122, 210402 (2019); arXiv: 1807.08773

Key solution references:

• H. Wilming, “Entropy and reversible catalysis”, Phys. Rev. Lett. 127, 260402 (2021); arXiv: 2012.05573
  (approximate variant, as recorded on the OQP page)
• H. Wilming, “Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture”, Quantum 6, 858 (2022); arXiv: 2205.08915

Related / partial results mentioned on the OQP page (useful background for a formalization roadmap):

• The original 2019 paper provides partial solutions for transitions of the form ρ ⊗ I/d → ρ' ⊗ I/d, and under additional assumptions on the catalyst system
• R. Duan, Y. Feng, X. Li, and M. Ying, “Multiple-copy entanglement transformation and entanglement catalysis”, Phys. Rev. A 71, 042319 (2005)
• M. Klimesh, “Inequalities that Collectively Completely Characterize the Catalytic Majorization Relation”, arXiv: 0709.3680
• S. Turgut, “Necessary and Sufficient Conditions for the Trumping Relation”, arXiv: 0707.0444

Prerequisites needed

• Finite-dimensional quantum mechanics: density matrices, tensor products, partial trace, unitary conjugation
• von Neumann entropy, rank/support, spectral decompositions
• Majorization theory for spectra / probability vectors, and its equivalent formulations
• Correlating catalysts: exact preservation of the catalyst marginal while allowing final system–catalyst correlations
• Trace norm / trace distance, if formalizing the approximate 2021 version as a stepping stone
• (Useful background) catalytic majorization / trumping and the classical probability-vector analogue

AMS categories

• ams-81 (Quantum theory)
• ams-15 (Linear and multilinear algebra; matrix theory)
• ams-94 (Information and communication theory)

Choose either option

• I plan on adding this conjecture to the repository
• This issue is up for grabs: I would like to see this conjecture added by somebody else

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