Catalytic entropy conjecture

Cite this problem as Problem 45.


Consider a density matrix \rho on a finite-dimensional system S. Let H be the von Neumann entropy. Prove or disprove that for any density matrix \rho' on S such that H(\rho')\geq H(\rho) and for any \epsilon>0 there exists a finite-dimensional system C (“catalyst”) with state \sigma_C and a unitary operator U on SC such that the following holds:

\| tr_C[U \rho\otimes \sigma_C U^\dagger] - \rho' \|_1 \leq \epsilon, tr_S[U\rho\otimes \sigma_C U^\dagger] = \sigma_C.

If appropriate \sigma and U can be found, we say that the catalytic transition \rho\rightarrow \rho' is possible.


It is general wisdom in quantum information theory that operational characterisations of standard entropic quantities, like von Neumann entropy, require an i.i.d. limit, while single-shot settings are characterized by (smoothed) Rényi entropies. This is made explicit in resource theories, such as those of entanglement, informational non-equilibrium or quantum thermodynamics.
Catalysts have played an important role in these resource theories and can significantly simplify the task of characterizing possible state-transitions (usually in terms of Rényi entropies or divergences, see solutions by Klimesh and Turgut [1, 2] to Problem 4 as well as [3, 4].).
A positive solution to this problem would provide an operational single-shot characterization of von Neumann entropy in terms of the catalytic transitions defined in the problem statement.

Partial results

In [5], it was shown that:

  1. Weak solution 1: The statement is true for state-transitions of the form \rho\otimes {\mathbb I}/d \rightarrow \rho'\otimes {\mathbb I}/d where d is a sufficiently large, (\rho,\rho')-dependent dimension of a further auxiliary system.
  2. Quasi-unique monotone: Since \rho_{12} \mapsto \rho_1\otimes \rho_2 is always possible using \sigma_C=\rho_2 and a SWAP unitary, any additive monotone of catalytic transitions has to be sub-additive. This can be used to show that von Neumann entropy is the only continuous and additive monotone (up to constants).
  3. Weak solution 2: The statement is true if we assume that the catalyst is returned in a state \hat \sigma_C that has the same diagonal as \sigma_C in the eigenbasis of \sigma_C instead of \hat\sigma_C=\sigma_C.

The above results rely on the following result, which was shown in [6]:

Let H(\rho')>H(\rho) and \mathrm{rank}(\rho')\geq \mathrm{rank}(\rho). Then there exists a finite-dimensional state \sigma such that \rho\otimes \sigma \succ \rho'\sigma, where \rho'\sigma denotes a bipartite quantum state with local marginals \rho' and \sigma and \succ denotes majorization.

Note that this implies that the problem statement is true if one replaces the unitary quantum channel with a random unitary quantum channel. Thus, while it provides a single-shot characterization of von Neumann entropy, it relies on an external source of randomness, in contrast to the problem statement above.


[1] S. Turgut, J. Phys. A 40, 12185 (2007).

[2] M. Klimesh, “Inequalities that collectively completely characterize the catalytic majorization relation,” (2007), arXiv:0709.3680v1.

[3] F. G. S. L. Brandao, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner, PNAS 112, 3275 (2015).

[4] G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N. Yunger Halpern, Phys. Rep. 583, 1 (2015).

[5] P. Boes, J. Eisert, R. Gallego, M. P. Müller, and H. Wilming, Physical Review Letters 122 (2019).

[6] M. P. Müller, Physical Review X 8 (2018).