# Catalytic entropy conjecture

Cite this problem as Problem 45.

### Problem

Consider a density matrix on a finite-dimensional system . Let be the von Neumann entropy. Prove or disprove that for any density matrix on such that and for any there exists a finite-dimensional system (“catalyst”) with state and a unitary operator on such that the following holds: If appropriate and can be found, we say that the catalytic transition is possible.

### Background

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 , it was shown that:

1. Weak solution 1: The statement is true for state-transitions of the form where is a sufficiently large, -dependent dimension of a further auxiliary system.
2. Quasi-unique monotone: Since is always possible using 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 that has the same diagonal as in the eigenbasis of instead of .

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

Let and . Then there exists a finite-dimensional state such that , where denotes a bipartite quantum state with local marginals and and 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.

### References

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

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

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

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

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

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