Catalytic entropy conjecture – Open Quantum Problems

Cite this problem as Problem 45.

Problem

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 $H(\rho') > H(\rho)$ and $\operatorname{rank}(\rho')\geq\operatorname{rank}(\rho)$ , there exists a finite-dimensional system $C$ (“catalyst”) with state $\sigma_C$ and a unitary operator $U$ on $SC$ such that the following holds:

$tr_C[U \rho\otimes \sigma_C U^\dagger] = \rho', \quad tr_S[U\rho\otimes \sigma_C U^\dagger] = \sigma_C.$

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

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.

The problem was originally proposed in [5], where partial solutions were provided for transformations of the form $\rho\otimes {\mathbb I}/d \rightarrow \rho'\otimes {\mathbb I}/d$ and under additional assumptions on the catalyst system.

Solution

First, a solution was given in [6] to an approximate variant of the conjecture, where the condition $tr_C[U \rho\otimes \sigma_C U^\dagger] = \rho'$ is replaced with the requirement that $\| tr_C[U \rho\otimes \sigma_C U^\dagger] - \rho' \|_1 \leq \epsilon$ for arbitrary $\epsilon > 0$ . The desired catalytic transformation is shown to exist through a construction of the catalyst dating back to [7].

The argument was then extended to the exact catalytic entropy conjecture in [6], giving an affirmative solution to the conjecture and closing Problem 45.

Of note is the fact that the case when $H(\rho') = H(\rho)$ remains open — it is out of scope of the conjecture, as such a transformation cannot be implemented with a finite-dimensional catalyst, but it is possible that a suitable infinite-dimensional catalyst may exist.

References

[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] H. Wilming, Phys. Rev. Lett. 127, 260402 (2021).

[7] R. Duan, Y. Feng, X. Li, and M. Ying, Phys. Rev. A 71, 042319 (2005).

[8] H. Wilming, Quantum 6, 858 (2022).