# Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture

Henrik Wilming

Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany

### Abstract

I show that if a finite-dimensional density matrix has strictly smaller von Neumann entropy than a second one of the same dimension (and the rank is not bigger), then sufficiently (but finitely) many tensor-copies of the first density matrix majorize a density matrix whose single-body marginals are all exactly equal to the second density matrix. This implies an affirmative solution of the exact catalytic entropy conjecture (CEC) introduced by Boes et al. [PRL 122, 210402 (2019)]. Both the Lemma and the solution to the CEC transfer to the classical setting of finite-dimensional probability vectors (with permutations of entries instead of unitary transformations for the CEC).

The entropy of a quantum state or probability distribution is an important quantity in physics. Among others, its applications range from being a central quantity of interest in statistical mechanics and thermodynamics, over quantifying how much one can compress a signal in (quantum) information theory to quantifying the amount of entanglement contained in a quantum state. However, typically it attains its physical meaning only in an "asymptotic limit" where many weakly correlated copies of the state are available such as in a large, thermodynamic system or when long messages need to be compressed.

In the paper, a conjecture is solved in the affirmative which implies that one can think of entropy without an asymptotic limit. Instead it is asked when it is the case that a system's statistical state (density matrix) can be transformed to a different one using unitary dynamics if one has access to a finite auxiliary system whose statistical state must not change in the process. The auxiliary system is refereed to as catalyst, as it enables state-transitions otherwise impossible while not changing its own state. The results of the paper show that a system's state can be transformed from one state to another using a suitable catalyst if and only if the entropy increases (and the rank of the density matrix does not decrease).

### ► References

[1] Paul Boes, Jens Eisert, Rodrigo Gallego, Markus P. Müller, and Henrik Wilming. Von neumann entropy from unitarity''. Physical Review Letters 122, 210402 (2019).
https:/​/​doi.org/​10.1103/​physrevlett.122.210402

[2] H. Wilming. Entropy and reversible catalysis''. Physical Review Letters 127, 260402 (2021).
https:/​/​doi.org/​10.1103/​physrevlett.127.260402

[3] Runyao Duan, Yuan Feng, Xin Li, and Mingsheng Ying. Multiple-copy entanglement transformation and entanglement catalysis''. Phys. Rev. A 71, 042319 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.042319

[4] Yuan Feng, Runyao Duan, and Mingsheng Ying. Relation between catalyst-assisted transformation and multiple-copy transformation for bipartite pure states''. Physical Review A 74, 042312 (2006).
https:/​/​doi.org/​10.1103/​physreva.74.042312

[5] Naoto Shiraishi and Takahiro Sagawa. Quantum thermodynamics of correlated-catalytic state conversion at small scale''. Physical Review Letters 126, 150502 (2021).
https:/​/​doi.org/​10.1103/​physrevlett.126.150502

[6] Rajendra Bhatia. Matrix analysis''. Springer New York. (1997).
https:/​/​doi.org/​10.1007/​978-1-4612-0653-8

[7] Albert W. Marshall, Ingram. Olkin, and Barry C. Arnold. Inequalities : theory of majorization and its applications''. Springer Science+Business Media, LLC. (2011).
https:/​/​doi.org/​10.1007/​978-0-387-68276-1

[8] Markus P. Müller. Correlating thermal machines and the second law at the nanoscale''. Physical Review X 8, 041051 (2018).
https:/​/​doi.org/​10.1103/​physrevx.8.041051

[9] Tulja Varun Kondra, Chandan Datta, and Alexander Streltsov. Catalytic transformations of pure entangled states''. Physical Review Letters 127, 150503 (2021).
https:/​/​doi.org/​10.1103/​physrevlett.127.150503

[10] Patryk Lipka-Bartosik and Paul Skrzypczyk. Catalytic quantum teleportation''. Physical Review Letters 127, 080502 (2021).
https:/​/​doi.org/​10.1103/​physrevlett.127.080502

[11] Roberto Rubboli and Marco Tomamichel. Fundamental limits on correlated catalytic state transformations''. Physical Review Letters 129, 120506 (2022).
https:/​/​doi.org/​10.1103/​physrevlett.129.120506

[12] Soorya Rethinasamy and Mark M. Wilde. Relative entropy and catalytic relative majorization''. Physical Review Research 2, 033455 (2020).
https:/​/​doi.org/​10.1103/​physrevresearch.2.033455

[13] Paul Boes, Nelly H.Y. Ng, and Henrik Wilming. Variance of relative surprisal as single-shot quantifier''. PRX Quantum 3, 010325 (2022).
https:/​/​doi.org/​10.1103/​prxquantum.3.010325

[14] Vjosa Blakaj and Michael M. Wolf. Transcendental properties of entropy-constrained sets'' (2021). arXiv:2111.10363.
arXiv:2111.10363

[15] R. Renner. Security of Quantum Key Distribution''. PhD thesis. ETH Zurich. (2005).

[16] Marco Tomamichel. Quantum information processing with finite resources''. Springer International Publishing. (2016).
https:/​/​doi.org/​10.1007/​978-3-319-21891-5

[17] T Holenstein and R Renner. On the randomness of independent experiments''. IEEE Transactions on Information Theory 57, 1865–1871 (2011).
https:/​/​doi.org/​10.1109/​tit.2011.2110230

[18] Noah Linden, Milán Mosonyi, and Andreas Winter. The structure of rényi entropic inequalities''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, 20120737 (2013).
https:/​/​doi.org/​10.1098/​rspa.2012.0737

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