Correlations In Typicality And An Affirmative Solution To The Exact Catalytic Entropy Conjecture

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Henrik Wilming

Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany

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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).

Featured image: Three conditions on a pair of finite-dimensional density matrices $(rho,rho’)$ are equivalent. Top: Entropy is strictly increasing and rank non-decreasing. Middle: There exists a finite-dimensional auxiliary system in state $sigma$ and a joint-unitary $U$ such that the transformed state $Urhootimes sigma U^dagger$ has reduced density matrices $rho’$ and $sigma$. This provides a single-shot characterization of entropy. Bottom: There exists a finite number $n$ such that $n$ copies of $rho$ majorize a state whose marginals all exactly coincide with $rho’$.

Popular summary

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).

► BibTeX data

@article{Wilming2022correlationsin, doi = {10.22331/q-2022-11-10-858}, url = {https://doi.org/10.22331/q-2022-11-10-858}, title = {Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture}, author = {Wilming, Henrik}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {6}, pages = {858}, month = nov, year = {2022} }

► References

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