Von Neumann Entropy from Unitarity

The von Neumann entropy is a key quantity in quantum information theory and, roughly speaking, quantifies the amount of quantum information contained in a state when many identical and independent (i.i.d.) copies of the state are available, in a regime that is often referred to as being asymptotic. In this Letter, we provide a new operational characterization of the von Neumann entropy which neither requires an i.i.d. limit nor any explicit randomness. We do so by showing that the von Neumann entropy fully characterizes single-shot state transitions in unitary quantum mechanics, as long as one has access to a catalyst—an ancillary system that can be reused after the transition—and an environment which has the effect of dephasing in a preferred basis. Building upon these insights, we formulate and provide evidence for the catalytic entropy conjecture, which states that the above result holds true even in the absence of decoherence. If true, this would prove an intimate connection between single-shot state transitions in unitary quantum mechanics and the von Neumann entropy. Our results add significant support to recent insights that, contrary to common wisdom, the standard von Neumann entropy also characterizes single-shot situations and opens up the possibility for operational single-shot interpretations of other standard entropic quantities. We discuss implications of these insights to readings of the third law of quantum thermodynamics and hint at potentially profound implications to holography.

Figure 1

Reusability of the auxiliary system for further transitions. Consider $N$ subsystems in an uncorrelated state ${\rho }_{A_1,\dots ,A_N}={\rho }^{\otimes N}$. Because of Theorem 1, for any transition ${\rho }_{A_1}\to {\rho }_{A_1}^{\prime }$ respecting the entropy and rank condition, it is possible to find an auxiliary system in state $\sigma$ that enables this transition. When brought back in contact with the environment, it dephases and returns to its initial state while establishing correlations with $A_1$. In spite of these correlations, it is reusable to implement the same transition on $A_2$. This is true, since in the second step one applies a local operation on $A_2$ and the auxiliary system, whose outcome is independent of the correlations with $A_1$. Repeating this process on the $N$ subsystems results in having performed locally transitions $\{{\rho }_{A_i}\to {\rho }_{A_i}^{\prime }{\}}_{i=1,\dots ,N}$ while using a single auxiliary system. At the end of the process, all the subsystems are possibly correlated. However, these correlations do not play any role if one intends to use each subsystem $A_i$ independently for further thermodynamic or information protocols, as is generally the case in a single-shot setting.

Figure 2

Comparison of various settings and results. Top: State transitions implementable using a source of randomness and an uncorrelated catalyst $\sigma$ are characterized by the trumping relations. Middle top: State transitions allowing for a source of randomness and a correlated catalyst, an auxiliary system that locally remains unchanged, are characterized by entropy and rank [18]. Middle bottom: By Theorem 1, state transitions using a correlated catalyst and a dephasing environment that acts on the catalyst (dashed boundary) are also characterized by entropy and rank. Bottom: State transitions using a correlated catalyst alone are characterized by entropy and rank. This is the content of Conjecture 1.