Delocalized and dynamical catalytic randomness and information flow

We generalize the theory of catalytic quantum randomness to distributed and dynamical settings. First, we expand the theory of catalytic quantum randomness by calculating the amount of (Rényi) entropy catalytically extractable from a distributed or dynamical randomness source. We show that no entropy can be catalytically extracted when one cannot implement local projective measurement on randomness source without altering its state. As an application, we prove that quantum operation cannot be hidden in correlation between two parties without using randomness, which is the dynamical generalization of the no-hiding theorem. Moreover, the formalism of distributed catalysis is applied to develop a formal definition of semantic quantum information and it follows that utilizing semantic information is equivalent to catalysis using a catalyst already correlated with the transforming system. By doing so, we unify the utilization of semantic and nonsemantic quantum information and conclude that one can always extract more information from an incompletely depleted classical randomness source, but it is not possible for quantum randomness sources.

Figure 1

A book is a randomness (equivalently an information) source, but not every usage of it is pure randomness utilization. For example, it is hard to say that burning a book utilizes only the randomness of the book, as it leaves evidently detectable physical traces on it. Intuitively, it is clear that any usage of a book that necessitates non-negligible physical alternation of the book is not a pure information utilization. Therefore, we claim that (pure) randomness utilization must not leave any locally detectable statistical change on the randomness source.

Figure 2

Catalytic decomposition of a density matrix. A superselection rule forbids between subspaces called superselection sectors, and each density matrix has an eigenspace for each distinct eigenvalue. The intersection of a superselection sector and an eigenspace is called a catalysis sector and it plays an important role in calculating the catalytic entropies.

Figure 3

Distributed catalysis of quantum randomness. Alice and Bob, separated in different laboratories, utilize the bipartite state ${\sigma }_{A_C{B}_C}$ as a catalyst to transform ${\rho }_{A_S{B}_S}$ into ${\rho }_{A_S{B}_S}^{\prime }=\mathcal{N}\left({\rho }_{A_S{B}_S}\right)$. On the right side, $A_S{B}_S$ and $A_C{B}_C$ could be correlated (depicted as a colored box between them) but the marginal state of $A_C{B}_C$ stays in the initial state ${\sigma }_{A_C{B}_C}$.

Figure 4

Dynamical catalysis of quantum randomness. Input and output systems of the target quantum channel $\mathcal{N}$ and the catalyst channel $\mathcal{R}$ unitarily interact. The resultant bipartite channel on $SC$ might correlate two systems, but the catalyst channel $\mathcal{R}$ stays in its original form when one ignores the system $S$.

Figure 5

Semantic information utilization demonstrated in Landauer's erasure principle. The information carrier $C$, depicted as a piece of paper, provides the information about the state of the gas molecule $G$. However, in the course of the interaction, no information flows from $G$ to $C$. Here, the fact that two systems $C$ and $G$ start as correlated systems is important for semantic information utilization.

Figure 6

Suppose that input and output systems of a given quantum operation are reversibly distributed to two systems. Is it possible to hide the identity of the operation from the respective systems? In other words, is it possible to implement quantum operations stealthily? The no-stealth theorem says that it is impossible.

Figure 7

Comparison of three types of catalysis of quantum randomness. Randomness represented by dice enters the interaction and leaves it locally unchanged but correlated with the system. On the other hand, distributed catalysis and dynamical catalysis of randomness are intimately related; rotating one diagram by ${90}^{\circ }$ makes it very similar to the other one.

Figure 8

Comparison of convex and concave resource theories. In a convex resource theory, a statistical mixture of two free objects is still free, and the action of free operation can only draw a resourceful object closer to the set of free objects. However, in a “concave” resource theory, any statistical mixture of two resourceful objects is resourceful, and there is no universal “resource destroying operation.” However, there are resourceful operations that never make a resourceful object free.