Distributed Private Randomness Distillation

We develop the resource theory of private randomness extraction in the distributed and device-dependent scenario. We begin by introducing the notion of independent random bits, which are bipartite states containing ideal private randomness for each party, and motivate the natural set of free operations. As a conceptual tool, we introduce virtual quantum state merging, which is essentially the flip side of quantum state merging, without communication. We focus on the bipartite case and find the rate regions achievable in different settings. Surprisingly, it turns out that local noise can boost randomness extraction. As a consequence of our analysis, we resolve a long-standing problem by giving an operational interpretation for the reverse coherent information (up to a constant term $\log d$) as the number of private random bits obtained by sending quantum states from one honest party (server) to another one (client) via the eavesdropped quantum channel.

Figure 1

The rate regions of $(R_A,R_B)$ when $S(A|B)>0>S(A)-\log |A|>S(B|A)$. The green lines show the rate region of setting (1), the purple lines of setting (2), the red lines of setting (3), and the blue lines of setting (4).