Tutte seminar - Ashwin Nayak

The Quantum Substate Theorem

Abstract:

Consider quantum states ρ, σ in the same finite dimensional Hilbert space. We say that ρ is a c-substate of σ if ρ &le 2^c σ, where &le represents the Lowner partial order. This may be used to construct the state ρ from σ, much like we may generate fair coin tosses from biased ones through rejection sampling. The success probability of the process is then 1/2^c. 

We are interested in how well σ simulates ρ in the above sense. In other words, we are interested in the least c such that ρ is a c-substate of σ. This quantity is the relative min-entropy of the two states. For typical applications, such as privacy trade-offs in communication protocols, it suffices to construct an approximation ρ' to ρ, with respect to a metric on quantum states. This leads us to the notion of the smooth relative min-entropy of the two states. 

If ρ is a c-substate of σ, i.e., their relative min-entropy is at most c, then their relative entropy is also at most c. Jain, Radhakrishnan and Sen (2002) gave a weak converse to this relation via the Quantum Substate Theorem. This gives a bound on the eps-smooth relative min-entropy in terms of the more familiar notion of relative entropy. We present alternative proofs of the Quantum Substate Theorem, also strengthening it in the process. The proofs that we present are both shorter and conceptually simpler than the original proof.



This is joint work with Rahul Jain, CQT and NUS, Singapore.