# A theorem about relative entropy of quantum states with an application to privacy in quantum communication - Quantum Physics

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Abstract: We prove the following theorem about relative entropy of quantum states.-Substate theorem: Let rho and sigma be quantum states in the same Hilbertspace with relative entropy S(rho|sigma) = Tr rho (log rho - log sigma) = c.Then for all epsilon > 0, there is a state rho- such that the trace distance||rho- rho|| t = Tr sqrt{(rho- rho)^2} <= epsilon, andrho-2^{O(c-epsilon^2)} <= sigma.-It states that if the relative entropy of rho and sigma is small, then thereis a state rho- close to rho, i.e. with small trace distance ||rho- rho|| t,that when scaled down by a factor 2^{O(c)} `sits inside-, or becomes a`substate- of, sigma. This result has several applications in quantumcommunication complexity and cryptography. Using the substate theorem, wederive a privacy trade-off for the set membership problem in the two-partyquantum communication model. Here Alice is given a subset A of n, Bob aninput i in n, and they need to determine if i in A.-Privacy trade-off for set membership: In any two-party quantum communicationprotocol for the set membership problem, if Bob reveals only k bits ofinformation about his input, then Alice must reveal at least n-2^{O(k)} bits ofinformation about her input.-We also discuss relationships between various information theoreticquantities that arise naturally in the context of the substate theorem.

Author: ** Rahul Jain, Jaikumar Radhakrishnan, Pranab Sen**

Source: https://arxiv.org/