A theorem about relative entropy of quantum states with an application to privacy in quantum communication
| dc.creator | Jain, Rahul | |
| dc.creator | Radhakrishnan, Jaikumar | |
| dc.creator | Sen, Pranab | |
| dc.date | 2007-05-16 | |
| dc.date.accessioned | 2026-07-07T08:01:58Z | |
| dc.date.available | 2026-07-07T08:01:58Z | |
| dc.description | We prove the following theorem about relative entropy of quantum states. "Substate theorem: Let rho and sigma be quantum states in the same Hilbert space 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, and rho'/2^{O(c/epsilon^2)} <= sigma." It states that if the relative entropy of rho and sigma is small, then there is 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 quantum communication complexity and cryptography. Using the substate theorem, we derive a privacy trade-off for the set membership problem in the two-party quantum communication model. Here Alice is given a subset A of [n], Bob an input i in [n], and they need to determine if i in A. "Privacy trade-off for set membership: In any two-party quantum communication protocol for the set membership problem, if Bob reveals only k bits of information about his input, then Alice must reveal at least n/2^{O(k)} bits of information about her input." We also discuss relationships between various information theoretic quantities that arise naturally in the context of the substate theorem. | |
| dc.description | 27 pages, no figures, full version | |
| dc.identifier | https://arxiv.org/abs/0705.2437 | |
| dc.identifier | http://arxiv.org/abs/0705.2437 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/129083 | |
| dc.subject | Quantum Physics | |
| dc.title | A theorem about relative entropy of quantum states with an application to privacy in quantum communication | |
| dc.type | text |