Southwest Quantum Information and Technology

Decay of quantum conditional mutual information in infinite uniform matrix product states.

Presenting Author: Pavel Svetlichnyy, Georgia Institute of Technology
Contributing Author(s): Shivan Mittal, T.A.B. Kennedy

The quantum conditional mutual information (QCMI), defined by $I(A:C|B)=S(AB)+S(BC)-S(B)-S(ABC)$, where $S(X)$ is the von Neumann entropy of the reduced density operator on $X$, is associated to the problem of state preparation by layers of quantum channels (quantum circuits of finite depth), also known as the quantum Markov property. We investigate the decay of QCMI for the reduced density operators of infinite uniform matrix product states (uMPS). We consider reduced density operators on the domain consisting of three consecutive contiguous regions, $A$, $B$, and $C$. We show that the QCMI is bounded by the function $C\exp(-q|B|+c\ln|B|)$, where $C$, $c$, and $q$ are constants, and $|B|$ is the size of the region $B$, in the asymptotic limit of large $|B|$. Notice that for uMPS QCMI converges to zero in general, unlike quantum correlations and quantum mutual information, for which the assumption of uMPS to be injective is necessary. It is known that for injective uMPS, the decay rate of correlations and quantum mutual information is determined by the spectral gap ($\nu_{\mathrm{gap}}$) of the transfer matrix. We prove that for any uMPS, not necessarily injective, the bound on the asymptotic decay rate of QCMI is $q>\ln{\nu_{\mathrm{gap}}^{-1}}/2$. Our numerical study of example cases suggests, that while the scaling of the decay rate with $\ln{\nu_{\mathrm{gap}}^{-1}}$ is correct, a stricter bound $q>2\ln{\nu_{\mathrm{gap}}^{-1}}$ may hold.

(Session 5 : Thursday from 12:00pm-2:00 pm)