Abstract
Herein we study the problem of recovering a density operator from a set of compatible marginals, motivated by limitations of physical observations. Given that the set of compatible density operators is not singular, we adopt Jaynes’ principle and wish to characterize a compatible density operator with maximum entropy. We first show that comparing the entropy of compatible density operators is complete for the quantum computational complexity class QSZK, even for the simplest case of 3-chains. Then, we focus on the particular case of quantum Markov chains and trees and establish that for these cases, there exists a procedure polynomial in the number of subsystems that constructs the maximum entropy compatible density operator. Moreover, we extend the Chow–Liu algorithm to the same subclass of quantum states.
Highlights
Quantum tomography [1] allows us to associate a unique quantum state over a finite-dimensional Hilbert space provided that multiple copies of the quantum system are available, together with a complete set of measurements
Given that the general problem of finding the maximum entropy state is hard, we focus on a well-behaved subset of density operators, namely quantum Markov trees (QMT)—
We are able to show that for QMTs, the MECM problem is in P and that there is a polynomial quantum circuit that constructs the Maximum entropy compatible density operator
Summary
Quantum tomography [1] allows us to associate a unique quantum state over a finite-dimensional Hilbert space provided that multiple copies of the quantum system are available, together with a complete set of measurements. The quantum marginal problem [11,12], that consists of determining whether a set of marginal quantum states has a global density operator compatible with them, and for which a solution is known just in some particular cases [13,14,15]. We consider quantum Markov trees, states for which each 3-subchains form a quantum Markov chain [28] In this case, we show that the maximum entropy compatible problem is in P, and that there exists a polynomial-time quantum circuit that constructs the maximal entropy compatible state. We show that the maximum entropy compatible problem is in P, and that there exists a polynomial-time quantum circuit that constructs the maximal entropy compatible state We use this result to extend the Chow–Liu algorithm [19].
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