Measuring energy by measuring any other observable

Abstract

We present a method to estimate the probabilities of outcomes of a quantum observable, its mean value, and higher moments by measuring any other observable. This method is general and can be applied to any quantum system. In the case of estimating the mean energy of an isolated system, the estimate can be further improved by measuring the other observable at different times. Intuitively, this method uses interplay and correlations between the measured observable, the estimated observable, and the state of the system. We provide two bounds: one that is looser but analytically computable and one that is tighter but requires solving a non-convex optimization problem. The method can be used to estimate expectation values and related quantities such as temperature and work in setups where performing measurements in a highly entangled basis is difficult, finding use in state-of-the-art quantum simulators. As a demonstration, we show that in Heisenberg and Ising models of ten sites in the localized phase, performing two-qubit measurements excludes 97.5% and 96.7% of the possible range of energies, respectively, when estimating the ground state energy.

7+9 pages, 6 figures, 3 tables. v2: significantly reworked structure and made it easier to read. Proofs simplified and moved to the main text. v3: title changed, cosmetic changes. Questions and comments are welcome

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Cited by in corpus (6)

• Inferring physical properties of symmetric states from the fewest copies
• Continuity bounds on observational entropy and measured relative entropies
• On the generic increase of observational entropy in isolated systems
• Observational entropy with general quantum priors
• Entropic partial orderings of quantum measurements
• Macroscopicity and observational deficit in states, operations, and correlations