We study asymptotically optimal statistical inference concerning the unknown state of $N$ identical quantum systems, using two complementary approaches: a “poor man’s approach” based on the van Trees inequality, and a rather more sophisticated approach using the recently developed quantum form of LeCam’s theory of Local Asymptotic Normality. We study asymptotically optimal statistical inference concerning the unknown state of $N$ identical quantum systems, using two complementary approaches: a “poor man’s approach” based on the van Trees inequality, and a rather more sophisticated approach using the recently developed quantum form of LeCam’s theory of Local Asymptotic Normality.

arXiv:1112.2078Institute of Mathematical Statistics collectionsJan 1, 201384 citations

Statistical Inference in Quantum Mechanics

Jean-Paul Marchand

Springer eBooksJan 1, 19816 citations

Identification of Causal Influences in Quantum Processes

Isaac Friend, Aleks Kissinger

Though the topic of causal inference is typically considered in the context of classical statistical models, recent years have seen great interest in extending causal inference techniques to quantum and generalized theories. Causal identification is a type of causal inference problem concerned with recovering from observational data and qualitative assumptions the causal mechanisms generating the data, and hence the effects of hypothetical interventions. A major obstacle to a theory of causal identification in the quantum setting is the question of what should play the role of "observational data," as any means of extracting data at a certain locus will almost certainly disturb the system. Hence, one might think a priori that quantum measurements are already too much like interventions, so that the problem of causal identification trivializes. This is not the case. Fixing a limited class of quantum instruments (namely the class of all projective measurements) to play the role of "observations," we note that as in the classical setting, there exist scenarios for which causal identification is not possible. We then present sufficient conditions for quantum causal identification, starting with a quantum analogue of the well-known "front-door criterion" and finishing with a broader class of scenarios for which the effect of a single intervention is identifiable. These results emerge from generalizing the process-theoretic account of classical causal inference due to Jacobs, Kissinger, and Zanasi beyond the setting of Markov categories, and thereby treating the classical and quantum problems uniformly.Though the topic of causal inference is typically considered in the context of classical statistical models, recent years have seen great interest in extending causal inference techniques to quantum and generalized theories. Causal identification is a type of causal inference problem concerned with recovering from observational data and qualitative assumptions the causal mechanisms generating the data, and hence the effects of hypothetical interventions. A major obstacle to a theory of causal identification in the quantum setting is the question of what should play the role of "observational data," as any means of extracting data at a certain locus will almost certainly disturb the system. Hence, one might think a priori that quantum measurements are already too much like interventions, so that the problem of causal identification trivializes. This is not the case. Fixing a limited class of quantum instruments (namely the class of all projective measurements) to play the role of "observations," we note that as in the classical setting, there exist scenarios for which causal identification is not possible. We then present sufficient conditions for quantum causal identification, starting with a quantum analogue of the well-known "front-door criterion" and finishing with a broader class of scenarios for which the effect of a single intervention is identifiable. These results emerge from generalizing the process-theoretic account of classical causal inference due to Jacobs, Kissinger, and Zanasi beyond the setting of Markov categories, and thereby treating the classical and quantum problems uniformly.

arXiv:2311.08882Electronic Proceedings in Theoretical Computer ScienceNov 14, 20230 citations

Statistical inference in quantum mechanics

Stanley Gudder

Reports on Mathematical PhysicsApr 1, 19804 citations

We study asymptotically optimal statistical inference concerning the unknown state of

N

identical quantum systems, using two complementary approaches: a “poor man’s approach” based on the van Trees inequality, and a rather more sophisticated approach using the recently developed quantum form of LeCam’s theory of Local Asymptotic Normality.