title: Quantum State Smoothing: General Properties and Applications to Linear Gaussian Systems
Creators: name: Laverick, Kiarn T
Publisher: name: Griffith University
Year: 2022
Subjects: subject: quantum state smoothing theory
subject: cost function
subject: sufficient condition
subject: linear Gaussian quantum (LGQ) systems

description: Filtering and smoothing are classical estimation techniques that provide an estimate of the state of a classical system based on measurement information in the past (prior to the estimation time) and the past-future (prior and posterior to the estimation time), respectively. However, with the advent of quantum technologies, the need to estimate the state of an individual quantum system has also arisen. While the ﬁltering technique is easily generalized and applied to quantum systems, it was not so simple for the smoothing technique. Applying the direct quantum analog of the smoothing theory often leads to an estimate that is unphysical, indicating that the classical theory is incompatible with quantum systems. The reason for this incompatibility is that the operators describing the future measurements on the system, called the retroﬁltered eﬀect, and the state conditioned on the past measurement, i.e., the ﬁltered state, do not necessarily commute. One way to solve this issue is the quantum state smoothing theory. In order to deal with past and future information, the theory introduces the concept of a hidden measurement record, gathered by a secondary observer, say Bob, in order to deﬁne the true quantum state, a state containing maximal information about the system. With this concept of a true state, it is then possible to construct a valid smoothed quantum state, that is, a state conditioned on a past-future measurement record. In this thesis I delve into the quantum state smoothing theory. I begin by reformu-lating the quantum state smoothing theory as an optimal estimation problem, that is, minimizing a particular expected cost function. I show that the smoothed state is the optimal estimator for two cost functions, the trace-square deviation from and relative entropy with the true state. Additionally, I show, for a closely related cost function, the linear inﬁdelity, that the smoothed state is suboptimal, while the pure state corre-sponding to the largest eigenvalue of the smoothed state is optimal. I then investigate under what conditions the smoothed quantum state reduces to a classically smoothed state, ﬁnding a suﬃcient condition. This suﬃcient condition requires the true state to be described probabilistically in a ﬁxed basis. Subsequently, in an attempt to remove some of the restrictions on how Bob measures the system, I hypothesize a weaker suﬃ-cient condition of only requiring the ﬁltered state and retroﬁltered eﬀect to be described probabilistically in a ﬁxed basis. This hypothesis is disproven with a counter example. The remainder of the thesis is dedicated to a particular class of quantum system, the linear Gaussian quantum (LGQ) systems. I apply the quantum state smoothing theory to the LGQ systems, obtaining closed-form expressions for the smoothed quantum state.These closed-form expressions allow for numerous properties of the smoothed quantum state to be determined that would otherwise be arduous to even verify in the general setting. In particular, I investigate the behaviour of the smoothed quantum state in the low and high measurement eﬃciency limit. Furthermore, I derive a necessary and suﬃcient condition on the true state of the system that, in the event that Bob’s measurement is unknown, restricts the possible true states of the system based on the observer’s, say Alice’s, measurement choice. From the dynamical form of the LGQ state smoothing equations, I derive a necessary and suﬃcient condition for diﬀerentiable evolution of the smoothed quantum state. Lastly, I investigate the optimal measurement strategy for Alice and Bob in order to maximize the relative purity recovery. I pose three hypotheses and test the validity of each against two physical systems. One of these hypotheses provides an approximately optimal solution. To further verify this hypothesis, I generalize the hypothesis to qubit systems and test against an example system, again verifying the hypothesis.