Designing an observer of an unknown quantum density operator is difficult because the operator must be Hermitian positive semidefinite with unit trace. In this paper, we derive a closed-form solution for projecting an arbitrary matrix onto the set of valid density operators. This allows us to design linear quantum state observers and retract the observer’s state back to this set while retaining the exponential convergence rate of the linear observer. The derived closed-form projection can be used alongside any quantum state estimation technique to produce a valid state estimate without increasing the state estimation error.
Designing an observer of an unknown quantum density operator is difficult because the operator must be Hermitian positive semidefinite with unit trace. In this paper, we derive a closed-form solution for projecting an arbitrary matrix onto the set of valid density operators. This allows us to design linear quantum state observers and retract the observer’s state back to this set while retaining the exponential convergence rate of the linear observer. The derived closed-form projection can be used alongside any quantum state estimation technique to produce a valid state estimate without increasing the state estimation error.
This paper considers the use of linear state observers to infer the unknown state (density matrix) of a closed quantum system using generalized quantum measurement—positive operator-valued measures (POVMs). An efficient test for observability is described in terms of two quantum observability matrices. The main contribution of this work is the development of a canonical quantum state observer, which may be utilized to estimate the state of any closed quantum system using POVM measurements. The observer ensures the state estimation error tends to the unobservable space of the system, which is exactly zero when the system is observable. The space of Hermitian matrices is shown to be invariant with respect to the dynamics of the canonical observer, which allows one to project the observer's state onto the space of valid quantum density operators while retaining convergence to the true state.
Time evolution of open quantum systems is governed by the master equation. The master equation, which is a matrix formalism, is the time derivative of the density matrix, which contains the complete information on the state of a quantum system. Instead of implementing successive measurements on repeated identically prepared systems, which are inevitably imperfect and can only be performed a limited number of times, a state estimator can be designed to obtain the whole information about the state of a quantum system represented in a density matrix. Trace-one and positive semi-definite properties of the density matrix arising from physical constraints have to be preserved during state estimation in quantum systems. To address this challenge, we present a projection technique that incorporates Dykstra’s algorithm and physical constraints into state estimation. This technique, which is an iterative method, ensures convergence and includes a designed oracle that projects the estimated state into intersections of admissible closed convex sets. The oracle structure is constructed using Hilbert projection, which looks for the best approximation of the projected estimated state within a Hilbert space into a closed convex set. According to the Hilbert projection theorem, this proposed oracle guarantees the existence and uniqueness of the best approximation of the projected state.
Abstract In this paper, the problem of constructing a measure which accepts values within a positive cone of bounded operators in a Hilbert space is considered. The assumption is made that a projection-valued function defined on a subset X0 of the original set X is initially given. The goal of this paper is to find such a scalar measure μ on set X as well as the extension of a projection-valued function from X0 to X, and, as a result, to obtain an operator-valued measure which has a projection-valued density relative to μ. In general, the solution to this problem is found for |X| = 4 and |X0| = 2. As an example, a function on X0 is considered which accepts values within a set of projections onto coherent states. For this case, the problem which concerns the informational completeness of the measurement determined by the constructed measure is investigated. In other words, the possibility is considered of restoring a quantum state (a positive unit-trace operator) based on trace values of the matrix formed from the product of a measure and a quantum state. In the case of the constructed measure, it is demonstrated that it is possible to restore a quantum state only if it represents a projection. A restriction imposed on the probability distribution is also found; with this restriction satisfied, the probability distribution can be obtained as a result of measuring a certain quantum state.
The density matrix ρ is defined and its basic properties are derived in detail: ρ is a Hermitian non-negative operator with unit trace, its eigenvalues are non-negative and lie in the closed interval [0, 1]; if ρ is a projector, it projects onto a one-dimensional sub-space, the trace of ρ 2 is ≤ 1, the equality holding only if ρ is a projector. A necessary and sufficient condition for ρ to be a projector is that all vectors in its definition are identical up to a phase. Pure cases are sets that can be described by a state vector, unlike mixtures . The density matrix of a mixture is not a projection operator, and this may be viewed as its distinguishing feature with respect to pure cases. The mean value of an observable can be reexpressed with the help of the density matrix, and a Schrödinger equation for density matrices holds. Last, but not least, pure cases cannot unitarily evolve into mixtures . The applications of the formalism presented in this chapter deal with orientation of spin-½ particles, polarization of pairs of photons, thermal-equilibrium states. In the second part, quantum entanglement is presented, with emphasis on the existence of state vectors for entangled pairs of particles that belong to the tensor product of Hilbert spaces H 1 ⊗ H 2 , but are not of the simple product form ψ 1 ⊗ ψ 2 . The only correct interpretation of such states has a statistical nature, associated to series of repeated measurements performed by two observers on the constituents of the entangled pair. Hidden-variable theories, with the associated Bell inequalities, are introduced, and a way of using entangled quantum states is described which makes it possible to transfer the polarization state of a photon to another photon. This property provides the foundation of the modern investigations of quantum teleportation.
Bayesian quantum estimation provides a robust formalism for quantum technologies, particularly in scenarios with limited data and minimal prior information. Yet, its application to continuous-variable systems has remained limited and largely numerical due to the difficulty of the underlying parameter integrals. Here, we introduce a variational framework that restricts the optimisation over all measurements and estimators to a finite-dimensional subspace, reducing the problem to a linear system of equations with closed-form solutions. These solutions have a geometric interpretation as orthogonal projections of the global optimum onto the chosen subspace. For Gaussian states, a natural subspace is spanned by operators polynomial in the canonical quadratures, whose moments are computable from the first and second moments of the state. We further derive a necessary and sufficient condition for global optimality, and show that post-processing with the posterior mean, rather than using the estimator obtained from the subspace, improves performance towards the global optimum. Through single-shot examples, we show that the framework yields experimentally feasible strategies based on Gaussian operations and quadrature measurements that are either optimal or near-optimal.
An essential quantity in quantum information theory is the von Neumann entropy which depends entirely on the quantum density operator. Once known, the density operator reveals the statistics of observables in a quantum process, and the corresponding von Neumann entropy yields the full information content. However, the state, or density operator, of a given system may be unknown. Quantum state observers have been proposed to infer the unknown state of a quantum system. In this note, we show (i) that the von Neumann entropy of the state estimate produced by our quantum state observer is exponentially convergent to that of the system’s true state, and (ii) the relative entropy between the system and observer’s state converges exponentially to zero as long as the system starts in a full-rank state.