Entanglement and the quantum brachistochrone problem

In the issue of quantum evolution, quantum evolution speed is usually quantified by the time rate of change of state distance between the initial sate and its time evolution. In this paper, the path distance of quantum evolution is introduced to study the evolution of a quantum system, through the approach combined with basic theory of quantum evolution and the linear algebra. In a quantum unitary system, the quantum evolution operator contains the path information of the quantum evolution, where the path distance is determined by the principal argument of the eigenvalues of the unitary operator. Accordingly, the instantaneous quantum evolution speed is proportional to the distance between the maximum and minimum eigenvalues of the Hamiltonian. As one of the applications, the path distance and the instantaneous quantum evolution speed could be used to form a new lower bound of the real evolution time, which depends on the evolution operator and Hamiltonian, and is independent of the initial state. It is found that the lower bound presented here is exactly equal to the real evolution time in the range <inline-formula><tex-math id="M2">\begin{document}$ \left[ {0, {\pi }/({{2{\omega _{\rm{H}}}}}}) \right]$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231009_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231009_M2.png"/></alternatives></inline-formula>. The tool of path distance and instantaneous quantum evolution speed introduced here provides new method for the related researches.

Controlling the evolution of photonic quantum states is crucial for most quantum information processing and metrology tasks. Due to its importance, many mechanisms of quantum state evolution have been tested in detail and are well understood; however, the fundamental phase anomaly of evolving waves, called the Gouy phase, has had a limited number of studies in the context of elementary quantum states of light, especially in the case of photon number states. Here we outline a simple method for calculating the quantum state evolution upon propagation and demonstrate experimentally how this quantum Gouy phase affects two-photon quantum states. Our results show that the increased phase sensitivity of multi-photon states also extends to this fundamental phase anomaly and has to be taken into account to fully understand the state evolution. We further demonstrate how the Gouy phase can be used as a tool for manipulating quantum states of any bosonic system in future quantum technologies, outline a possible application in quantum-enhanced sensing, and dispel a common misconception attributing the increased phase sensitivity of multi-photon quantum states solely to an effective de Broglie wavelength.

The quantum speed limit is a fundamental upper bound on the speed of quantum evolution. However, the actual mathematical expression of this fundamental limit depends on the choice of a measure of distinguishability of quantum states. We show that quantum speed limits are qualitatively governed by the Schatten-p-norm of the generator of quantum dynamics. Since computing Schatten-p-norms can be mathematically involved, we then develop an alternative approach in Wigner phase space. We find that the quantum speed limit in Wigner space is fully equivalent to expressions in density operator space, but that the new bound is significantly easier to compute. Our results are illustrated for the parametric harmonic oscillator and for quantum Brownian motion.

We present an operational reconstruction of the well-known two-to-one homomorphism between the groups SU(2) and SO(3), grounded in the physical description of quantum state preparation and evolution. Starting from the connection between vectors in three-dimensional physical space and quantum states of two-level systems, we investigate how reversible transformations–modeled as completely positive and trace-preserving maps–give rise to a correspondence between spatial rotations and unitary operations. Our approach reveals how this group-theoretic structure naturally emerges from physical constraints, particularly the preservation of purity and reversibility in quantum processes. Beyond its theoretical relevance, the construction offers a pedagogically accessible framework for introducing core ideas in quantum mechanics and symmetry groups, making the abstract correspondence between SU(2) and SO(3) tangible through experimentally meaningful procedures.

There exists an interesting relationship between entanglement and the time evolution of composite quantum systems: quantum entanglement enhances the ‘speed’ of evolution of certain quantum states, as measured by the time needed to reach an orthogonal state. Previous research done on this subject has been focused upon comparing extreme cases (highly entangled states versus separable states) or upon bi-partite systems. In the present contribution we explore the aforementioned connection (between entanglement and time evolution) in the cases of two-qubits and N-qubits systems, taking into account states of intermediate entanglement. In particular, we investigate a family of energetically symmetric states of low entanglement that saturate the quantum speed bound. We show that, as the number of qubits increases, very little entanglement is needed to reach the quantum speed limit.

The detection of microwave states is complicated by strong thermal noise, which is inevitably introduced by linear amplifiers. We show how to extract from measured data normally or anti-normally ordered moments of photon creation and annihilation operators, the set of which contains complete information on the quantum state of an electromagnetic field. Equations for the evolution of the quantum state are derived in terms of moments. Using this approach, we consider in detail issues of decoherence and thermalization of microwave quantum states. Results are illustrated using the examples of Fock, coherent, squeezed, thermal, and even and odd coherent states (Schrodinger cat states).

Time fractional evolution of a single quantum state and entangled state

Quantum gyroscopes have the advantages of high precision and drift-free, and are extensively applied in high accuracy spacecraft attitude control, but it is extremely sensitive to quantum noise, zero-bias drift and nonlinear errors under complex space conditions, which directly affect the accuracy of attitude solving. According to the well-established quantum gyro error models, including the random walk model and Markov error modeling, We propose a multi-channel error physics model, which integrates the non-Gaussian error modeling, multi-scale quantum state perturbation analysis and sensor-environment interaction coupling mechanism (MCEAM-AKF). In this section we use the error propagation equation to obtain the response matrix as influenced by noise under the evolution of quantum state, not only the noise responses of device linear part but also the nonlinear terms of state transition, which is the internal state decoherence phenomenon of the device. Then a model and augmented state based adaptive Kalman filter algorithm is designed, which integrates the time-varying error covariance estimation mechanism and the multi-layer scale observation update strategy and achieves the real-time dynamic correction of the attitude angle output error. Furthermore, external reference sources (e.g., star sensors) for closed-loop joint calibration accordingly to the movements to enhance the anti-interference ability and fast convergence performance are introduced in the model. Simulation results demonstrate that the average attitude angle error of MCEAM-AKF model can be decreased by $38.2 \%$ for different orbit and disturbance scenarios.

Chapter 8 - Cavity QED of quantum dots with dielectric microspheres

The restricted numerical range WR(A) of an operator A acting on a D-dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset R of the set of pure quantum states of dimension D. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of A—a normalized probability distribution on the complex plane supported in WR(A). Its value at point is equal to the probability that the expectation value 〈ψ|A|ψ〉 is equal to z, where |ψ〉 represents a random quantum state in subset R distributed according to the natural measure on this set, induced by the unitarily invariant Fubini–Study measure. Studying restricted shadows of operators of a fixed size D = NANB we analyse the geometry of sets of separable and maximally entangled states of the NA × NB composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow, we study the dynamics of quantum entanglement. A similar analysis extended for operators on D = 23-dimensional Hilbert space allows us to investigate the structure of the orbits of GHZ and W quantum states of a three-qubit system.

Describing the dynamics of strong-laser driven open quantum systems is a very challenging task that requires the solution of highly involved equations of motion. While machine learning techniques are being applied with some success to simulate the time evolution of individual quantum states, their use to approximate time-dependent operators (that can evolve various states) remains largely unexplored. In this work, we develop driven (NQP), a universal neural network framework that solves driven-dissipative quantum dynamics by approximating propagators rather than wave functions or density matrices. NQP can handle arbitrary initial quantum states, adapt to various external fields, and simulate long-time dynamics, even when trained on far shorter time windows. Furthermore, by appropriately configuring the external fields, our trained NQP can be transferred to systems governed by different Hamiltonians. We demonstrate the effectiveness of our approach by studying the spin-boson and the three-state transition Gamma models.

The entanglement evolution of multipartite quantum states under aspin environment is analyzed. Due to interaction, the extent towhich the entanglement vanishes depends not only on the size ofsystem and the structure of quantum states, but also on the exchangecouplings with environment. The decoherence-free subspaces have beenidentified by using the linear entropy.

The dynamics of Hermitian many-body quantum systems has long been a challenging subject due to the complexity induced by the particle-particle interactions. In contrast, this difficulty may be avoided in a well-designed non-Hermitian system. The exceptional point (EP) in a non-Hermitian system admits a peculiar dynamics: the final state being a particular eigenstate, coalescing state. In this work, we study the dynamic transition from a trivial insulating state to an $\ensuremath{\eta}$-pairing state in a composite non-Hermitian Hubbard system. The system consists of two subsystems, A and B, which are connected by unidirectional hoppings. We show that the dynamic transition from an insulating state to an $\ensuremath{\eta}$-pairing state occurs by the probability flow from A to B: the initial state is prepared as an insulating state of A, while B is left empty. The final state is an $\ensuremath{\eta}$-pairing state in B but empty in A. Analytical analyses and numerical simulations show that the speed of relaxation of the off-diagonal long-range order pair state depends on the order of the EP, which is determined by the number of pairs and the fidelity of the scheme is immune to the irregularity of the lattice.

The speed of evolution between perfectly distinguishable states is thoroughly analyzed in a closed three-level (qutrit) quantum system. Considering an evolution under an arbitrary time-independent Hamiltonian, we fully characterize the relevant parameters according to whether the corresponding quantum speed limit is given by the Mandelstam-Tamm, the Margolus-Levitin, or the Ness-Alberti-Sagi (dual) bound, thereby elucidating their hierarchy and relative importance. We revisit the necessary and sufficient conditions that guarantee the evolution of the initial state towards an orthogonal one in a finite time, and pay special attention to the full characterization of the speed of evolution, offering a speed map in parameter space that highlights regions associated with faster or slower dynamics. The general analysis is applied to concrete physical settings, particularly a pair of bosons governed by an extended Bose-Hubbard Hamiltonian, and a single particle in a triple-well potential. Our findings provide a framework to explore how the energetic resources and the initial configurations shape the pace of the dynamics in higher-dimensional systems.

We study the internal dynamics of an elementary quantum system placed close to a body held at a temperature different from that of the surrounding radiation. We derive general expressions for lifetime and density matrix valid for bodies of arbitrary geometry and dielectric permittivity. Out of equilibrium, the thermalization process and steady states become both qualitatively and quantitatively significantly different from the case of radiation at thermal equilibrium. For the case of a three-level atom close to a slab of finite thickness, we predict the occurrence of population inversion and an efficient cooling mechanism for the quantum system, whose effective internal temperature can be driven to values much lower than both involved temperatures. Our results show that non-equilibrium configurations provide new promising ways to control the state of an atomic system.