The problem of geometric phase for an open quantum system is reinvestigated in a unifying approach. Two of existing methods to define geometric phase, one by Uhlmann's approach and the other by kinematic approach, which have been considered to be distinct, are shown to be related in this framework. The method is based upon purification of a density matrix by its uniform decomposition and a generalization of the parallel transport condition obtained from this decomposition. It is shown that the generalized parallel transport condition can be satisfied when Uhlmann's condition holds. However, it does not mean that all solutions of the generalized parallel transport condition are compatible with those of Uhlmann's one. It is also shown how to recover the earlier known definitions of geometric phase as well as how to generalize them when degeneracy exists and varies in time.
A kinematic approach to the geometric phase for mixed quantal states in nonunitary evolution is proposed. This phase is manifestly gauge invariant and can be experimentally tested in interferometry. It leads to well-known results when the evolution is unitary.
Practical implementations of quantum computing are always done in the presence of decoherence. Geometric phase is useful in the context of quantum computing as a tool to achieve fault tolerance. Recent experimental progresses on coherent control of single electron have suggested that electron in quantum dot systems is a promising candidate of qubit in future quantum information processing devices. In this paper, by considering a feasible quantum dot model, we calculate the geometric phase of the quantum dot system in nonunitary evolution and investigate the effect of environment parameters on …
Geometric phases, arising from cyclic evolutions in a curved parameter space, appear in a wealth of physical settings. Recently, and largely motivated by the need of an experimentally realistic definition for quantum computing applications, the quantum geometric phase was generalized to open systems. The definition takes a kinematical approach, with an initial state that is evolved cyclically but coupled to an environment --- leading to a correction of the geometric phase with respect to the uncoupled case. We obtain this correction by measuring the nonunitary evolution of the reduced density …
We describe the decoherence process induced on a two-level quantum system in direct interaction with a non-equilibrium environment. The non-equilibrium feature is represented by a non-stationary random function corresponding to the fluctuating transition frequency between two quantum states coupled to the surroundings. In this framework, we compute the decoherence factors which have a characteristic "dip" related to the initial phases of the bath modes. We therefore study different types of environments, namely ohmic and supra-ohmic. These environments present different decoherence time-scales…
We calculate the geometric phase for an open system (spin-boson model) which interacts with an environment (ohmic or nonohmic) at arbitrary temperature. However there have been many assumptions about the time scale at which the geometric phase can be measured, there has been no reported observation yet for mixed states under nonunitary evolution. We study not only how they are corrected by the presence of the different type of environments but also estimate the corresponding times at which decoherence becomes effective. These estimations should be taken into account when planning experimental …
In a differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and momentum operators or quadrature components. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state, unitarily evolving according to a Hamiltonian $\hat{H}$. We also present the corresponding differential equations which describe the nonun…
We first discuss the geometrical construction and the main mathematical features of the maximum-entropy-production/steepest-entropy-ascent nonlinear evolution equation proposed long ago by this author in the framework of a fully quantum theory of irreversibility and thermodynamics for a single isolated or adiabatic particle, qubit, or qudit, and recently rediscovered by other authors. The nonlinear equation generates a dynamical group, not just a semigroup, providing a deterministic description of irreversible conservative relaxation towards equilibrium from any non-equilibrium density operato…
The metric underlying the mixed state geometric phase in unitary and nonunitary evolution [Phys. Rev. Lett. {\bf 85}, 2845 (2000); Phys. Rev. Lett. {\bf 93}, 080405 (2004)] is delineated. An explicit form for the line element is derived and shown to be related to an averaged energy dispersion in the case of unitary evolution. The line element is measurable in interferometry involving nearby internal states. Explicit geodesics are found in the single qubit case. It is shown how the Bures line element can be obtained by extending our approach to arbitrary decompositions of density operators. The…
We calculate the geometric phase of a spin-1/2 particle coupled to an external environment comprising N spin-1/2 particle in the framework of open quantum systems. We analyze the decoherence factor and the deviation of the geometric phase under a nonunitary evolution from the one gained under an unitary one. We show the dependence upon the system's and bath's parameter and analyze the range of validity of the perturbative approximation. Finally, we discuss the implications of our results.
Using a kinematic approach we show that the non-adiabatic, non-cyclic, geometric phase corresponding to the radiation emitted by a three level cascade system provides a sensitive diagnostic tool for determining the entanglement properties of the two modes of radiation. The nonunitary, noncyclic path in the state space may be realized through the same control parameters which control the purity/mixedness and entanglement. We show analytically that the geometric phase is related to concurrence in certain region of the parameter space. We further show that the rate of change of the geometric phas…
We compute the geometric phase for a spin-1/2 particle under the presence of a composite environment, composed of an external bath (modeled by an infinite set of harmonic oscillators) and another spin-1/2 particle. We consider both cases: an initial entanglement between the spin-1/2 particles and an initial product state in order to see if the initial entanglement has an enhancement effect on the geometric phase of one of the spins. We follow the nonunitary evolution of the reduced density matrix and evaluate the geometric phase for a single two-level system. We also show that the initial enta…
The geometric phase is of fundamental interest and plays an important role in quantum information processing. However, the definition and calculation of this phase for open systems remains a problem due to the lack of agreement on generalizations of the parallel transport condition to mixed state nonunity evolutions. Here we tackle this problem by associating the open system geometric phase with the parallel transport of the joint system-reservoir state. Our approach not only provides a way around the nonunitary evolution obstacle, but also sheds light on the relation between the geometric pha…
In our previous work it has been shown the possibility to use the Aharonov-Anandan invariant as a tool in the analysis of disparate systems, including Hawking and Unruh effects, as well as graphene physics and thermal states. We show that the vacuum condensation, characterizing such systems, is also related with geometric phases and we analyze the properties of the geometric phase of systems represented by mixed state and undergoing a nonunitary evolution. In particular, we consider two level atoms accelerated by an external potential and interacting with a thermal state. We propose the realiz…
We study, in the framework of open quantum systems, the geometric phase acquired by a uniformly accelerated two-level atom undergoing nonunitary evolution due to its coupling to a bath of fluctuating vacuum electromagnetic fields in the multipolar scheme. We find that the phase variation due to the acceleration can be in principle observed via atomic interferometry between the accelerated atom and the inertial one, thus providing an evidence of the Unruh effect.
We calculate the geometric phase of a bipartite two-level system coupled to an external environment. We analyze the reduced density matrix for an arbitrary initial state of the composite system and compute the correction to the unitary geometric phase through a kinematic approach. In all cases considered, we observe a similar structure as a function of the degree of the entanglement of the initial state. Further, we compute the entanglement entropy and concurrence of the bipartite state and analyze if there is any relation among these quantities and the geometric phase acquired during the nonu…
Geometric phase has been proposed as one of the promising methodologies to perform fault tolerant quantum computations. However, since decoherence plays a crucial role in such studies, understanding of mixed state geometric phase has become important. While mixed state geometric phase was first introduced mathematically by Uhlmann, recently Sjoqvist et al. [Phys. Rev. Lett. 85(14), 2845 (2000)] have described the mixed state geometric phase in the context of quantum interference and shown theoretically that the visibility as well as the shift of the interference pattern are functions of geomet…
The geometric measure of entanglement of a pure state, defined by its distance to the set of pure separable states, is extended to multipartite mixed states. We characterize the nearest disentangled mixed state to a given mixed state with respect to this measure by a system of equations. The entanglement eigenvalue for a mixed state is introduced. For a given mixed state, we show that its nearest disentangled mixed state is associated with its entanglement eigenvalue.
A new definition and interpretation of geometric phase for mixed state cyclic unitary evolution in quantum mechanics are presented. The pure state case is formulated in a framework involving three selected Principal Fibre Bundles, and the well known Kostant-Kirillov-Souriau symplectic structure on (co) adjoint orbits associated with Lie groups. It is shown that this framework generalises in a natural and simple manner to the mixed state case. For simplicity, only the case of rank two mixed state density matrices is considered in detail. The extensions of the ideas of Null Phase Curves and Panc…
We extend the off-diagonal geometric phase [Phys. Rev. Lett. {\bf 85}, 3067 (2000)] to mixed quantal states. The nodal structure of this phase in the qubit (two-level) case is compared with that of the diagonal mixed state geometric phase [Phys. Rev. Lett. {\bf 85}, 2845 (2000)]. Extension to higher dimensional Hilbert spaces is delineated. A physical scenario for the off-diagonal mixed state geometric phase in polarization-entangled two-photon interferometry is proposed.
We calculate the geometric phase for different open systems (spin-boson and spin-spin models). We study not only how they are corrected by the presence of the different type of environments but also discuss the appearence of decoherence effects. These should be taken into account when planning experimental setups to study the geometric phase in the nonunitary regime. We propose a model with slow decoherence rate in which the geometric phase is still modified and might be measured.
In this paper we introduce a geometric framework for mixed quantum states based on a Kähler structure. The geometric framework includes a symplectic form, an almost complex structure, and a Riemannian metric that characterize the space of mixed quantum states. We argue that the almost complex structure is integrable. We also in detail discuss a visualizing application of this geometric framework by deriving a geometric uncertainty relation for mixed quantum states. The framework is computationally effective and it provides us with a better understanding of general quantum mechanical systems.
We consider arbitrary mixed state in unitary evolution and provide a comprehensive description of corresponding geometric phase in which two different points of view prevailing currently can be unified. Introducing an ancillary system and considering the purification of given mixed state, we find that different results of mixed state geometric phase correspond to different choice of the representation of Hilbert space of the ancilla. Moreover we demonstrate that in order to obtain Uhlmann's geometric phase it is not necessary to resort to the unitary evolution of ancilla.
We first propose a new separability criterion based on algebraic-geometric invariants of bipartite mixed states introduced in [1], then prove that for all low ranks r <m+n-2, generic rank r mixed states in mxn systems have relatively high Schmidt numbers (thus entangled) by this separability criterion. This also means that the algebraic-geometric separability criterion proposed here can be used to dectect all low rank entangled mixed states outside a measure zero set.
We show that the geometric phase for mixed state during a cyclic evolution suggested in 2004 J. Phys. A 37 3699 is U(1) invariant and can be observed by nowaday techniques.
It is shown that a recently suggested concept of mixed state geometric phase in cyclic evolutions [2004 {\it J. Phys. A} {\bf 37} 3699] is gauge dependent.
The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical one-form is defined whose line integral gives the geometric phase which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment [Phys. Rev. Lett. \textbf{85}, 2845 (2000)] for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mi…
We provide a physical prescription based on interferometry for introducing the total phase of a mixed state undergoing unitary evolution, which has been an elusive concept in the past. We define the parallel transport condition that provides a connection-form for obtaining the geometric phase for mixed states. The expression for the geometric phase for mixed state reduces to well known formulas in the pure state case when a system undergoes noncyclic and unitary quantum evolution.
Off-diagonal geometric phases have been developed in order to provide information of the geometry of paths that connect noninterfering quantal states. We propose a kinematic approach to off-diagonal geometric phases for pure and mixed states. We further extend the mixed state concept proposed in [Phys. Rev. Lett. {\bf 90}, 050403 (2003)] to degenerate density operators. The first and second order off-diagonal geometric phases are analyzed for unitarily evolving pairs of pseudopure states.
Computing entanglement of an arbitrary bipartite or multipartite mixed state is in general not an easy task as it usually involves complex optimization. Here we show that exploiting symmetries of certain mixed states, we can compute a genuine multiparty entanglement measure, the generalized geometric measure for these classes of mixed states. The chosen states have different ranks and consist of an arbitrary number of parties.
The geometric measure of entanglement, originated by Shimony and by Barnum and Linden, is determined for a family of tripartite mixed states consisting of aribitrary mixtures of GHZ, W, and inverted-W states. For this family of states, other measures of entanglement, such as entanglement of formation and relative entropy of entanglement, have not been determined. The results for the geometric measure of entanglement are compared with those for negativity, which are also determined. The results for the geometric measure of entanglement and the negativity provide examples of the determination of…
Geometric phase may enable inherently fault-tolerant quantum computation. However, due to potential decoherence effects, it is important to understand how such phases arise for {\it mixed} input states. We report the first experiment to measure mixed-state geometric phases in optics, using a Mach-Zehnder interferometer, and polarization mixed states that are produced in two different ways: decohering pure states with birefringent elements; and producing a nonmaximally entangled state of two photons and tracing over one of them, a form of remote state preparation.
The concept of off-diagonal geometric phases for mixed quantal states in unitary evolution is developed. We show that these phases arise from three basic ideas: (1) fulfillment of quantum parallel transport of a complete basis, (2) a concept of mixed state orthogonality adapted to unitary evolution, and (3) a normalization condition. We provide a method for computing the off-diagonal mixed state phases to any order for unitarities that divide the parallel transported basis of Hilbert space into two parts: one part where each basis vector undergoes cyclic evolution and one part where all basis …
Geometrical phases have been applied in virtually every major branch of physics and they play an important role in topology and knot theory in mathematics and quantum computation. However, most of the early works focus on pure quantum states which are very unrealistic and almost never occur in practice. The two existed definitions -Uhlmann's and Sjöqvist's- would result in different values for the geometric phase in general. The definition of geometric phase in mixed state scenario is still an open question. Here we present a unified framework for both approaches within one and the same formal…
This paper focuses on the geometric phase of general mixed states under unitary evolution. Here we analyze both non-degenerate as well as degenerate states. Starting with the non-degenerate case, we show that the usual procedure of subtracting the dynamical phase from the total phase to yield the geometric phase for pure states, does not hold for mixed states. To this end, we furnish an expression for the geometric phase that is gauge invariant. The parallelity conditions are shown to be easily derivable from this expression. We also extend our formalism to states that exhibit degeneracies. He…
In astrophysical environments, neutrinos may propagate over a long distance in a magnetic field. In the presence of a rotating magentic field, the neutrino spin can flip from left-handed neutrino to right-handed neutrino. Smirnov demonstrated that the pure state geometric phase due to the neutrino spin precession may cause resonantg spin conversion inside the Sun. However, in general, the neutrinos may in an ensemble of thermal state. In this article, the corresponding mixed state geometric phases will be formulated, including the off-diagonal casse and diagonal ones. The spefic features towar…
The geometric phase for a pure quantal state undergoing an arbitrary evolution is a ``memory'' of the geometry of the path in the projective Hilbert space of the system. We find that Uhlmann's geometric phase for a mixed quantal state undergoing unitary evolution not only depends on the geometry of the path of the system alone but also on a constrained bi-local unitary evolution of the purified entangled state. We analyze this in general, illustrate it for the qubit case, and propose an experiment to test this effect. We also show that the mixed state geometric phase proposed recently in the c…
It is shown that Uhlmann's parallel transport of purifications along a path of mixed states represented by $2\times 2$ density matrices is just the path ordered product of Thomas rotations. These rotations are invariant under hyperbolic translations inside the Bloch sphere that can be regarded as the Poincaré ball model of hyperbolic geometry. A general expression for the mixed state geometric phase for an {\it arbitrary} geodesic triangle in terms of the Bures fidelities is derived. The formula gives back the solid angle result well-known from studies of the pure state geometric phase. It is …
We quantify the geometric measure of entanglement in terms of mean values of observables of entangled system. For pure states we find the relation of geometric measure of entanglement with the mean value of spin one-half for the system composed of spin and arbitrary quantum system. The geometric measure of entanglement for mixed states of rank-2 is studied as well. We find the explicit expression for geometric entanglement and the relation of entanglement in this case with the values of spin correlations. These results allow to find experimentally the value of entanglement by measuring a value…
Examples of geometric phases abound in many areas of physics. They offer both fundamental insights into many physical phenomena and lead to interesting practical implementations. One of them, as indicated recently, might be an inherently fault-tolerant quantum computation. This, however, requires to deal with geometric phases in the presence of noise and interactions between different physical subsystems. Despite the wealth of literature on the subject of geometric phases very little is known about this very important case. Here we report the first experimental study of geometric phases for mi…
We investigate the geometric phase of a two-level atom (qubit) coupled to a bosonic reservoir with Lorentzian spectral density, and find that for the non-Markovian dynamics in which rotating-wave approximation (RWA) is performed, geometric phase has a $π$-phase jump at the nodal point. However, the exact result without RWA given by hierarchy equation of motion method shows that there is no such a phase jump or nodal structure in geometric phase. Thus our results demonstrate that the counter-rotating terms significantly contribute to the geometric phase in multi-mode Hamiltonian under certain c…
We study a kind of geometric phases for entangled quantum systems, and particularly a spin driven by a magnetic field and entangled with another spin. The new kind of geometric phase is based on an analogy between open quantum systems and dissipative quantum systems which uses a C\ast-module structure. We show that the system presents from the viewpoint of their geometric phases, two behaviours. The first one is identical to the behaviour of an isolated spin driven by a magnetic field, as the problem originally treated by Berry. The second one is specific to the decoherence process. The gauge …
Geometric phase of an open quantum system that is interacting with a thermal environment (bath) is studied through some simple examples. The system is considered to be a simple spin-half particle which is weakly coupled to the bath. It is seen that even in this regime the geometric phase can vary with temperature. In addition, we also consider the system under an adiabatically time-varying magnetic field which is weakly coupled to the bath. An important feature of this model is that it reveals existence of a temperature-scale in which adiabaticity condition is preserved and beyond which the ge…
Information geometry promotes an investigation of the geometric structure of statistical manifolds, providing a series of elucidations in various areas of scientific knowledge. In the physical sciences, especially in quantum theory, this geometric method has an incredible parallel with the distinguishability of states, an ability of great value for determining the effectiveness in implementing physical processes. This gives us the context for this work. Here we will approach a problem of uniparametric statistical inference from an information-geometric perspective. We will obtain the generalis…
The geometric phase of a bi-particle model is discussed. For different initial states, especially when the initial state is pure or mixed, the geometric phase will show different properties. The relationship between the geometric phase and the entanglement or mixedness of the initial state is considered.
We examine evolutions where each component of a given decomposition of a mixed quantal state evolves independently in a unitary fashion. The geometric phase and parallel transport conditions for this type of decomposition dependent evolution are delineated. We compare this geometric phase with those previously defined for unitarily evolving mixed states, and mixed state evolutions governed by completely positive maps.
We describe quantum and classical Hamiltonian dynamics in a common Hilbert space framework, that allows the treatment of mixed quantum-classical systems. The analysis of some examples illustrates the possibility of entanglement between classical and quantum systems. We give a summary of the main tools of Berezin-Toeplitz and geometric quantization, that provide a relation between the classical and the quantum models, based essentially on the selection of a subspace of the classical Hilbert space. Coherent states provide a systematic tool for the inverse process, called dequantization, that ass…
The attempt to gain a theoretical understanding of the concept of time in quantum mechanics has triggered significant progress towards the search for faster and more efficient quantum technologies. One of such advances consists in the interpretation of the time-energy uncertainty relations as lower bounds for the minimal evolution time between two distinguishable states of a quantum system, also known as quantum speed limits. We investigate how the non uniqueness of a bona fide measure of distinguishability defined on the quantum state space affects the quantum speed limits and can be exploite…
Quantifying coherence and entanglement is extremely important in quantum information processing. Here, we present numerical and analytical results for the geometric measure of coherence, and also present numerical results for the geometric measure of entanglement. On the one hand, we first provide a semidefinite algorithm to numerically calculate geometric measure of coherence for arbitrary finite-dimensional mixed states. Based on this semidefinite algorithm, we test randomly generated single-qubit states, single-qutrit states, and a special kind of $d$-dimensional mixed states. Moreover, we …
We analyze the geometric aspects of unitary evolution of general states for a multilevel quantum system by exploiting the structure of coadjoint orbits in the unitary group Lie algebra. Using the same method in the case of SU(3) we study the effect of degeneracies on geometric phases for three-level systems. This is shown to lead to a highly nontrivial generalization of the result for two-level systems in which degeneracy results in a "monopole" structure in parameter space. The rich structures that arise are related to the geometry of adjoint orbits in SU(3). The limiting case of a two-level …
We study the entanglement between a certain qubit and the remaining system in rank- 2 mixed states prepared on the quantum computer. The protocol, which we propose for this purpose, is based on the relation of geometric measure of entanglement with correlations between qubits. As a special case, we consider a two-qubit rank-2 mixed state and find the relation of concurrence with the geometric measure of entanglement. On the ibmq-melbourne quantum computer we measure the geometric measure of entanglement in the cases of 2- and 4-qubit mixed quantum states which consist of Schrödinger cat states…
