Aharonov and Anandan's claim that the notion of ``proper mixture'' is improper is shown to be unjustified. The point is made that if a purely operationalist standpoint is taken the three difficulties these authors describe relatively to the conventional interpretation of density matrices in fact vanish. It is noted that nevertheless it is very difficult for us to do without any form of realism, in particular when the quantum measurement problem is considered, and it is stressed that the proper mixture notion comes in precisely at this level. The more general question of the real bearing of Aharonov and Anandan's ideas on the interpretation of quantum mechanics problem is considered.
The quantum states which satisfy the equality in the generalised uncertainty relation are called intelligent states. We prove the existence of intelligent states for the Anandan-Aharonov uncertainty relation based on the geometry of the quantum state space for arbitrary parametric evolutions of quantum states when the initial and final states are non-orthogonal.
We present a method for calculating the Aharonov-Anandan phase for time-independent Hamiltonians that avoids the calculation of evolution operators. We compare the generic method used to calculate the Aharonov-Anandan phase with the method proposed here through four examples; a spin-1/2 particle in a constant magnetic field, an arbitrary infinite-sized Hamiltonian with two known eigenvalues, a Fabry-Perot cavity with one movable mirror and a three mirrors cavity with a slightly transmissive movable middle mirror.
In the system of several interacting spins, geometric phases have been researched intensively.However, the studies are mainly focused on the adiabatic case (Berry phase), so it is necessary for us to study the non-adiabatic counterpart (Aharonov and Anandan phase). In this paper, we analyze both the non-degenerate and degenerate geometric phase of Lipkin-Meskov-Glick type model, which has many application in Bose-Einstein condensates and entanglement theory. Furthermore, in order to calculate degenerate geometric phases, the Floquet theorem and decomposition of operator are generalized. And th…
We construct an operator for the Aharonov-Anandan phase for time independent Hamiltonians. This operator is shown to generate the motion of cyclic quantum systems through an equation of evolution involving only geometric quantities, i.e. the distance between quantum states, the geometric phase and the total length of evolution. From this equation, we derive an operator for the Samuel and Bhandari phase (SB-phase) for non cyclic evolutions. Finally we show how the SB-phase can be used to construct an operator corresponding to a quantum clock which commutator with the Hamiltonian has a canonical…
We study quantum decoherence of single-qubit and two-qubit Aharonov-Anandan (AA) geometric phase gates realized in a multistep scheme. Each AA gate is also compared with the dynamical phase gate performing the same unitary transformation within the same time period and coupled with the same environment, which is modeled as harmonic oscillators. It is found that the fidelities and the entanglement protection of the AA phase gates are enhanced by the states being superpositions of different eigenstates of the environmental coupling, and the noncommutativity between the qubit interaction and the …
The Aharonov-Anandan phase is a description of the geometric nature in non-adiabatic cyclic evolutions of quantum states. Here we report on a measurement of the Aharonov-Anandan phase in photonics. We consider a time-independent quantum driven harmonic oscillator that is initially prepared at the vacuum state. We utilize evanescently coupled waveguides to realize this physical model and achieve a measurement of the Aharonov-Anandan phase via integrated interferometry.
We argue that a complete description of quantum annealing (QA) implemented with continuous variables must take into account the non-adiabatic Aharonov-Anandan geometric phase that arises when the system Hamiltonian changes during the anneal. We show that this geometric effect leads to the appearance of non-stoquastic terms in the effective quantum Ising Hamiltonians that are typically used to describe QA with flux-qubits. We explicitly demonstrate the effect of these geometric interactions when QA is performed with a system of one and two coupled flux-qubits. The realization of non-stoquastic …
Quantum computing in terms of geometric phases, i.e. Berry or Aharonov-Anandan phases, is fault-tolerant to a certain degree. We examine its implementation based on Zeeman coupling with a rotating field and isotropic Heisenberg interaction, which describe NMR and can also be realized in quantum dots and cold atoms. Using a novel physical representation of the qubit basis states, we construct $π/8$ and Hadamard gates based on Berry and Aharonov-Anandan phases. For two interacting qubits in a rotating field, we find that it is always impossible to construct a two-qubit gate based on Berry phases…
In the standard physical interpretation of quantum theory, prediction and retrodiction are not symmetric. The opposite assertion by some authors results from their use of non-standard interpretations of the theory.
We present the time evolution of the electronic two-level model in the rotating wave approximation (RWA). We calculate the Aharonov-Anandan phase of the vector states with cyclic evolution either within the period of an external monochromatic electric field or the period corresponding to Rabi's frequency. The Aharonov-Anandan phase is shown to be dependent on the initial vector state, unless the system evolves in the adiabatic regime; in the latter case, the Aharonov-Anandan phase recovers Berry's phase. Our results are also discussed in the quasi-resonant regime.
Preceding Comment challenged my claim that potentials might be just auxiliary mathematical tools and that they are not necessary for explaining physical phenomena. The Comment did not confront my explanation without potentials of the Aharonov-Bohm effects appeared in the original paper, but stated that I cannot apply this explanation for seven other examples. In my reply I provide explanations using my method, of one of the examples, show that two other example are not relevant, and agree that the remaining examples require further analysis. However, I argue that none of the examples provide r…
Recently, it is proposed to do quantum computation through the Berry's phase(adiabatic cyclic geometric phase) shift with NMR (Jones et al, Nature, 403, 869(2000)). This geometric quantum gate is hopefully to be fault tolerant to certain types of errors because of the geometric property of the Berry phase. Here we give a scheme to realize the NMR C-NOT gate through Aharonov-Anandan's phase(non-adiabatic cyclic phase) shift on the dynamic phase free evolution loop. In our scheme, the gate is run non-adiabatically, thus it is less affected by the decoherence. And, in the scheme we have chosen th…
In this Reply we argue that (i) the Hamiltonian, Eq. (17) in our paper (Phys. Rev. Lett. 108, 070405 (2012)), is definitely Lorentz invariant; (ii) the conditions of generating topological Aharonov-Casher(AC) and Scalar Aharonov-Bohm (SAB) effects are essential and physically meaningful; (iii) the Hamiltonians both in Phys. Rev. Lett. 74, 2847 (1995) and arXiv:1311.4011 are not suitable to describe the polarized spinor particles.
The aim of the present paper is to propose experiments for observing the significant features of Berry's phases for S>1, generated by spin-Hamiltonians endowed with two couplings, a magnetic dipole and an electric quadrupole one with external B and E fields, as theoretically studied in our previous work. The fields are assumed orthogonal, this mild restriction leading to geometric and algebraic simplifications. Alkali atoms appear as good candidates for interferometric measurements but there are challenges to be overcome. The only practical way to generate a suitable E-field is to use the ac S…
It is shown that the results of our paper are correct, although the use of the chirality operator $γ_5=σ_1$ in the paper is not correct, as fairly the authors of the Comment stated.
Geometric (Aharonov--Anandan) phases in neutrino oscillations have been claimed [Phys. Lett. B 780 (2018) 216] to be sensitive to the Majorana phases in neutrino mixing. More recently, however, it has been pointed out [Phys. Lett. B 818 (2021) 136376] that the proposed phases are not gauge invariant. Using both kinematic and geometric approaches, we show that all gauge-invariant Aharonov--Anandan phases (including the off-diagonal geometric phases associated with flavor transitions) are independent of the Majorana phases. This finding, which generalizes the well-known fact that conventional os…
In this letter, the generalization of geometric phase in density matrix is presented, we show that the extended sub-geometric phase have unified expression whatever in adiabatic or nonadiabatic procedure, the relations between them and the usual Berry phase or Aharonov-Anandan phase are established. We also demonstrated the influence of sub-geometric phases on the physical observables. Finally, our treatment is naturally used to investigate the geometric phase in mixed state.
A new type of procedures, called protective measurements, has been proposed by Aharonov, Anandan and Vaidman. These authors argue that a protective measurement allows the determination of arbitrary observables of a single quantum system and claim that this favors a realistic interpretation of the quantum state. This paper proves that only observables that commute with the system's Hamiltonian can be measured protectively. It is argued that this restriction saves the coherence of alternative interpretations.
The dynamics of a spin--1/2 neutral particle possessing electric and magnetic dipole moments interacting with external electric and magnetic fields in noncommutative coordinates is obtained. Noncommutativity of space is interposed in terms of a semiclassical constrained Hamiltonian system. The relation between the quantum phase acquired by a particle interacting with an electromagnetic field and the (semi)classical force acting on the system is examined and generalized to establish a formulation of the quantum phases in noncommutative coordinates. The general formalism is applied to physical s…
This paper is the answer to the paper by Kastner [Found. Phys., to be published, quant-ph/9807037] in which she continued the criticism of the counterfactual usage of the Aharonov-Bergman-Lebowitz rule in the framework of the time-symmetrized quantum theory, in particular, by analyzing the three-box ``paradox''. It is argued that the criticism is not sound. Paradoxical features of the three-box example has been explained. It is explained that the elements of reality in the framework of time-symmetrized quantum theory are counterfactual statements and, therefore, even conflicting elements of re…
Despite their important applications in metrology and in spite of numerous experimental demonstrations, weak measurements are still confusing for part of the community. This sometimes leads to unjustified criticism. Recent papers have experimentally clarified the meaning and practical significance of weak measurements, yet in [R.E. Kastner, Found. Phys. 47, 697-707 (2017)], Kastner seems to take us many years backwards in the debate, casting doubt on the very term "weak value" and the meaning of weak measurements. Kastner appears to ignore both the basics and frontiers of weak measurements and…
The Aharonov-Anandan and Berry phases are determined for the cyclic motions of a non-relativistic charged spinless particle evolving in the superposition of the fields produced by a Penning trap and a rotating magnetic field. Discussion about the selection of the parameter's space and the relationship between the Berry phase and the symmetry of the binding potential is given.
The (Berry-Aharonov-Anandan) geometric phase acquired during a cyclic quantum evolution of finite-dimensional quantum systems is studied. It is shown that a pure quantum state in a (2J+1)-dimensional Hilbert space (or, equivalently, of a spin-J system) can be mapped onto the partition function of a gas of independent Dirac strings moving on a sphere and subject to the Coulomb repulsion of 2J fixed test charges (the Majorana stars) characterizing the quantum state. The geometric phase may be viewed as the Aharonov-Bohm phase acquired by the Majorana stars as they move through the gas of Dirac s…
The PT-symmetric (PTS) quantum brachistochrone problem is reanalyzed as quantum system consisting of a non-Hermitian PTS component and a purely Hermitian component simultaneously. Interpreting this specific setup as subsystem of a larger Hermitian system, we find non-unitary operator equivalence classes (conjugacy classes) as natural ingredient which contain at least one Dirac-Hermitian representative. With the help of a geometric analysis the compatibility of the vanishing passage time solution of a PTS brachistochrone with the Anandan-Aharonov lower bound for passage times of Hermitian brach…
We derive differential equations for the modified Feynman propagator and for the density operator describing time-dependent measurements or histories continuous in time. We obtain an exact series solution and discuss its applications. Suppose the system is initially in a state with density operator $ρ(0)$ and the projection operator $E(t) = U(t) E U^\dagger(t)$ is measured continuously from $t = 0$ to $T$, where $E$ is a projector obeying $Eρ(0) E = ρ(0)$ and $U(t)$ a unitary operator obeying $U(0) = 1$ and some smoothness conditions in $t$. Then the probability of always finding $E(t) = 1$ fr…
A notion of monitored recurrence for discrete-time quantum processes was recently introduced in [Commun. Math. Phys., DOI 10.1007/s00220-012-1645-2] (see also arXiv:1202.3903) taking the initial state as an absorbing one. We extend this notion of monitored recurrence to absorbing subspaces of arbitrary finite dimension. The generating function approach leads to a connection with the well-known theory of operator-valued Schur functions. This is the cornerstone of a spectral characterization of subspace recurrence that generalizes some of the main results in the above mentioned paper. The spectr…
