## Quantum Mechanics Research Papers

We are pleased to announce the appointment of Valerio Scarani National University of Singapore as the new section editor of the Quantum mechanics and quantum information theory section of *Journal of Physics A: Mathematical and Theoretical*.

To celebrate this appointment, please find below some recently published papers from the section which are now available * free to read!*

### Scope

The Quantum mechanics and quantum information theory section of *Journal of Physics A: Mathematical and Theoretical* publishes high quality, innovative and significant new results in areas including:

- coherent states
- eigenvalue problems
- supersymmetric quantum mechanics
- scattering theory
- relativistic quantum mechanics
- semiclassical approximations
- foundations of quantum mechanics and measurement theory
- entanglement and quantum nonlocality
- geometric phases and quantum tomography
- quantum tunnelling
- decoherence and open systems
- quantum cryptography, communication and computation
- theoretical quantum optics

### Editorial Board

The Editorial Board has significant expertise in these fields. Board members who work in the area are:

Section Editor Valerio Scarani National University of Singapore

M V Berry University of Bristol, UKD Bruß Heinrich-Heine-Universität Dusseldorf, Germany

S Coppersmith University of Wisconsin, Madison, USA

J Eisert University of Potsdam, Germany

U Günther Forschungszentrum Rossendorf, Dresden, Germany

**M Horodecki**Gdańsk University, Poland

**M Saraceno**Comision Nacional de Energia Atomica—Departamento de Fisica, Buenos Aires, Argentina

P Schmelcher Universität Hamburg, Germany

**H-Q Zhou**Chongqing University, People's Republic of China

### Selected papers—*free to read!*

A flavour of the Quantum mechanics and quantum information theory section can be obtained from the following recent papers. We are delighted to make these papers *free to read!*

### Fast Track Communications: Short Innovative Papers

### Topical Review

### Special Issue Articles

#### From the Special Issue Quantum Phases: 50 years of the Aharonov–Bohm effect and 25 years of the Berry phase

#### From the Special Issue Spectral and transport properties of quantum systems: in memory of Pierre Duclos (1948-2010)

### Regular Articles

### Submission Information

We invite you to submit your quantum mechanics or quantum information theory research article to *Journal of Physics A: Mathematical and Theoretical*. You can benefit from fast publication, with receipt to first decision usually in under 50 days, a thorough and constructive peer review from experts in your field, a truly global reach for your paper with over 2000 institutions worldwide having access to our journal's current content and our outstanding author service where you can follow your article's progress from start to finish with our online author enquiry service.

For details on how to submit, including author guidelines and submission addresses, please follow this link.

Boris F Samsonov 2010 *J. Phys. A: Math. Theor.***43** 402006

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Supersymmetric (SUSY) transformation operators with complex factorization constants are analyzed as operators acting in the Hilbert space of functions square integrable on the positive semiaxis. The obtained results are applied to Hamiltonians possessing spectral singularities which are non-Hermitian SUSY partners of self-adjoint operators. A new regularization procedure for the resolution of the identity operator in terms of a continuous biorthonormal set of the non-Hermitian Hamiltonian eigenfunctions is proposed. It is also argued that if the binorm of continuous spectrum eigenfunctions is interpreted in the same way as the norm of similar functions in the usual Hermitian case, then one can state that the function corresponding to a spectral singularity has zero binorm.

https://doi.org/10.1088/1751-8113/43/40/402006Cited byReferences

Uwe Günther and Sergii Kuzhel 2010 *J. Phys. A: Math. Theor.***43** 392002

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Gauged quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie-triple structure is found and an interpretation as -symmetrically generalized Jaynes–Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space-related *J*-self-adjoint extensions for PTQM setups with ultra-localized potentials.

https://doi.org/10.1088/1751-8113/43/39/392002Cited byReferences

Yan-Wei Dai *et al* 2010 *J. Phys. A: Math. Theor.***43** 372001

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The ground-state fidelity per lattice site is computed for the quantum three-state Potts model in a transverse magnetic field on an infinite-size lattice in one spatial dimension in terms of the infinite matrix product state algorithm. It is found that, on the one hand, a pinch point is identified on the fidelity surface around the critical point, and on the other hand, the ground-state fidelity per lattice site exhibits bifurcations at pseudo critical points for different values of the truncation dimension, which in turn approach the critical point as the truncation dimension becomes large. This implies that the ground-state fidelity per lattice site enables us to capture spontaneous symmetry breaking when the control parameter crosses the critical value. In addition, a finite-entanglement scaling of the von Neumann entropy is performed with respect to the truncation dimension, resulting in a precise determination of the central charge at the critical point. Finally, we compute the transverse magnetization, from which the critical exponent β is extracted from the numerical data.

https://doi.org/10.1088/1751-8113/43/37/372001Cited byReferences

MD. Rajjak Gazi *et al* 2010 *J. Phys. A: Math. Theor.***43** 452001

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Study of non-local correlations in terms of Hardy's argument has been quite popular in quantum mechanics. Hardy's non-locality argument depends on some kind of asymmetry, but a two-qubit maximally entangled state, being symmetric, does not exhibit this kind of non-locality. Here we ask the following question: can this feature be explained by some principle outside quantum mechanics? The no-signaling condition does not provide a solution. But, interestingly, the information causality principle (Pawlowski *et al* 2009 *Nature***461** 1101) offers an explanation. It shows that any generalized probability theory which gives completely random results for local dichotomic observable, cannot provide Hardy's non-local correlation if it is restricted by a necessary condition for respecting the information causality principle. In fact, the applied necessary condition imposes even more restrictions on the local randomness of measured observable. Still, there are some restrictions imposed by quantum mechanics that are not reproduced from the considered information causality condition.

https://doi.org/10.1088/1751-8113/43/45/452001Cited byReferences

Dorje C Brody and Eva-Maria Graefe 2011 *J. Phys. A: Math. Theor.***44** 072001

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While real Hamiltonian mechanics and Hermitian quantum mechanics can both be cast in the framework of complex canonical equations, their complex generalizations have hitherto remained tangential. In this communication, quaternionic and coquaternionic (split-signature analogue of quaternions) extensions of Hamiltonian mechanics are introduced and are shown to offer a unifying framework for complexified classical and quantum mechanics. In particular, quantum theories characterized by complex Hamiltonians invariant under spacetime reflection are shown to be equivalent to certain coquaternionic extensions of Hermitian quantum theories. One of the interesting consequences is that the spacetime dimension of these systems is six, not four, on account of the structures of coquaternionic quantum mechanics.

https://doi.org/10.1088/1751-8113/44/7/072001Cited byReferences

Ingrid Rotter 2009 *J. Phys. A: Math. Theor.***42** 153001

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The Hamiltonian *H*_{eff} of an open quantum system consists formally of a first-order interaction term describing the closed (isolated) system with discrete states and a second-order term caused by the interaction of the discrete states via the common continuum of scattering states. Under certain conditions, the last term may be dominant. Due to this term, *H*_{eff} is non-Hermitian. Using the Feshbach projection operator formalism, the solution Ψ ^{E}_{c} of the Schrödinger equation in the whole function space (with discrete as well as scattering states, and the Hermitian Hamilton operator *H*) can be represented in the interior of the localized part of the system in the set of eigenfunctions _{λ} of *H*_{eff}. Hence, the characteristics of the eigenvalues and eigenfunctions of the non-Hermitian operator *H*_{eff} are contained in observable quantities. Controlling the characteristics by means of external parameters, quantum systems can be manipulated. This holds, in particular, for small quantum systems coupled to a small number of channels. The paper consists of three parts. In the first part, the eigenvalues and eigenfunctions of non-Hermitian operators are considered. Most important are the true and avoided crossings of the eigenvalue trajectories. In approaching them, the phases of the _{λ} lose their rigidity and the values of observables may be enhanced. Here the second-order term of *H*_{eff} determines decisively the dynamics of the system. The time evolution operator is related to the non-Hermiticity of *H*_{eff}. In the second part of the paper, the solution Ψ ^{E}_{c} and the *S* matrix are derived by using the Feshbach projection operator formalism. The regime of overlapping resonances is characterized by non-rigid phases of the Ψ ^{E}_{c} (expressed quantitatively by the phase rigidity ρ). They determine the *internal impurity* of an open quantum system. Here, level repulsion passes into width bifurcation (resonance trapping): a dynamical phase transition takes place which is caused by the feedback between environment and system. In the third part, the internal impurity of open quantum systems is considered by means of concrete examples. Bound states in the continuum appearing at certain parameter values can be used in order to stabilize open quantum systems. Of special interest are the consequences of the non-rigidity of the phases of _{λ} not only for the problem of dephasing, but also for the dynamical phase transitions and questions related to them such as phase lapses and enhancement of observables.

https://doi.org/10.1088/1751-8113/42/15/153001Cited byReferences

T Kaufherr and Y Aharonov 2010 *J. Phys. A: Math. Theor.***43** 354012

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The answer to the question 'when does the AB effect occur?' is elusive, for in every gauge the relative phase between the two wave packets evolves differently. Considering gauge-invariant modulo momentum, i.e., the displacement operator or its Hermitian counterpart , it is found that when the external particle's two wave packets become co-linear with the solenoid, an abrupt nonlocal exchange of the conserved quantity occurs. Using the Heisenberg picture, we show that this exchange is responsible for the shift of the interference pattern of the AB effect. We also describe a gedanken experiment that shows that our prediction can, in principle, be tested experimentally. Finally, this exchange gives new insight into the famous two-slit quantum interference experiment.

https://doi.org/10.1088/1751-8113/43/35/354012Cited byReferences

M V Berry and S Popescu 2010 *J. Phys. A: Math. Theor.***43** 354005

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Every energy level of a charged quantum particle confined in a region threaded by a magnetic flux line with quantum flux one-half must be degenerate for some position of the semifluxon within the boundary B. This is illustrated by computations for which B is a circle and a conformal transformation of a circle without symmetry. As the shape of B is varied, two degeneracies between the same pair of levels can collide and annihilate. Degeneracy of three levels requires three shape parameters, or the positions of three semifluxons; degeneracy of *N* levels can be generated by int{ *N*( *N* + 1)/4} semifluxons. The force on the semifluxon is derived.

https://doi.org/10.1088/1751-8113/43/35/354005Cited byReferences

Somshubhro Bandyopadhyay *et al* 2010 *J. Phys. A: Math. Theor.***43** 455303

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The 'entanglement cost' of a bipartite measurement is the amount of shared entanglement two participants need to use up in order to carry out the given measurement by means of local operations and classical communication. We numerically investigate the entanglement cost of generic orthogonal measurements on two qubits. Our results strongly suggest that for almost all measurements of this kind, the entanglement cost is strictly greater than the average entanglement of the eigenstates associated with the measurements, implying that the nonseparability of a two-qubit orthogonal measurement is generically distinct from the nonseparability of its eigenstates.

https://doi.org/10.1088/1751-8113/43/45/455303Cited byReferences

*n*noisy Popescu–Rohrlich boxes

Matthias Fitzi *et al* 2010 *J. Phys. A: Math. Theor.***43** 465305

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We quantify the amount of *non-locality* contained in *n* noisy versions of the so-called *Popescu–Rohrlich boxes* ( *PRBs*), i.e. bipartite systems violating the CHSH Bell inequality maximally. Following the approach by Elitzur, Popescu and Rohrlich, we measure the amount of non-locality of a system by representing it as a convex combination of a *local* behaviour, with maximal possible weight, and a *non-signalling* system. We show that the local part of *n* systems, each of which approximates a PRB with probability 1 − ε, is of order Θ(ε ^{n/2 }) in the isotropic, and equal to (3ε) ^{n} in the maximally biased case.

https://doi.org/10.1088/1751-8113/43/46/465305Cited byReferences

Jean-Daniel Bancal *et al* 2010 *J. Phys. A: Math. Theor.***43** 385303

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Finding all Bell inequalities for a given number of parties, measurement settings and measurement outcomes is in general a computationally hard task. We show that all Bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full Bell polytope. As an illustration of our method, we generate 238 885 new Bell inequalities and 1085 new Svetlichny inequalities. We find, in particular, facet inequalities for Bell experiments involving two parties and two measurement settings that are not of the Collins–Gisin–Linden–Massar–Popescu type.

https://doi.org/10.1088/1751-8113/43/38/385303Cited byReferences

J J Halliwell and J M Yearsley 2010 *J. Phys. A: Math. Theor.***43** 445303

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It is of interest in a variety of contexts, and in particular in the arrival time problem, to consider the quantum state obtained through unitary evolution of an initial state regularly interspersed with periodic projections onto the positive *x*-axis (pulsed measurements). Echanobe, del Campo and Muga have given a compelling but heuristic argument that the state thus obtained is approximately equivalent to the state obtained by evolving in the presence of a certain complex potential of step-function form. In this paper, with the help of the path decomposition expansion of the associated propagators, we give a detailed derivation of this approximate equivalence. The propagator for the complex potential is known so the bulk of the derivation consists of an approximate evaluation of the propagator for the free particle interspersed with periodic position projections. This approximate equivalence may be used to show that to produce significant reflection, the projections must act at time spacing less than / *E*, where *E* is the energy scale of the initial state.

https://doi.org/10.1088/1751-8113/43/44/445303Cited byReferences

*k*-entangled?

Stanisław J Szarek *et al* 2011 *J. Phys. A: Math. Theor.***44** 045303

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The set of trace-preserving, positive maps acting on density matrices of size *d* forms a convex body. We investigate its nested subsets consisting of *k*-positive maps, where *k* = 2, ..., *d*. Working with the measure induced by the Hilbert–Schmidt distance we derive asymptotically tight bounds for the volumes of these sets. Our results strongly suggest that the inner set of ( *k* + 1) -positive maps forms a small fraction of the outer set of *k*-positive maps. These results are related to analogous bounds for the relative volume of the sets of *k*-entangled states describing a bipartite *d* × *d* system.

https://doi.org/10.1088/1751-8113/44/4/045303Cited byReferences

Łukasz Pankowski and Michał Horodecki 2011 *J. Phys. A: Math. Theor.***44** 035301

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We provide a class of bound entangled states that have a positive distillable secure key rate. The smallest state of this kind is 4⊗4. Our class is a generalization of the class presented in Horodecki *et al* (2008 *IEEE Trans. Inf. Theory***54** 2621–5). It is much wider, containing, in particular, states from the boundary of PPT entangled states (all of the states in the previous class were of this kind) and also states inside the set of PPT entangled states, even approaching the separable states. This generalization comes at a price: for the wider class, a positive key rate requires, in general, apart from the *one-way* Devetak–Winter protocol (used in the previous case) also the recurrence preprocessing and thus is effectively a *two-way* protocol. We also analyze the amount of noise that can be admixtured to the states of our class without losing the key distillability property which may be crucial for experimental realization. The wider class contains key-distillable states with higher entropy (up to 3.524, as opposed to 2.564 for the previous class).

https://doi.org/10.1088/1751-8113/44/3/035301Cited byReferences

Carl M Bender and Hugh F Jones 2011 *J. Phys. A: Math. Theor.***44** 015301

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Colin Wilmott *et al* 2010 *J. Phys. A: Math. Theor.***43** 505306

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Convex polytopes are convex hulls of point sets in the *n*-dimensional space that generalize two-dimensional convex polygons and three-dimensional convex polyhedra. We concentrate on the class of *n*-dimensional polytopes in called sign permutation polytopes. We characterize sign permutation polytopes before relating their construction to constructions over the space of quantum density matrices. Finally, we consider the problem of state identification and show how sign permutation polytopes may be useful in addressing issues of robustness.

https://doi.org/10.1088/1751-8113/43/50/505306Cited byReferences

M V Berry 2010 *J. Phys. A: Math. Theor.***43** 415302

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For wavefunctions whose fourier spectrum (wavenumber or frequency) is positive, the local phase gradient can sometimes be negative; examples of this 'backflow' occur in quantum mechanics and optics. The backflow probability *P* (fraction of the region that is backflowing) is calculated for several cases. For waves that are superpositions of many uncorrelated components, *P* = (1 − *r*)/2, where *r* is a measure of the dispersion (mean/r.m.s.) of the component frequencies or wavenumbers. In two dimensions (backflow in spacetime, or wave propagation in the plane) the boundary of the backflowing region includes the phase singularities of the wave.

https://doi.org/10.1088/1751-8113/43/41/415302Cited byReferences

**Quantum theory**, the modern physical theory concerned with the emission and absorption of energy by matter and with the motion of material particles, is one of the most important theories devised in the 20th century. The theory is revolutionary as it replaces classical physics in the description of events at the microscopic level and now the theory provides the foundation for modern physics and chemistry.

The person who formed the basis of the quantum theory was Max Planck. In the 19th century, scientists used laws of classical physics to explain the relationship between matter and energy. Toward the end of the 19th century, various experimental results were obtained that could not be explained by classical physics. One of the failures of classical physics was the inability to explain the observed frequency distribution of radiant energy emitted by a hot blackbody. Classical physics predicts that when a blackbody is heated, the frequencies of the light radiated will take on a continuous range of values from zero to infinity. However, from experimental observations, the frequency distribution reaches a maximum and then falls off to zero as the frequency increases. In 1900, Max Planck announced a theory to explain the observed frequency distribution of blackbody radiation. He suggested that a blackbody atom radiating light of frequency v is restricted to emitting an amount of energy given by hv (where h is the Planck’s constant). Planck called this definite amount of energy a quantum of energy. In classical physics, energy is a continuous variable. In quantum physics, energy is quantized, meaning that energy can take on only certain values.

After Planck announced his theory, Albert Einstein applied the concept of energy quantization to the explanation of the experimental observations in the photoelectric effect. The photoelectric effect is a phenomenon when electrons are ejected from a substance exposed to electromagnetic radiation. According to classical physics, the average energy carried by an ejected electron should increase with the intensity of the incident radiation and not the frequency. However, from experimental observations, the energy of electrons ejected depends on the frequency of the incident radiation. Increasing the intensity of the incident radiation would only increase the amount and not the average energy of the electrons ejected. Also, for every substance irradiated, there is a threshold frequency below which no electrons are ejected irrespective of the light intensity. In 1905, Einstein explained the photoelectric effect by extending Planck’s concept of energy quantization to electromagnetic radiation. He proposed that besides having wavelike properties, electromagnetic radiation can be considered to consist of individual quanta, called photons, which interact with the electrons in the substance like discrete particles. For a given frequency v of the incident radiation, each photon carries a definite amount of energy given by hv, where h is the Planck’s constant. The threshold frequency is explained by the different nature of the materials. For each material there is a certain minimum energy, called the work function F, necessary to liberate an electron. Thus the threshold frequency, v0, corresponds to a minimum energy packet, hv0 (=F), required to liberate the electron.

The next major contribution to the quantum theory was Niels Bohr’s model of the hydrogen atom. When hydrogen gas is heated, the hydrogen atoms emit electromagnetic radiation of only certain distinct frequencies. During 1885 to 1910, Rydberg and Balmer independently found an empirical formula, called the Rydberg equation, which correctly reproduces the observed hydrogen atom spectral frequencies. However, there was no explanation for this formula. Meanwhile, in 1911, Rutherford introduced his atomic model, a dense, positively charged nucleus surrounded by a revolving, negatively charged electron cloud. According to classical physics, Rutherford’s atom is unstable because the negative electrons are attracted by the positive nucleus. As a result, the electrons will spiral into the nucleus releasing huge amounts of energy and the electrons’ spectral frequencies will change continuously. In 1913, Bohr introduced his theory of the hydrogen atom by applying quantum theory to Rutherford’s electron cloud. In his theory, Bohr postulated that the electrons can only revolve about the nucleus in fixed orbits of different energy values, such that the angular momentum of the revolving electron are quantized. When an electron is in an allowed orbit, the atom does not radiate energy. Such an electron is said to be in the stationary state and it has a certain amount of energy. If the electron makes its transitions from one energy level to another, photons of energies corresponding to the difference between the initial and final energy levels are emitted or absorbed. This gives rise to the set of characteristic line spectra and the Rydberg equation can finally be explained.

After Bohr announced his theory of the hydrogen atom, attempts were made to apply Bohr’s theory to atoms with more than one electron and to molecules. However, all attempts to derive the spectra of such systems using extensions of Bohr’s theory failed. A key idea towards resolving these difficulties was advanced by Louis de Broglie in 1923. He proposed that just as light shows both wave and particle like behaviours, matter also has a “dual” nature. He assumed that any particle, for example, an electron, an atom, etc, has a wavelength l which is given by h/p , where h is the Planck’s constant and p is the particles’ momentum. De Broglie obtained this equation by reasoning in analogy with photons. Although photons don’t have mass, but they do have energy. As Einstein famously proved, mass and energy are related in the equation E = mc2 where E is the energy, m is the mass and c is the speed of light. At speed c, a photon has a nonzero mass m. So by combining the 2 formulae, E = mc2 and E = hv , de Broglie obtained the equation l = h/p . In 1927, his hypothesis was experimentally confirmed by Davisson and Germer, who observed diffraction effects when an electron beam was reflected from a crystal of nickel. Since then, similar diffraction effects have been observed with neutrons, protons, helium atoms, and hydrogen molecules, indicating that the de Broglie hypothesis applies to all material particles, not just electrons.

After the 1920s, more and more observations were found to prove the validity of the quantum theory. Soon, the quantum theory led to the modern theory of the interaction between matter and radiation known as quantum mechanics, which generalized and replaced classical mechanics and Maxwell’s electromagnetic theory. Since then, science never looked back, as the quest for the understanding of how everything works continued…

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