Probability density in the complex plane

The correspondence principle asserts that quantum mechanics resembles classical mechanics in the high-quantum-number limit. In the past few years, many papers have been published on the extension of both quantum mechanics and classical mechanics into the complex domain. However, the question of whether complex quantum mechanics resembles complex classical mechanics at high energy has not yet been studied. This paper introduces the concept of a local quantum probability density ρ(z) in the complex plane. It is shown that there exist infinitely many complex contours C of infinite length on which ρ(z) dz is real and positive. Furthermore, the probability integral is finite. Demonstrating the existence of such contours is the essential element in establishing the correspondence between complex quantum and classical mechanics. The mathematics needed to analyze these contours is subtle and involves the use of asymptotics beyond all orders.     

A Dynamics for Discrete Quantum Gravity

This paper is based on the causal set approach to discrete quantum gravity. We first describe a classical sequential growth process (CSGP) in which the universe grows one element at a time in discrete steps. At each step the process has the form of a causal set (causet) and the “completed” universe is given by a path through a discretely growing chain of causets. We then quantize the CSGP by forming a Hilbert space H on the set of paths. The quantum dynamics is governed by a sequence of positive operators ρ _n on H that satisfy normalization and consistency conditions. The pair (H,{ρ _n }) is called a quantum sequential growth process (QSGP). We next discuss a concrete realization of a QSGP in terms of a natural quantum action. This gives an amplitude process related to the “sum over histories” approach to quantum mechanics. Finally, we briefly discuss a discrete form of Einstein’s field equation and speculate how this may be employed to compare the present framework with classical general relativity theory.     

A Generalization of Quantum Stein’s Lemma

Given many independent and identically-distributed (i.i.d.) copies of a quantum system described either by the state ρ or σ (called null and alternative hypotheses, respectively), what is the optimal measurement to learn the identity of the true state? In asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying ρ instead of σ, while requiring that the probability that σ is identified in the place of ρ is bounded by a small fixed number. Quantum Stein’s Lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of ρ and σ. We present a generalization of quantum Stein’s Lemma to the situation in which the alternative hypothesis is formed by a family of states, which can moreover be non-i.i.d. We consider sets of states which satisfy a few natural properties, the most important being the closedness under permutations of the copies. We then determine the error rate function in a very similar fashion to quantum Stein’s Lemma, in terms of the quantum relative entropy. Our result has two applications to entanglement theory. First it gives an operational meaning to an entanglement measure known as regularized relative entropy of entanglement. Second, it shows that this measure is faithful, being strictly positive on every entangled state. This implies, in particular, that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero. Therefore, the operational definition of multipartite entanglement is equivalent to its mathematical definition.     

How behavior of systems with sparse spectrum can be predicted on a quantum computer

Call a spectrum of Hamiltonian H sparse if each eigenvalue can be quickly restored within ε from its rough approximation within ε₁ by means of some classical algorithm. It is shown how the behavior of a system with a sparse spectrum up to time T=(1?ρ)/14ε can be predicted on a quantum computer with the time complexity t=4/(1?ρ)ε₁ plus the time of classical algorithm, where ρ is the fidelity. The quantum knowledge of Hamiltonian eigenvalues is considered as the new Hamiltonian W _H whose action on each eigenvector of H gives the corresponding eigenvalue. Speedup of evolution for systems with a sparse spectrum is possible, because, for such systems, the Hamiltonian W _H can be quickly simulated on the quantum computer. For an arbitrary system (even in the classical case), its behavior cannot be predicted on a quantum computer even for one step ahead. By this method, we can also restore the history with the same efficiency.     

On the complexity of search for keys in quantum cryptography

The trace distance is used as a security criterion in proofs of security of keys in quantum cryptography. Some authors doubted that this criterion can be reduced to criteria used in classical cryptography. The following question has been answered in this work. Let a quantum cryptography system provide an ε-secure key such that ½‖ρ_XE ? ρ_U ? ρ_E‖₁ < ε, which will be repeatedly used in classical encryption algorithms. To what extent does the ε-secure key reduce the number of search steps (guesswork) as compared to the use of ideal keys? A direct relation has been demonstrated between the complexity of the complete consideration of keys, which is one of the main security criteria in classical systems, and the trace distance used in quantum cryptography. Bounds for the minimum and maximum numbers of search steps for the determination of the actual key have been presented.     

Bohr correspondence principle for large quantum numbers

Periodic systems are considered whose increments in quantum energy grow with quantum number. In the limit of large quantum number, systems are found to give correspondence in form between classical and quantum frequency-energy dependences. Solely passing to large quantum numbers, however, does not guarantee the classical spectrum. For the examples cited, successive quantum frequencies remain separated by the incrementhI ^?1, whereI is independent of quantum number. Frequency correspondence follows in Planck's limit,h → 0. The first example is that of a particle in a cubical box with impenetrable walls. The quantum emission spectrum is found to be uniformly discrete over the whole frequency range. This quality holds in the limitn → ∞. The discrete spectrum due to transitions in the high-quantum-number bound states of a particle in a box with penetrable walls is shown to grow uniformly discrete in the limit that the well becomes infinitely deep. For the infinitely deep spherical well, on the other hand, correspondence is found to be obeyed both in emission and configuration. In all cases studied the classical ensemble gives a continuum of frequencies.     

Quantum Hypothesis Testing and Sufficient Subalgebras

We introduce a new notion of a sufficient subalgebra for quantum states: a subalgebra is 2-sufficient for a pair of states {ρ ₀, ρ ₁} if it contains all Bayes optimal tests of ρ ₀ against ρ ₁. In classical statistics, this corresponds to the usual definition of sufficiency. We show this correspondence in the quantum setting for some special cases. Furthermore, we show that sufficiency is equivalent to 2-sufficiency, if the latter is required for \({\{\rho_0^{\otimes n},\rho_1^{\otimes n}\}}\), for all n.     

Critical behaviour at the displacive limit of structural phase transitions

For a special critical point at zero temperature,T _c =0, which is called the displacive limit of a classical or of a quantum-mechanical model showing displacive phase transitions, we derive a set of static critical exponents in the large-n limit. Due to zero-point motions and quantum fluctuations at low temperatures, the exponents of the quantum model are different from those of the classical model. Moreover, we report results on scaling functions, corrections to scaling, and logarithmic factors which appear ford=2 in the classical case and ford=3 in the quantum-mechanical case. Zero-point motions cause a decrease of the critical temperature of the quantum model with respect to the classicalT _c , which implies a difference between the classical and the quantum displacive limit. However, finite critical temperatures are found in both cases ford>2, while critical fluctuations still occur atT _c =0 for 0<d≦2 in the classical case and for 1 <d≦2 in the quantum model. Further we derive the slope of the critical curve at the classical displacive limit exactly. The absence of 1/n-corrections to the exponents of the classical model is also discussed.     

On the Distribution of the Wave Function for Systems in Thermal Equilibrium

For a quantum system, a density matrix ρ that is not pure can arise, via averaging, from a distribution μ of its wave function, a normalized vector belonging to its Hilbert space ?. While ρ itself does not determine a unique μ, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which μ, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix ρ, a natural measure on the unit sphere in ?, denoted GAP(ρ). We do this using a suitable projection of the Gaussian measure on ? with covariance ρ. We establish some nice properties of GAP(ρ) and show that this measure arises naturally when considering macroscopic systems. In particular, we argue that it is the most appropriate choice for systems in thermal equilibrium, described by the canonical ensemble density matrix ρ_β = (1/Z) exp (?β H). GAP(ρ) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on ? are often used.     

A remark on the concavity of entropy

We investigate to what extent theorems about quantum mechanical or classical entropy can be generalized to functionals of the type ρ→Tr f(ρ), or ψ→∫f(ψ)dμ, respectively, wheref is an arbitrary concave function.     

Distinguishing Quantum States with Holevo Bound and Its Application to Spatially Separated Bell States

Consider a system prepared in one of the quantum states of the ensemble {ρ _i } with a priori probability p _i . We prove that the quantum state can be deterministically distinguished if and only if the information gain from the measurement reaches the Holevo bound. We find it can be applied to distinguish spatially separated Bell states. A single copy of spatially separated Bell state cannot be distinguished under local operation and classical communication (LOCC), but it can be identified with an ancillary qubit. When two ancillary qubits are used, a spatially separated Bell state can be identified without demolition.     

A Lie algebraic approach to the Kondo problem

The Kondo problem is approached using the unitary Lie algebra of spin-singlet fermion bilinears. In the limit when the number of values of the spin N goes to infinity the theory approaches a classical limit, which still requires a renormalization. We determine the ground state of this renormalized theory. Then we construct a quantum theory around this classical limit, which amounts to recovering the case of finite N.     

The quantum 2-sphere as a complex quantum manifold

We describe the quantum sphere of Podles for c = 0 by means of a stereographic projection which is analogous to that which exibits the classical sphere as a complex manifold. We show that the algebra of functions and the differential calculus on the sphere are covariant under the coaction of fractional transformations with SU _q(2) coefficients as well as under the action of SU _q(2) vector fields. Going to the classical limit we obtain the Poisson sphere. Finally, we study the invariant integration of functions on the sphere and find its relation with the translationally invariant integration on the complex quantum plane.     

Structural transitions in granular packs: statistical mechanics and statistical geometry investigations

We investigate equal spheres packings generated from several experiments and from a large number of different numerical simulations. The structural organization of these disordered packings is studied in terms of the network of common neighbours. This geometrical analysis reveals sharp changes in the network’s clustering occurring at the packing fractions (fraction of volume occupied by the spheres respect to the total volume, ρ) corresponding to the so called Random Loose Packing limit (RLP, ρ ～ 0.555) and Random Close Packing limit (RCP, ρ ～ 0.645). At these packing fractions we also observe abrupt changes in the fluctuations of the portion of free volume around each sphere. We analyze such fluctuations by means of a statistical mechanics approach and we show that these anomalies are associated to sharp variations in a generalized thermodynamical variable which is the analogous for these a-thermal systems to the specific heat in thermal systems.     

Von Neumann entropy-preserving quantum operations

For a given quantum state ρ and two quantum operations Φ and Ψ, the information encoded in the quantum state ρ is quantified by its von Neumann entropy S(ρ). By the famous Choi-Jamio?kowski isomorphism, the quantum operation Φ can be transformed into a bipartite state, the von Neumann entropy Smap(Φ) of the bipartite state describes the decoherence induced by Φ. In this Letter, we characterize not only the pairs (Φ,ρ) which satisfy S(Φ(ρ))=S(ρ), but also the pairs (Φ,Ψ) which satisfy Smap(Φ°Ψ)=Smap(Ψ).     

Electronic and superconducting properties of a superlattice of quantum stripes at the atomic limit

The superconducting gap, the critical temperature and the isotope coefficient in a superlattice of metallic quantum stripes is calculated as a function of the electron number density. We show that it is possible to design a particular artificial superlattice of quantum stripes that exhibits the curves of T _c and of the isotope coefficient as a function of the charge density as in cuprate superconductors. The shape of the superlattice is designed in order to tune the chemical potential near the bottom of the third subband for an electron number density of ρ ～ 5:810-²Å^-2. The superconducting critical temperature shows a resonant amplification as a function of electron number density ρ with a maximum at a critical value ρ _c. The isotope coefficient shows a sharp drop from a regime where α > 0:5 at ρ < ρ _c to a regime where α < 0:2 at ρ ≥ ρ _c. The underdoped and overdoped regime in cuprate superconductors is associated with a transition from a quasi 1D behavior for ρ > ρ _c to quasi 2D behavior for ρ < ρ _c with opening of a pseudogap at ρ ～ ρ _c.     

Quantum nonintegrability and the classical limit for usp(4) systems

We investigate the transition from integrability to chaos in a system built of usp(4) elements, both in the quantum case and in its classical limit, obtained using coherent states. This algebraic Hamiltonian consists in an integrable term plus a nonlinear perturbation, and we see that the level spacing distribution for the quantum system is well approximated by the Berry-Robnik-Brody distribution, and accordingly the classical limit displays mixed dynamics.     

Classial and quantum solitons in the transverse field Ising model

We investigate the shape and the dynamics of domain walls in the one-dimensional Ising model with spin S, exchange constant J and external transverse field Γ using numerical calculations up to S = 20 and analytical approximations. For $\tfrac{\Gamma } {{JS}}$ \] we describe classical domain walls as strongly localized excitations, which have either central spin or central bond symmetry. These symmetries are identified also in the quantum case, when solitary excitations develop into energy bands. In the classical limit S → ∞ localization results from the exponential vanishing of the bandwidth for the lowest bands. We describe the relation between the spectrum of moving classical solitons and the quantum band structure.     

The classical limit for quantum mechanical correlation functions

For quantum systems of finitely many particles as well as for boson quantum field theories, the classical limit of the expectation values of products of Weyl operators, translated in time by the quantum mechanical Hamiltonian and taken in coherent states centered inx- andp-space around? ^?1/2 (coordinates of a point in classical phase space) are shown to become the exponentials of coordinate functions of the classical orbit in phase space. In the same sense,? ^?1/2 [(quantum operator) (t) — (classical function) (t)] converges to the solution of the linear quantum mechanical system, which is obtained by linearizing the non-linear Heisenberg equations of motion around the classical orbit.