Extending SDL and LMC complexity measures to quantum states

An extension of SDL (Shiner, Davison, Landsberg) and LMC (López-Ruiz, Mancini, Calbet) complexity measures is proposed for the quantum information context, considering that Von Neumann entropy is a natural disorder quantifier for quantum states. As a first example of application, the simple qubit was studied, presenting results similar to that obtained by applying SDL and LMC measures to a classical probability distribution. Then, for the Werner state, a mixture of Bell states, SDL and LMC measures were calculated, depending on the mixing factor γ$\gamma$, providing some conjectures concerning quantum systems.     

Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states

Starting from classical lattice systems ind2 dimensions with a regular zerotemperature phase diagram, involving a finite number of periodic ground states, we prove that adding a small quantum perturbation and/or increasing the temperature produce only smooth deformations of their phase diagrams. The quantum perturbations can involve bosons or fermions and can be of infinite range but decaying exponentially fast with the size of the bonds. For fermions, the interactions must be given by monomials of even degree in creation and annihilation operators. Our methods can be applied to some anyonic systems as well. Our analysis is based on an extension of Pirogov-Sinai theory to contour expansions ind+1 dimensions obtained by iteration of the Duhamel formula.     

Algorithmic complexity and entanglement of quantum states

We define the algorithmic complexity of a quantum state relative to a given precision parameter, and give upper bounds for various examples of states. We also establish a connection between the entanglement of a quantum state and its algorithmic complexity.     

Auxiliary field Monte-Carlo for quantum many-body ground states

We develop an algorithm for determining the exact ground state properties of quantum many-body systems which is equally applicable to bosons and fermions. The Schroedinger eigenvalue equation for the ground state energy is recast as a many-dimensional integral using the Hubbard-Stratonovitch representation of the imaginary-time many-body evolution operator. The integral is then evaluated stochastically. We test the algorithm for an exactly soluble boson system with an attractive potential and then extend it to fermions and repulsive potentials. Importance sampling is crucial to the success of the method, particularly for more complex systems. Computational efficiency is improved by performing the calculations in Fourier space.     

Distinguishability of quantum states and shannon complexity in quantum cryptography

The proof of the security of quantum key distribution is a rather complex problem. Security is defined in terms different from the requirements imposed on keys in classical cryptography. In quantum cryptography, the security of keys is expressed in terms of the closeness of the quantum state of an eavesdropper after key distribution to an ideal quantum state that is uncorrelated to the key of legitimate users. A metric of closeness between two quantum states is given by the trace metric. In classical cryptography, the security of keys is understood in terms of, say, the complexity of key search in the presence of side information. In quantum cryptography, side information for the eavesdropper is given by the whole volume of information on keys obtained from both quantum and classical channels. The fact that the mathematical apparatuses used in the proof of key security in classical and quantum cryptography are essentially different leads to misunderstanding and emotional discussions [1]. Therefore, one should be able to answer the question of how different cryptographic robustness criteria are related to each other. In the present study, it is shown that there is a direct relationship between the security criterion in quantum cryptography, which is based on the trace distance determining the distinguishability of quantum states, and the criterion in classical cryptography, which uses guesswork on the determination of a key in the presence of side information.     

Gibbs measure as quantum ground states

We study certain quantum spin systems which are equivalent to stochastic Ising models. We show that any translationally invariant quantum ground state is given by integration of Gibbs measure. The existence of mass gap is shown to be the same as exponentially fast convergence of stochastic models to invariant states.     

Passive states and KMS states for general quantum systems

We characterize equilibrium states of quantum systems by a condition of passivity suggested by the second principle of thermodynamics. Ground states and -KMS states for all inverse temperatures 0 are completely passive. We prove that these states are the only completely passive ones. For the special case of states describing pure phases, assuming the passivity we reproduce the results of Haag et al.     

Scale-renormalized matrix-product states for correlated quantum systems

A generalization of matrix product states (MPS) for interacting quantum systems in two and three dimensions is introduced. These scale-renormalized matrix-product states (SR-MPS) are based on a coarse graining of the lattice in which the blocks at each level are associated with matrix products that are further transformed (scale renormalized) with other matrices before they are assembled to form blocks at the next level. Using the two-dimensional transverse-field Ising model as a test, it is shown that the SR-MPS converge much more rapidly with the matrix size than a standard MPS. It is also shown that the use of lattice symmetries speeds up the convergence very significantly.     

Ground states of quantum spin systems

We prove that ground states of quantum spin systems are characterized by a principle of minimum local energy and that translationally invariant ground states are characterized by the principle of minimum energy per unit volume.     

Metastable states of quantum lattice systems

We extend the characterisation of metastability, as given in [1] for classical systems, to show that quantal lattice systems with suitable long range forces can support metastable states.     

Quasidistributions and coherent states for finite-dimensional quantum systems

In recent years, an approach to discrete quantum phase spaces which comprehends all the main quasiprobability distributions known has been developed. It is the research that started with the pioneering work of Galetti and Piza, where the idea of operator bases constructed of discrete Fourier transforms of unitary displacement operators was first introduced. Subsequently, the discrete coherent states were introduced, and finally, the s-parametrized distributions, that include the Wigner, Husimi, and Glauber–Sudarshan distribution functions as particular cases. In the present work, we adapt its formulation to encompass some additional discrete symmetries, achieving an elegant yet physically sound formalism.     

Valence bond ground states in isotropic quantum antiferromagnets

Haldane predicted that the isotropic quantum Heisenberg spin chain is in a massive phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exactSO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.Supported in part by N.S.F. Grant PHY-80-19754. Fellow of the A.P. Sloan Foundation and the Canadian Institute for Advanced ResearchN.S.F. Post-doctoral FellowSupported in part by N.S.F. Grant PHY-85-15288-A01     

Low-temperature expansion for lattice systems with many ground states

Low-temperature expansion for systems with many ground states is discussed. It is pointed out that, in general, different ground states may yield different formal perturbation expansions, and that the right expansion of the free energy is provided by ground states called here dominant.Supported in part by Grant AFOR-78-3522.     

Quantum states and potentialities of quantum systems

In a previous article it was shown that in general quantum states represent perspectives on the potentialities of quantum systems, rather than the potentialities themselves. In the present paper the following questions are investigated in the context of this result: (1) How do quantum states which undergo collapse transform under pure translations? (2) Under what conditions do quantum states represent the potentialities themselves? Two alternatives are presented in response to the first question: (1) Quantum states are scalars under translations. (2) The collapse of a quantum state propagates between frames of reference at the speed of light. The advantages and disadvantages of the two alternatives are discussed. The response to the second question is shown to depend on the chosen alternative. In addition, the second alternative is shown to lead to a consistent view of quantum states as “potential perspectives on potentialities.”     

Entanglement and factorized ground states in two-dimensional quantum antiferromagnets

Making use of exact results and quantum Monte Carlo data for the entanglement of formation, we show that the ground state of anisotropic two-dimensional S=1/2 antiferromagnets in a uniform field takes the classical-like form of a product state for a particular value and orientation of the field, at which the purely quantum correlations due to entanglement disappear. Analytical expressions for the energy and the form of such states are given, and a novel type of exactly solvable two-dimensional quantum models is therefore singled out. Moreover, we show that the field-induced quantum phase transition present in the models is unambiguously characterized by a cusp minimum in the pairwise-to-global entanglement ratio R, marking the quantum-critical enhancement of multipartite entanglement.     

Some connections between ground states and temperature states of thermodynamical systems

We investigate the properties of ground states of thermodynamical systems as limits of temperature states; we enlarge the algebra of observables in order to define a *-automorphism relevant to the study of K.M.S.-states and derive an usefull theorem of transitivity of *-automorphisms within the locally normal pure states of a quasilocal algebra.Chargé de Recherches au C.N.R.S.Maître-Assistant — Université d'Aix-Marseille II — Centre de Luminy.