Finite Quantum Tomography and Semidefinite Programming

Using the convex semidefinite programming method and superoperator formalism we obtain the finite quantum tomography of some mixed quantum states such as: truncated coherent states tomography, phase tomography and coherent spin state tomography, qudit tomography, N-qubit tomography, where that obtained results are in agreement with those of References (Buzek et al., Chaos, Solitons and Fractals 10 (1999) 981; Schack and Caves, Separable states of N quantum bits. In: Proceedings of the X. International Symposium on Theoretical Electrical Engineering, 73. W. Mathis and T. Schindler, eds. Otto-von-Guericke University of Magdeburg, Germany (1999); Pegg and Barnett Physical Review A 39 (1989) 1665; Barnett and Pegg Journal of Modern Optics 36 (1989) 7; St. Weigert Acta Physica Slov. 4 (1999) 613). PACs index: 03.65.Ud     

Quantum State Merging and Negative Information

We consider a quantum state shared between many distant locations, and define a quantum information processing primitive, state merging, that optimally merges the state into one location. As announced in [Horodecki, Oppenheim, Winter, Nature 436, 673 (2005)], the optimal entanglement cost of this task is the conditional entropy if classical communication is free. Since this quantity can be negative, and the state merging rate measures partial quantum information, we find that quantum information can be negative. The classical communication rate also has a minimum rate: a certain quantum mutual information. State merging enabled one to solve a number of open problems: distributed quantum data compression, quantum coding with side information at the decoder and sender, multi-party entanglement of assistance, and the capacity of the quantum multiple access channel. It also provides an operational proof of strong subadditivity. Here, we give precise definitions and prove these results rigorously.     

Quantum Ergodicity for Graphs Related to Interval Maps

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakónski et al (J. Phys. A, 34, 9303-9317 (2001)). As observables we take the L ² functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.     

Classical and Quantum Coherent States

p-Mechanics is a consistent physical theory which describes both quantum and classical mechanics simultaneously (V. V. Kisil, p-Mechanics as a physical theory. An Introduction, E-print:arXiv:quant-ph/0212101, 2002; International Journal of Theoretical Physics 41(1), 63–77, 2002). We continue the development of p-mechanics by introducing the concept of states. The set of coherent states we introduce allows us to evaluate classical observables at any point of phase space and simultaneously to evaluate quantum probability amplitudes. The example of the forced harmonic oscillator is used to demonstrate these concepts.     

Discrimination Among Quantum States

It has recently been shown that quantum measurement is not always useful for discrimination among quantum states (K. Hunter (2003). Physical Review A 68, 012306). This paper provides another proof of the necessary and sufficient condition for quantum measurement not to be useful and examines the condition by considering quantum measurement which discriminates between two spin-1/2 states in thermal equilibrium.     

Quantum Structures in Einstein General Relativity

We introduce the notion of a quantum structure on an Einstein general relativistic classical spacetime M. It consists of a line bundle over M equipped with a connection fulfilling certain conditions. We give a necessary and sufficient condition for the existence of quantum structures and classify them. The existence and classification results are analogous to those of geometric quantisation (Kostant and Souriau), but they involve the topology of spacetime, rather than the topology of the configuration space. We provide physically relevant examples, such as the Dirac monopole, the Aharonov–Bohm effect and the Kerr–Newman spacetime. Our formulation is carried out by analogy with the geometric approach to quantum mechanics on a spacetime with absolute time, given by Jadczyk and Modugno.     

Quantum broadcast communication with authentication

Two simple quantum broadcast communication schemes are proposed.A central party can broadcast his secret message to all the legitimate receivers simultaneously.Compared with the three schemes proposed recently (Wang et al.2007 Chin.Phys.16 1868),the proposed schemes have the advantages of consuming fewer quantum and classical resources,lessening the difficulty and intensity of necessary operations,and having higher efficiency.     

Early Universe Quantum Processes in BEC Collapse Experiments

We show that in the collapse of a Bose–Einstein condensate (BEC)⁴certain processes involved and mechanisms at work share a common origin with corresponding quantum field processes in the early universe such as particle creation, structure formation, and spinodal instability. Phenomena associated with the controlled BEC collapse observed in the experiment of Donley et al. (Donley, E., et al. (2001), Nature 412, 295; Claussen, N. (2003), PhD Thesis, University of Colorado; Claussen, N., et al. (2003), Physical Review A 67, 060701(R))(they call it “Bose–Nova,” see also Chin, J., Vogels, J., and Ketterle, W. (2003), Physical Review Letters 90, 160405) such as the appearance of bursts and jets can be explained as a consequence of the squeezing and amplification of quantum fluctuations above the condensate by the dynamics of the condensate. Using the physical insight gained in depicting these cosmological processes, our analysis of the changing amplitude and particle contents of quantum excitations in these BEC dynamics provides excellent quantitative fits with the experimental data on the scaling behavior of the collapse time and the amount of particles emitted in the jets. Because of the coherence properties of BEC and the high degree of control and measurement precision in atomic and optical systems, we see great potential in the design of tabletop experiments for testing out general ideas and specific (quantum field) processes in the early universe, thus opening up the possibility for implementing “laboratory cosmology.” This essay has the same content as v2 of Calzetta and Hu (2002), with a few references updated. For more details, see Calzetta and Hu (2002). ⁴For an excellent introduction to BEC theory, see Pethick and Smith (2002).     

PT Symmetry in Quantum Field Theory

The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. We show that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are H=p ²+ix ³ and H=p ²-x ⁴. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry that we call C. Using C, we show how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.     

Symmetry and Topology in Quantum Logic

A test space is a collection of non-empty sets, usually construed as the catalogue of (discrete) outcome sets associated with a family of experiments. Subject to a simple combinatorial condition called algebraicity, a test space gives rise to a “quantum logic”—that is, an orthoalgebra. Conversely, all orthoalgebras arise naturally from algebraic test spaces. In non-relativistic quantum mechanics, the relevant test space is the set ℱ F(H) of frames (unordered orthonormal bases) of a Hilbert space H. The corresponding logic is the usual one, i.e., the projection lattice L(H) of H. The test space ℱ F(H) has a strong symmetry property with respect to the unitary group of H, namely, that any bijection between two frames lifts to a unitary operator. In this paper, we consider test spaces enjoying the same symmetry property relative to an action by a compact topological group. We show that such a test space, if algebraic, gives rise to a compact, atomistic topological orthoalgebra. We also present a construction that generates such a test space from purely group-theoretic data, and obtain a simple criterion for this test space to be algebraic. PACS: 02.10.Ab; 02.20.Bb; 03.65.Ta.     

Pascal's Theorem and Quantum Deformation

By a transfer principle, Pascal's Theorem is equivalent to a theorem about point pairs on the real line. It appears that Pascal's Theorem is equivalent to the vanishing of a common invariant of six quadratic forms. Using the q-deformed invariant theory of Leitenberger (J. Algebra 222 (1999), 82), we construct corresponding quantum invariants by a computer calculation.     

The Kantowski-Sachs Space-Time in Loop Quantum Gravity

We extend the ideas introduced in the previous work to a more general space-time. In particular we consider the Kantowski-Sachs space time with space section with topology . In this way we want to study a general space time that we think to be the space time inside the horizon of a black hole. In this case the phase space is four dimensional and we simply apply the quantization procedure suggested by loop quantum gravity and based on an alternative to the Schroedinger representation introduced by H. Halvorson. Through this quantization procedure we show that the inverse of the volume density and the Schwarzschild curvature invariant are upper bounded and so the space time is singularity free. Also in this case we can extend dynamically the space time beyond the classical singularity. PACS number: 04.60.Pp, 04.70.Dy     

Two-Tape Finite Automata with Quantum and Classical States

Two-way finite automata with quantum and classical states (2QCFA) were introduced by Ambainis and Watrous, and two-way two-tape deterministic finite automata (2TFA) were introduced by Rabin and Scott. In this paper we study 2TFA and propose a new computing model called two-way two-tape finite automata with quantum and classical states (2TQCFA). First, we give efficient 2TFA algorithms for identifying languages which can be recognized by 2QCFA. Second, we give efficient 2TQCFA algorithms to recognize several languages whose status vis-a-vis 2QCFA have been posed as open questions, such as L_square={aⁿbⁿ² | n ? N}L_{\mathit{square}}=\{a^{n}b^{n^{2}}\mid n\in \mathbf{N}\}. Third, we show that {aⁿb^nk | n ? N}\{a^{n}b^{n^{k}}\mid n\in \mathbf{N}\} can be recognized by (k+1)-tape deterministic finite automata ((k+1)TFA). Finally, we introduce k-tape automata with quantum and classical states (kTQCFA) and prove that {aⁿb^nk | n ? N}\{a^{n}b^{n^{k}}\mid n\in \mathbf{N}\} can be recognized by kTQCFA.     

On Quantum Phase Transition. I. Spinless Electrons Strongly Correlated with Ions

We study a large class F of models of the quantum statistical mechanics dealing with two types of particles. First the spinless electrons are quantum particles obeying to the Fermi statistics, they can hop. Secondly the ions which cannot move, are classical particles. The Falicov–Kimball (FK) model⁽¹⁾ is a well known model belonging to F, for which the existence of an antiferomagnetic phase transition was proven in the seminal paper of Kennedy and Lieb.⁽²⁾ This result was extended by Lebowitz and Macris.⁽³⁾ A new approach to this problem based on quantum selection of the ground states was proposed in ref. 4. In this paper we extend this approach to show that, under the strong insulating condition, any hamiltonian of the class F admits, at every temperature, an effective hamiltonian, which governs the behaviour of the ions interacting through forces mediated by the electrons. The effective hamiltonians are long range many body Ising hamiltonians, which can be computed by a cluster expansion expressed in term of the quantum fluctuations. Our main result is that we can apply the powerfull results of the classical statistical mechanics to our quantum models. In particular we can use the classical Pirogov–Sinai theory to establish a hierarchy of phase diagrams, we can also study of the behaviour of the quantum inter- faces,⁽²⁹⁾ and so on...     

Complete Axiomatizations for Quantum Actions

We present two equivalent axiomatizations for a logic of quantum actions: one in terms of quantum transition systems, and the other in terms of quantum dynamic algebras. The main contribution of the paper is conceptual, offering a new view of quantum structures in terms of their underlying logical dynamics. We also prove Representation Theorems, showing these axiomatizations to be complete with respect to the natural Hilbert-space semantics. The advantages of this setting are many: (1) it provides a clear and intuitive dynamic-operational meaning to key postulates (e.g. Orthomodularity, Covering Law); (2) it reduces the complexity of the Solèr–Mayet axiomatization by replacing some of their key higher-order concepts (e.g. “automorphisms of the ortholattice”) by first-order objects (“actions”) in our structure; (3) it provides a link between traditional quantum logic and the needs of quantum computation. PACS: 02.10.-v Logic; set theory and algebra; 03.65.-w Quantum mechanics; 03.65.Fd Algebraic methods; 03.67.-a Quantum information.     

Nonrelativistic Quantum Mechanics with Fundamental Environment

Spontaneous transitions between bound states of an atomic system, “Lamb Shift” of energy levels and many other phenomena in real nonrelativistic quantum systems are connected within the influence of the quantum vacuum fluctuations (fundamental environment (FE)) which are impossible to consider in the limits of standard quantum-mechanical approaches. The joint system “quantum system (QS) + FE” is described in the framework of the stochastic differential equation (SDE) of Langevin-Schr?dinger (L-Sch) type, and is defined on the extended space R ³ ⊗ R _{ξ}, where R ³ and R _{ξ} are the Euclidean and functional spaces, respectively. The density matrix for single QS in FE is defined. The entropy of QS entangled with FE is defined and investigated in detail. It is proved that as a result of interaction of QS with environment there arise structures of various topologies which are a new quantum property of the system.     

Quantum melting of mesoscopic clusters

The phase diagram of a two-dimensional mesoscopic system of charges or dipoles, whose realizations could be electrons in a semiconductor quantum dot or indirect excitons in a system of two vertically coupled quantum dots, is investigated. Quantum calculations using ab initio Monte Carlo integration along trajectories determine the properties of such objects in the temperature-quantum de-Boer-parameter plane. At zero (sufficiently low) temperature, as the quantum fluctuations of the particles increase, two types of quantum disordering phenomena occur with increasing quantum de Boer parameter q: first, for q∼10⁻⁵ the systems transform into a radially ordered but orientationally disordered state wherein various shells of the “atom” rotate relative to one another. For much larger q∼0.1, a transition occurs to a disordered state (a superfluid in the case of a system of bosons). Fiz. Tverd. Tela (St. Petersburg) 41, 1856–1862 (October 1999)     

Generalizations of Quantum Mechanics Induced by Classical Statistical Field Theory

No Heading We show that the Dirac-von Neumann formalism for quantum mechanics can be obtained as an approximation of classical statistical field theory. This approximation is based on the Taylor expansion (up to terms of the second order) of classical physical variables – maps f : Ω → R, where Ω is the infinite-dimensional Hilbert space. The space of classical statistical states consists of Gaussian measures ρ on Ω having zero mean value and dispersion σ²(ρ) ≈ h. This viewpoint to the conventional quantum formalism gives the possibility to create generalized quantum formalisms based on expansions of classical physical variables in the Taylor series up to terms of nth order and considering statistical states ρ having dispersion σ²(ρ) = hⁿ (for n = 2 we obtain the conventional quantum formalism).