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This is a review of the geometry of quantum states using elementary methods and pictures. Quantum states are represented by a convex body, often in high dimensions. In the case of $n$ qubits, the dimension is exponentially large in $n$. The space of states can be visualized, to some extent, by its simple cross sections: Regular simplexes, balls and hyper-octahedra.^a When the dimension gets large, there is a precise sense in which the space of states resembles, almost in every direction, a ball. The ball turns out to be a ball of rather low purity states. We also address some of the corresponding, but harder, geometric properties of separable and entangled states and entanglement witnesses.

“All convex bodies behave a bit like Euclidean balls.”

Keith Ball

We show that any unitary transformation performed on the quantum state of a closed quantum system describes an inner, reversible, generalized quantum measurement.

We also show that under some specific conditions it is possible to perform a unitary transformation on the state of the closed quantum system by means of a collection of generalized measurement operators. In particular, given a complete set of orthogonal projectors, it is possible to implement a reversible quantum measurement that preserves the probabilities.

In this context, we introduce the concept of "Truth-Observable", which is the physical counterpart of an inner logical truth.

We give an algebraic formulation based on Clifford algebras and algebraic spinors for quantum information. In this context, logic gates and concepts such as chirality, charge conjugation, parity and time reversal are introduced and explored in connection with states of qubits. Supersymmetry and M-superalgebra are also analyzed with our formalism. Specifically we use extensively the algebras $C{l}_{3,0}$ and $C{l}_{1,3}$ as well as tensor products of Clifford algebras.

We investigate, in this paper, the possibilities of generating qubits and qutrits in strongly correlated systems described by a modified of Hubbard Hamiltonian. Out of the complete set of commutating operators that form a close Lie algebra with this Hamiltonian, one can generate a particular operator, the expectation values of which, with respect to the density matrix generated from the Gibbs entropy by Maximum Entropy Principle (MEP), are 0 and ±1 near a particular temperature. This density matrix is generated by the superposition of highly coherent two-electronic states, analogous to the BCS ones. The concurrent existence of the expectation values of 0, +1 and -1 of this operator with respect to the density matrix occurs near the phase transition of aligned states to anti-aligned states. These are qutrits, which in the absence of a magnetic field reduces to qubits. We also present the general uncertainty principle (GUP) valid for the set of these operators, evaluate its value for specific heat, and examine the behavior of the specific heat and the related GUP as a function of the temperature. This temperature dependence of the specific heat, exhibits the expected trend of phase transition near the transition temperature. For the chosen Hamiltonian, we present the derivation of the postulate of Weiss' mean field theory. This relation points to the fact that to generate qubits and qutrits for the system investigated here, it must have an intrinsic magnetic field and be a strongly correlated system such as manganites. This investigation further points to the fact that the qutrits gate may be a suitable quantum computing algorithm for systems with intrinsic magnetic and applied electromagnetic fields, since in the presence of such fields z-projections of the state with spin-1 are no longer degenerate. This investigation establishes that the thermodynamical evolution of fermion pair in the presence of an interaction with its environment are different for qubits and qutrits, particularly in the presence of an internal and external magnetic field and possibly for the general case of electro-magnetic field.

The history of complementary observables and mutual unbiased bases is shortly reviewed. A characterization is given in terms of conditional entropy of subalgebras due to Connes and Størmer. The extension of complementarity to noncommutative subalgebras is considered as well. Possible complementary decompositions of a four-level quantum system are described and a characterization of the Bell basis is obtained: The MASA generated by the Bell basis is complementary to both M₂(ℂ) tensor factors.

Finite projective geometries, especially the Fano plane, have been observed to arise in the context of certain quantum gate operators. We use Clifford algebras to explain why these geometries, both planar and higher dimensional, appear in the context of multi-qubit composite systems.

We consider the quantum computational process as viewed by an insider observer: this is equivalent to an isomorphism between the quantum computer and a quantum space, namely the fuzzy sphere. The result is the formulation of a reversible quantum measurement scheme, with no hidden information.

We obtain the most general ensemble of qubits for which it is possible to design a universal Hadamard gate. These states, when geometrically represented on the Bloch sphere, give a new trajectory. We further consider some Hadamard "type" operations and find ensembles of states for which such transformations hold. The unequal superposition of a qubit and its orthogonal complement is also investigated.

In a number of Internet applications, we need to search for objects to download them. This includes peer-to-peer (P2P) file sharing, grid computing and content distribution networks. Here the single object will be searched for in multiple servers. There are many searching algorithms in existence today for this purpose based on the concept of classical physics and classical algorithms. The principles of quantum mechanics can be used to build and analyze a quantum computer and its algorithms. Quantum searching is such an algorithm. In this paper we are proposing a search method based on the quantum physics and quantum algorithms. Our method, the targeted quantum search is found to be more cost effective than any other classical searching algorithms like linear and two-way linear, simulated annealing, including broadcast based searching. Our targeted quantum search method is analyzed and simulated to show the best results.

We study the dynamical properties of a cavity field coupling to a Cooper pair box (CPB). We assumed that the CPB is prepared initially in a mixed state with a coherent state for the field. By solving the time-dependent equations using the evolution operator, it shows that mean numbers of Cooper pairs is affected by the detuning. The mean number of Cooper pairs is further enhanced by the multi-photon processes in commonly used cavity field.

A conjecture of Ramanujan that was later proved by Nagell is used to show on the basis of matching dimensions that only three n-qubit systems, for n = 1, 2, 6, can possibly share an isomorphism of their symmetry algebras with those of rotations in corresponding dimensions 3, 6, 91. Such isomorphisms are valuable for use in quantum information. Simple algebraic analysis, however, already rules out the last case so that one and two qubits are the only instances of such isomorphism of the algebras and of a local homomorphism of the corresponding symmetry groups. A more mathematical topological analysis of the group spaces is also provided demonstrating their topological inequivalence.

Explicitly separable density matrices are constructed for all separable two-qubits states based on Hilbert–Schmidt (HS) decompositions. For density matrices which include only two-qubits correlations the number of HS parameters is reduced to 3 by using local rotations, and for two-qubits states which include single qubit measurements, the number of parameters is reduced to 4 by local Lorentz transformations. For both cases, we related the absolute values of the HS parameters to probabilities, and the outer products of various Pauli matrices were transformed to pure state density matrices products. We discuss related problems for three-qubits. For n-qubits correlation systems ($n\ge 2$) the sufficient condition for separability may be improved by local transformations, related to high order singular value decompositions (SVDs).

We discussed the entanglement generated by the quantum annealing processor in the thermal state. The quantum annealing processor is modeled using the spin-chain model. The system is analytically solved using the unitary operator method and generated correlations (Von Neuman, Shanonn entropies and Purity) are discussed. The effect of the system parameters such as coupling constant, strength coupling and bias parameter, on the dynamics of the generated entanglement is studied. It is shown that the system parameters can be used as a controller of the entanglement.

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schrödinger framework from this perspective and provide a description of the Weyl–Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses.

The quantum computer is supposed to process information by applying unitary transformations to 2^N complex amplitudes defining the state of N qubits. A useful machine needing N~10³ or more, the number of continuous parameters describing the state of a quantum computer at any given moment is at least 2¹⁰⁰⁰ ~10³⁰⁰ which is much greater than the number of protons in the Universe. However, the theorists believe that the feasibility of large-scale quantum computing has been proved via the “threshold theorem”. Like for any theorem, the proof is based on a number of assumptions considered as axioms. However, in the physical world none of these assumptions can be fulfilled exactly. Any assumption can be only approached with some limited precision. So, the rather meaningless “error per qubit per gate” threshold must be supplemented by a list of the precisions with which all assumptions behind the threshold theorem should hold. Such a list still does not exist. The theory also seems to ignore the undesired free evolution of the quantum computer caused by the energy differences of quantum states entering any given superposition. Another important point is that the hypothetical quantum computer will be a system of 10³ –10⁶ qubits PLUS an extremely complex and monstrously sophisticated classical apparatus. This huge and strongly nonlinear system will generally exhibit instabilities and chaotic behavior.

Elements of quantum computing and quantum infromation processing are introduced for nonspecialists. Subjects inclulde quantum physics, qubits, quantum gates, quantum algorithms, decoherece, quantum error correcting codes and physical realizations. Presentations of these subjects are as pedagogical as possible. Some sections are meant to be brief introductions to contributions by other lecturers.

We can use a physical system for information processing and computing. If a quantum system is employed for these purposes, we are able to use powerful tools, which are beyond our imagination cultivated in the macroscopic world. Subjects introduced in this overview include qubits, quantum gates, quantum algorithms and physical realizations.

First I give a brief description of the classical Hopfield model introducing the fundamental concepts of patterns, retrieval, pattern recognition, neural dynamics, capacity and describe the fundamental results obtained in this field by Amit, Gutfreund and Sompolinsky,¹ using the non rigorous method of replica and the rigorous version given by Pastur, Shcherbina, Tirozzi² using the cavity method. Then I give a formulation of the theory of Quantum Neural Networks (QNN) in terms of the XY model with Hebbian interaction. The problem of retrieval and storage is discussed. The retrieval states are the states of the minimum energy. I apply the estimates found by Lieb³ which give lower and upper bound of the free-energy and expectation of the observables of the quantum model. I discuss also some experiment and the search of ground state using Monte Carlo Dynamics applied to the equivalent classical two dimensional Ising model constructed by Suzuki et al.⁶ At the end there is a list of open problems.