Narrow Results

Let ${\mathscr{H}}$ be a linear space equipped with an indefinite inner product $[\cdot ,\cdot ]$. Denote by ${{ℱ}}_{++}=\{f\in {\mathscr{H}}:[f,f]>0\}$ the nonlinear set of positive vectors in ${\mathscr{H}}$. We demonstrate that the properties of a linear operator $W$ in ${\mathscr{H}}$ can be uniquely determined by its restriction to ${{ℱ}}_{++}$. In particular, we prove that the bijectivity of $W$ on ${{ℱ}}_{++}$ is equivalent to $W$ being close to a unitary operator with respect to $[\cdot ,\cdot ]$. Furthermore, we consider a one-parameter semi-group of operators $W_{+}=\{W(t):t\ge 0\}$, where each $W(t)$ maps ${{ℱ}}_{++}$ onto itself in a one-to-one manner. We show that, under this natural restriction, the semi-group $W_{+}$ can be transformed into a one-parameter group $U=\{U(t):t\in ℝ\}$ of operators that are unitary with respect to $[\cdot ,\cdot ]$. By imposing additional conditions, we show how to construct a suitable definite inner product $〈\cdot ,\cdot 〉$, based on $[\cdot ,\cdot ]$, which guarantees the unitarity of the operators $U(t)$ in the Hilbert space obtained by completing ${\mathscr{H}}$ with respect to $〈\cdot ,\cdot 〉$.

We present a suitable framework for the definition of quantum time delay in terms of sojourn times for unitary operators in a two-Hilbert spaces setting. We prove that this time delay defined in terms of sojourn times (time-dependent definition) exists and coincides with the expectation value of a unitary analogue of the Eisenbud–Wigner time delay operator (time-independent definition). Our proofs rely on a new summation formula relating localization operators to time operators and on various tools from functional analysis such as Mackey’s imprimitivity theorem, Trotter–Kato formula and commutator methods for unitary operators. Our approach is general and model-independent.

In this paper the existence of quantum wave-packets in neural structures, such as automata and associative memories, is studied. The synaptic weights are considered to be stochastic variables, with probability density functions given by the solution of Schrödinger's equation. It is shown that the weights' update can be performed with the use of unitary operators, as indicated by the postulates of quantum mechanics. Moreover, it is proved that the number of attractors of quantum associative memories increases exponentially with respect to conventional associative memories. Finally, simulation tests are used to demonstrate the improved pattern storage capabilities of quantum associative memories.

In linear quantum optics we consider phase shifters, beam splitters, displacement operations, squeezing operations etc. The evolution can be described by unitary operators using Bose creation and annihilation operators. This evolution can be reduced to matrix multiplication using unitary matrices. We derive these evolutions for the different unitary operators. Finally a computer algebra implementation is provided.

It is widely understood that quantum computing — quantum gates upon qubits — is the general case, encompassing computing by classical means, viz. Boolean logic upon classical bits. It also seems reasonable that Quantum Software should encompass Classical Software. However, to accept such a statement regarding software, the feeling that it seems reasonable is not enough. One needs clear-cut definitions and formal conclusions. This is exactly the purpose of this paper. Previously, we have represented Classical Software by the Laplacian Matrix. More recently, we have shown that Quantum Software is faithfully represented by Density Matrices. It turns out that a Laplacian Matrix normalized by the Laplacian Trace easily obtains a Density Matrix. This opens the horizons for Quantum Software operations — such as unitary and reversible evolution — not naturally available with the classical Laplacian. This paper provides the necessary definitions and conclusions, illustrating the more general Quantum operations with a relevant case study, playing the double role of both classical and quantum software.

A recurrence scheme is presented to decompose an n-qubit unitary gate to the product of no more than N(N - 1)/2 single qubit gates with small number of controls, where N = 2ⁿ. Detailed description of the recurrence steps and formulas for the number of k-controlled single qubit gates in the decomposition are given. Comparison of the result to a previous scheme is presented, and future research directions are discussed.

We present a case study of quantum-like probabilities that are motivated by quantum cognition. We introduce quantum-like evolution that is l₂ norm preserving but in which the matrix does not need to be unitary. We show how to map any 2 × 2 stochastic matrix to an l₂ norm preserving balanced phase matrix that maps real vectors of length one into complex vectors of length one. Quantum-like evolution can simulate a probability distribution of open system in which the operator is not unitary but norm preserving. Such a kind of behaviour is studied in quantum cognition. By tensor product higher dimensional balanced phase matrices can be built. Quantum-like evolution can simulate either unitary open one by coding the phase of input vector into the phase of a balanced phase matrix, a Markov chain or an alternative evolution that can lead to fixed, periodic or chaotic behaviour resulting in strange oscillations.