Entanglement witness is a Hermitian operator that is useful for detecting the genuine multipartit... more Entanglement witness is a Hermitian operator that is useful for detecting the genuine multipartite entanglement of mixed states. Nonlinear entanglement witnesses have the advantage of a wider detection range in the entangled region. We construct genuine entanglement witnesses for four qubits density matrices in the mutually unbiased basis. First, we find the convex feasible region with positive partial transpose states. Then, to reveal the entangled regions, we present some appropriate linear entanglement witnesses and, we find the envelope of this family of linear witnesses as a nonlinear witness. Finally, we study thermal entanglement and we show for some Hamiltonians the witnesses can detect the entanglement at all temperatures.
Quantum pattern recognition algorithm for two-qubit systems has been implemented by quantum adiab... more Quantum pattern recognition algorithm for two-qubit systems has been implemented by quantum adiabatic evolution. We will estimate required running time for this algorithm by means of an analytical solution of time-dependent Hamiltonian since the time complexity of adiabatic quantum evolution is a limitation on the quantum computing. These results can be useful for experimental implementation. Adiabatic quantum computation (AQC) [1,2] is a model of quantum computing designed to solve optimization problem [3,4] and then as an alternative method for standard circuit model of quantum computing. [5] By applying the AQC, several quantum algorithms such as Deutsch, [6] Detutsch-joza, [7] Bernstein-Vazirani [8] and Grover [9] are redesigned and the required running time for adiabatic evolution has been estimated. Quantum pattern recognition for two-qubit systems is another algorithm that has been designed by AQC. [10] In Ref. [10], a non-training quantum neural network (QNN) has been presented by the combination of classical Hopfield network and the AQC method for pattern recognition and associative memory. Moreover, an experimental proof of their model for the two-qubit system has been implemented by liquid state NMR. In this Letter, we estimate a lower bound of required computation time for the two-qubit quantum pattern recognition algorithm. [10] For this purpose, we use the adiabatic theorem condition analytically and discuss the runtime of the quantum system. This estimation may be useful in experimental implementation in the different quantum systems. Let us consider a quantum system which is described by a quantum state |ψ t to solve a problem instance. Suppose that H 0 and H T are initial and final Hamiltonians of the system. Suppose that |ψ 0 , the ground state of H 0 , is the initial state of the system and |ψ T , the ground state of H T is the final state of the system that encodes the solution of the problem. The evolution of the system can be derived according to a path Hamiltonian H(t) = (1 − s)H 0 + sH T , (1) where s = s(T) is a time-dependent parameter with s(0) = 0 and s(T) = 1, with T being the total running time of the evolution. Suppose that |E 0 (t) and |E 1 (t) are the ground and first excited states of the Hamiltonian at time t, corresponding to eigen-values E 0 (t) and E 1 (t), respectively. The adiabatic theorem [11,12] guarantees that if the system evolves slowly enough, at time T the system would be close to E 0 (T) with a high probability
Here we consider a class of $2\otimes2\otimes d$ chessboard density matrices starting with three-... more Here we consider a class of $2\otimes2\otimes d$ chessboard density matrices starting with three-qubit ones which have positive partial transposes with respect to all subsystems. To investigate the entanglement of these density matrices, we use the entanglement witness approach. For constructing entanglement witnesses (EWs) detecting these density matrices, we attempt to convert the problem to an exact convex optimization problem. To this aim, we map the convex set of separable states into a convex region, named feasible region, and consider cases that the exact geometrical shape of feasible region can be obtained. In this way, various linear and non-linear EWs are constructed. The optimality and decomposability of some of introduced EWs are also considered. Furthermore, the detection of the density matrices by introduced EWs are discussed analytically and numerically. {\bf Keywords: chessboard density matrices, optimal non-linear entanglement witnesses, convex optimization} {\bf PA...
Nonlinear entanglement witnesses have wider range entanglement detection ability relative to the ... more Nonlinear entanglement witnesses have wider range entanglement detection ability relative to the linear ones. Here we construct them for a four qubit mutually unbiased basis Bell diagonal density matrices. To this aim we specify the convex feasible region with positive partial transpose states. To reveal the entangled regions, we present some appropriate optimal non-decomposable linear entanglement witnesses as the envelope of family of linear entanglement witnesses and in this case, the nonlinear witnesses have been nonlinear functional of linear witnesses. The results of the study are evaluated and assessed in the light of the problems of four qubit entanglement detection.
If energy ensemble average is less than the minimum energy of separable states, the system is ent... more If energy ensemble average is less than the minimum energy of separable states, the system is entangled. In this study, we consider energy as an entanglement witness for one dimensional XYZ Heisenberg lattice up to ten qubits analytically. We find minimum of energy using Lagrange undetermined multipliers to construct the entanglement witness. We also find threshold temperature and magnetic field, for which below them the system is entangled. The results are in good agreement with the literature. For systems with more than six qubits, the results show temperature gets a stable value for a zero magnetic field.
In the paper Hao Wang et al (2012 Phys. Scr. 86 015001) have studied the entangled quantum Otto e... more In the paper Hao Wang et al (2012 Phys. Scr. 86 015001) have studied the entangled quantum Otto engine and have derived some thermodynamics quantities like the amount of heat energy exchange, the net work output and the efficiency. Our investigations reveal that although the numeric results (figures) are correct, but the analytic formulas for heat, work, and efficiency are not correct.
Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed... more Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed. Using single particle feasible region, a way of constructing effective entanglement witnesses for bipartite systems is provided by exact convex optimization. Examples for some well known two qutrit quantum systems show these entanglement witnesses in most cases, provide necessary and sufficient conditions for separability of given bipartite system. Also this method is applied to a class of bipartite qudit quantum systems with details for d=3, 4 and 5. Keywords: non-linear and linear entanglement witnesses PACS number(s): 03.67.Mn, 03.65.Ud
We consider the problem of obtaining the entanglement witnesses for a given quantum system. We pr... more We consider the problem of obtaining the entanglement witnesses for a given quantum system. We present a method to convert this problem to a standard convex optimization problem by defining a feasible region. Then we develop a generic two-step algorithm for this problem which can be applied to the entanglement detection of N -partite quantum system. This permits an organized
Here we consider a class of 2 ⊗ 2 ⊗ d chessboard density matrices starting with threequbit ones w... more Here we consider a class of 2 ⊗ 2 ⊗ d chessboard density matrices starting with threequbit ones which have positive partial transposes with respect to all subsystems. To investigate the entanglement of these density matrices, we use the entanglement witness approach. For constructing entanglement witnesses (EWs) detecting these density matrices, we attempt to convert the problem to an exact convex optimization problem. To this aim, we map the convex set of separable states into a convex region, named feasible region, and consider cases that the exact geometrical shape of feasible region can be obtained. In this way, various linear and non-linear EWs are constructed. The optimality and decomposability of some of introduced EWs are also considered. Furthermore, the detection of the density matrices by introduced EWs are discussed analytically and numerically.
We present a quantum neural network with multi-neuron interactions for pattern recognition tasks ... more We present a quantum neural network with multi-neuron interactions for pattern recognition tasks by a combination of extended classic Hopfield network and adiabatic quantum computation. This scheme can be used as an associative memory to retrieve partial patterns with any number of unknown bits. Also, we propose a preprocessing approach to classifying the pattern space S to suppress spurious patterns. The results of pattern clustering show that for pattern association, the number of weights (η) should equal the numbers of unknown bits in the input pattern (d). It is also remarkable that associative memory function depends on the location of unknown bits apart from the d and load parameter α.
Entanglement witness is a Hermitian operator that is useful for detecting the genuine multipartit... more Entanglement witness is a Hermitian operator that is useful for detecting the genuine multipartite entanglement of mixed states. Nonlinear entanglement witnesses have the advantage of a wider detection range in the entangled region. We construct genuine entanglement witnesses for four qubits density matrices in the mutually unbiased basis. First, we find the convex feasible region with positive partial transpose states. Then, to reveal the entangled regions, we present some appropriate linear entanglement witnesses and, we find the envelope of this family of linear witnesses as a nonlinear witness. Finally, we study thermal entanglement and we show for some Hamiltonians the witnesses can detect the entanglement at all temperatures.
Quantum pattern recognition algorithm for two-qubit systems has been implemented by quantum adiab... more Quantum pattern recognition algorithm for two-qubit systems has been implemented by quantum adiabatic evolution. We will estimate required running time for this algorithm by means of an analytical solution of time-dependent Hamiltonian since the time complexity of adiabatic quantum evolution is a limitation on the quantum computing. These results can be useful for experimental implementation. Adiabatic quantum computation (AQC) [1,2] is a model of quantum computing designed to solve optimization problem [3,4] and then as an alternative method for standard circuit model of quantum computing. [5] By applying the AQC, several quantum algorithms such as Deutsch, [6] Detutsch-joza, [7] Bernstein-Vazirani [8] and Grover [9] are redesigned and the required running time for adiabatic evolution has been estimated. Quantum pattern recognition for two-qubit systems is another algorithm that has been designed by AQC. [10] In Ref. [10], a non-training quantum neural network (QNN) has been presented by the combination of classical Hopfield network and the AQC method for pattern recognition and associative memory. Moreover, an experimental proof of their model for the two-qubit system has been implemented by liquid state NMR. In this Letter, we estimate a lower bound of required computation time for the two-qubit quantum pattern recognition algorithm. [10] For this purpose, we use the adiabatic theorem condition analytically and discuss the runtime of the quantum system. This estimation may be useful in experimental implementation in the different quantum systems. Let us consider a quantum system which is described by a quantum state |ψ t to solve a problem instance. Suppose that H 0 and H T are initial and final Hamiltonians of the system. Suppose that |ψ 0 , the ground state of H 0 , is the initial state of the system and |ψ T , the ground state of H T is the final state of the system that encodes the solution of the problem. The evolution of the system can be derived according to a path Hamiltonian H(t) = (1 − s)H 0 + sH T , (1) where s = s(T) is a time-dependent parameter with s(0) = 0 and s(T) = 1, with T being the total running time of the evolution. Suppose that |E 0 (t) and |E 1 (t) are the ground and first excited states of the Hamiltonian at time t, corresponding to eigen-values E 0 (t) and E 1 (t), respectively. The adiabatic theorem [11,12] guarantees that if the system evolves slowly enough, at time T the system would be close to E 0 (T) with a high probability
Here we consider a class of $2\otimes2\otimes d$ chessboard density matrices starting with three-... more Here we consider a class of $2\otimes2\otimes d$ chessboard density matrices starting with three-qubit ones which have positive partial transposes with respect to all subsystems. To investigate the entanglement of these density matrices, we use the entanglement witness approach. For constructing entanglement witnesses (EWs) detecting these density matrices, we attempt to convert the problem to an exact convex optimization problem. To this aim, we map the convex set of separable states into a convex region, named feasible region, and consider cases that the exact geometrical shape of feasible region can be obtained. In this way, various linear and non-linear EWs are constructed. The optimality and decomposability of some of introduced EWs are also considered. Furthermore, the detection of the density matrices by introduced EWs are discussed analytically and numerically. {\bf Keywords: chessboard density matrices, optimal non-linear entanglement witnesses, convex optimization} {\bf PA...
Nonlinear entanglement witnesses have wider range entanglement detection ability relative to the ... more Nonlinear entanglement witnesses have wider range entanglement detection ability relative to the linear ones. Here we construct them for a four qubit mutually unbiased basis Bell diagonal density matrices. To this aim we specify the convex feasible region with positive partial transpose states. To reveal the entangled regions, we present some appropriate optimal non-decomposable linear entanglement witnesses as the envelope of family of linear entanglement witnesses and in this case, the nonlinear witnesses have been nonlinear functional of linear witnesses. The results of the study are evaluated and assessed in the light of the problems of four qubit entanglement detection.
If energy ensemble average is less than the minimum energy of separable states, the system is ent... more If energy ensemble average is less than the minimum energy of separable states, the system is entangled. In this study, we consider energy as an entanglement witness for one dimensional XYZ Heisenberg lattice up to ten qubits analytically. We find minimum of energy using Lagrange undetermined multipliers to construct the entanglement witness. We also find threshold temperature and magnetic field, for which below them the system is entangled. The results are in good agreement with the literature. For systems with more than six qubits, the results show temperature gets a stable value for a zero magnetic field.
In the paper Hao Wang et al (2012 Phys. Scr. 86 015001) have studied the entangled quantum Otto e... more In the paper Hao Wang et al (2012 Phys. Scr. 86 015001) have studied the entangled quantum Otto engine and have derived some thermodynamics quantities like the amount of heat energy exchange, the net work output and the efficiency. Our investigations reveal that although the numeric results (figures) are correct, but the analytic formulas for heat, work, and efficiency are not correct.
Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed... more Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed. Using single particle feasible region, a way of constructing effective entanglement witnesses for bipartite systems is provided by exact convex optimization. Examples for some well known two qutrit quantum systems show these entanglement witnesses in most cases, provide necessary and sufficient conditions for separability of given bipartite system. Also this method is applied to a class of bipartite qudit quantum systems with details for d=3, 4 and 5. Keywords: non-linear and linear entanglement witnesses PACS number(s): 03.67.Mn, 03.65.Ud
We consider the problem of obtaining the entanglement witnesses for a given quantum system. We pr... more We consider the problem of obtaining the entanglement witnesses for a given quantum system. We present a method to convert this problem to a standard convex optimization problem by defining a feasible region. Then we develop a generic two-step algorithm for this problem which can be applied to the entanglement detection of N -partite quantum system. This permits an organized
Here we consider a class of 2 ⊗ 2 ⊗ d chessboard density matrices starting with threequbit ones w... more Here we consider a class of 2 ⊗ 2 ⊗ d chessboard density matrices starting with threequbit ones which have positive partial transposes with respect to all subsystems. To investigate the entanglement of these density matrices, we use the entanglement witness approach. For constructing entanglement witnesses (EWs) detecting these density matrices, we attempt to convert the problem to an exact convex optimization problem. To this aim, we map the convex set of separable states into a convex region, named feasible region, and consider cases that the exact geometrical shape of feasible region can be obtained. In this way, various linear and non-linear EWs are constructed. The optimality and decomposability of some of introduced EWs are also considered. Furthermore, the detection of the density matrices by introduced EWs are discussed analytically and numerically.
We present a quantum neural network with multi-neuron interactions for pattern recognition tasks ... more We present a quantum neural network with multi-neuron interactions for pattern recognition tasks by a combination of extended classic Hopfield network and adiabatic quantum computation. This scheme can be used as an associative memory to retrieve partial patterns with any number of unknown bits. Also, we propose a preprocessing approach to classifying the pattern space S to suppress spurious patterns. The results of pattern clustering show that for pattern association, the number of weights (η) should equal the numbers of unknown bits in the input pattern (d). It is also remarkable that associative memory function depends on the location of unknown bits apart from the d and load parameter α.
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Papers by K. Aghayar