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Ali Saif M. Hassan

Publications and source records attributed to Ali Saif M. Hassan.

15 recordsLinked to original sources
arXiv2024

Multipartite Generalization of Geometric measure of Discord

Radhakrishnan et.al [Phys. Rev. Lett. 124, 110401 (2020)] proposed a generalization of quantum discord to multipartite systems, which is consistency with the conventional definition of discord in bipartite systems and derived explicit formulae for any states. These results are significant in capturing quantum correlations for multi-qubit systems. We propose a generalization of geometric measure of multipartite quantum discord in the same manner. We find generic forms of the generalization of geometric measure of quantum discord in a general N-partite quantum state. Further, we obtain computable exact formulas for the generalization of geometric measure of quantum discord in an N-qubit quantum state, which include the results obtained in [arXiv:2104.12344].

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arXiv2021

Quantum Correlation Dynamics of Three Qubits in Non-Markovian Environments

We investigate the quantum correlation dynamics of three independent qubits each locally interacting with a zero temperature non-Markovian reservoir by using the Geometric measure of quantum discord (GQD). The dependence of quantum correlation dynamics on amount of non-Markovian, the degree of initial quantum correlation and purity of the initial states are studied in detail. It is found that the quantum correlation of such three qubits system revives after instantaneous disappearance period when a proper amount of non-Markovian is present. A comparison to the pairwise quantum discord and entanglement dynamics in three qubits system is also made.

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arXiv2020

Squeezing and Entanglement of two-modes Quantum $\mathrm{X}$ Waves

quantum theory of generalized $\mathrm{X}$ waves with orbital angular momentum in dispersive media, and the interaction of quantized $\mathrm{X}$ waves in quadratic nonlinear media were studied in (J. opt,20,065201(2018)). We present a kind of phase matching, which is called velocity phase matching, and this phase matching can be used for determining the length of the nonlinear crystal or the interaction time in the experiment setup, to produce $\mathrm{X}$ waves with particular velocity $v$. Moreover, we introduce more analysis for the dependence of squeezing of $\mathrm{X}$ waves on its spectral order, and for spectral orders $j>0$, we predict the existence of a characteristic axicon aperture for maximal squeezing. Then we find the quantum squeezed state of down-converted state generated by the $χ^{2}$-nonlinear process. Finally, we detect their entanglement using a criterion of separability.

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arXiv2013

Measurement-Induced Nonlocality in an $n$-partite quantum state

We generalize the concept of measurement-induced non-locality (MiN) to $n$-partite quantum states. We get exact analytical expressions for MiN in an $n$-partite pure and $n$-qubit mixed state. We obtain the conditions under which MiN equals geometric quantum discord in an $n$-partite pure state and an $n$-qubit mixed state.

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arXiv2012

Geometric measure of quantum discord and total quantum correlations in a N-partite quantum state

Quantum discord, as introduced by Olliver and Zurek [Phys. Rev. Lett. \textbf{88}, 017901 (2001)], is a measure of the discrepancy between quantum versions of two classically equivalent expressions for mutual information and is found to be useful in quantification and application of quantum correlations in mixed states. It is viewed as a key resource present in certain quantum communication tasks and quantum computational models without containing much entanglement. An early step toward the quantification of quantum discord in a quantum state was by Dakic, Vedral, and Brukner [Phys. Rev. Lett. 105,190502 (2010)] who introduced a geometric measure of quantum discord and derived an explicit formula for any two-qubit state. Recently, Luo and Fu [Phys. Rev. A \textbf{82}, 034302 (2010)] introduced a generic form of the geometric measure of quantum discord for a bipartite quantum state. We extend these results and find generic forms of the geometric measure of quantum discord and total quantum correlations in a general N-partite quantum state. Further, we obtain computable exact formulas for the geometric measure of quantum discord and total quantum correlations in a N-qubit quantum state. The exact formulas for the $N$-qubit quantum state are experimentally implementable.

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arXiv2012

Tight lower bound to the geometric measure of quantum discord

Dakic, Vedral and Brukner [Physical Review Letters \tf{105},190502 (2010)] gave a geometric measure of quantum discord in a bipartite quantum state as the distance of the state from the closest classical quantum (or zero discord) state and derived an explicit formula for a two qubit state. Further, S.Luo and S.Fu [Physical Review A \tf{82}, 034302 (2010)] obtained a generic form of this geometric measure for a general bipartite state and established a lower bound. In this brief report we obtain a rigorous lower bound to the geometric measure of quantum discord in a general bipartite state which dominates that obtained by S.Luo and S.Fu.

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arXiv2010

Thermal quantum and classical correlations in two qubit XX model in a nonuniform external magnetic field

We investigate how thermal quantum discord (QD) and classical correlations (CC) of a two-qubit one-dimensional XX Heisenberg chain in thermal equilibrium depend on the temperature of the bath as well as on nonuniform external magnetic fields applied to two qubits and varied separately. We show that the behavior of QD differs in many unexpected ways from the thermal entanglement (EOF). For the nonuniform case (B1 = -B2), we find that QD and CC are equal for all values of (B1 = -B2) and for different temperatures. We show that, in this case, the thermal states of the system belong to a class of mixed states and satisfy certain conditions under which QD and CC are equal. The specification of this class and the corresponding conditions are completely general and apply to any quantum system in a state in this class satisfying these conditions. We further find that the relative contributions of QD and CC can be controlled easily by changing the relative magnitudes of B1 and B2. Finally, we connect our results with the monogamy relations between the EOF, CC and the QD of two qubits and the environment.

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arXiv2010

Entanglement Capacity of Nonlocal Hamiltonians : A Geometric Approach

We develop a geometric approach to quantify the capability of creating entanglement for a general physical interaction acting on two qubits. We use the entanglement measure proposed by us for $N$-qubit pure states (PRA \textbf{77}, 062334 (2008)). Our procedure reproduces the earlier results (PRL \textbf{87}, 137901 (2001)). The geometric method has the distinct advantage that it gives an experimental way to monitor the process of optimizing entanglement production.

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arXiv2009

Detection and Quantification of Entanglement in Multipartite Quantum Systems Using Weighted Graph and Bloch Representation of States

This thesis is an attempt to enhance understanding of the following questions A- Given a multipartite quantum state (possibly mixed), how to find out whether it is entangled or separable? (Detection of entanglement.) B- Given an entangled state, how to decide how much entangled it is? (Measure of entanglement.), in the context of multipartite quantum states. We have explored two approaches. In the first approach, we assign a weighted graph with multipartite quantum state and address the question of separability in terms of these graphs and various operations involving them. In the second approach we use the so called Bloch representation of multipartite quantum states to establish new criteria for detection of multipartite entangled states. We further give a new measure for entanglement in $N$-qubit entangled pure state and formally extend it to cover $N$-qubit mixed states.

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arXiv2009

An Experimentally accessible geometric measure for entanglement in $N$-qudit pure states

We present a multipartite entanglement measure for $N$-qudit pure states, using the norm of the correlation tensor which occurs in the Bloch representation of the state. We compute this measure for important class of $N$-qutrit pure states, namely general GHZ states. We prove that this measure possesses almost all the properties expected of a good entanglement measure, including monotonicity. Finally, we extend this measure to $N$-qudit mixed states via convex roof construction and establish its various properties, including its monotonicity.

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arXiv2008

An Experimentally accessible geometric measure for entanglement in $N$-qubit pure states

We present a multipartite entanglement measure for $N$-qubit pure states, using the norm of the correlation tensor which occurs in the Bloch representation of the state. We compute this measure for several important classes of $N$-qubit pure states such as GHZ states, W states and their superpositions. We compute this measure for interesting applications like one dimensional Heisenberg antiferromagnet. We use this measure to follow the entanglement dynamics of Grover's algorithm. We prove that this measure possesses almost all the properties expected of a good entanglement measure, including monotonicity. Finally, we extend this measure to $N$-qubit mixed states via convex roof construction and establish its various properties, including its monotonicity. We also introduce a related measure which has all properties of the above measure and is also additive.

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arXiv2008

Separability Criterion for multipartite quantum states based on the Bloch representation of density matrices

We give a new separability criterion, a necessary condition for separability of $N$-partite quantum states. The criterion is based on the Bloch representation of a $N$-partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to {\it arbitrary} $N$-partite quantum states in $\mathcal{H}=\mathcal{H}^{d_1}\otimes \mathcal{H}^{d_2} \otimes ... \otimes \mathcal{H}^{d_N}.$ The criterion can test whether a $N$-partite state is entangled and can be applied to different partitions of the $N$-partite system. We provide examples that show the ability of this criterion to detect entanglement. We show that this criterion can detect bound entangled states. We prove a sufficiency condition for separability of a 3-partite state, straightforwardly generalizable to the case $N > 3,$ under certain condition. We also give a necessary and sufficient condition for separability of a class of $N$-qubit states which includes $N$-qubit PPT states.

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arXiv2008

On the degree conjecture for separability of multipartite quantum states

We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein {\it et al.} [Phys. Rev. A \textbf{73}, 012320 (2006)]. The conjecture states that a multipartite quantum state is separable if and only if the degree matrix of the graph associated with the state is equal to the degree matrix of the partial transpose of this graph. We call this statement to be the strong form of the conjecture. In its weak version, the conjecture requires only the necessity, that is, if the state is separable, the corresponding degree matrices match. We prove the strong form of the conjecture for {\it pure} multipartite quantum states, using the modified tensor product of graphs defined in [J. Phys. A: Math. Theor. \textbf{40}, 10251 (2007)], as both necessary and sufficient condition for separability. Based on this proof, we give a polynomial-time algorithm for completely factorizing any pure multipartite quantum state. By polynomial-time algorithm we mean that the execution time of this algorithm increases as a polynomial in $m,$ where $m$ is the number of parts of the quantum system. We give a counter-example to show that the conjecture fails, in general, even in its weak form, for multipartite mixed states. Finally, we prove this conjecture, in its weak form, for a class of multipartite mixed states, giving only a necessary condition for separability.

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arXiv2007

Combinatorial approach to multipartite quantum systems:basic formulation

In this paper we give a method to associate a graph with an arbitrary density matrix referred to a standard orthonormal basis in the Hilbert space of a finite dimensional quantum system. We study the related issues like classification of pure and mixed states, Von-Neumann entropy, separability of multipartite quantum states and quantum operations in terms of the graphs associated with quantum states. In order to address the separability and entanglement questions using graphs, we introduce a modified tensor product of weighted graphs, and establish its algebraic properties. In particular, we show that Werner's definition [1] of a separable state can be written in terms of a graphs, for the states in a real or complex Hilbert space. We generalize the separability criterion (degree criterion) due to S.L. Braunstein, S. Ghosh, T. Mansour, S. Severini, R.C. Wilson [2], to a class of weighted graphs with real weights. We have given some criteria for the Laplacian associated with a weighted graph to be positive semidefinite.

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