ali hassan | Univesity Of Amran (original) (raw)

Papers by ali hassan

Physical Review A, 2008

We present a multipartite entanglement measure for N-qudit pure states, using the norm of the cor... more 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.

Journal of Physics A: Mathematical and Theoretical, 2010

We investigate how thermal quantum discord (QD) and classical correlations (CC) of a two qubit on... more 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 temperature of the bath as well as on nonuniform external magnetic fields applied to two qubits and varied separately. We show that the behaviour of QD differs in many unexpected ways from 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 and 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, classical correlations and the quantum discord of two qubits and the environment.

Arxiv preprint arXiv:0905.0312, 2009

Arxiv preprint arXiv:1010.1920, 2010

Ali Saif M. Hassan,1, ∗ Behzad Lari,2, † and Pramod S. Joag2, ‡ 1Department of Physics, Universit... more Ali Saif M. Hassan,1, ∗ Behzad Lari,2, † and Pramod S. Joag2, ‡ 1Department of Physics, University of Amran, Amran, Yemen 2Department of Physics, University of Pune, Pune, India-411007. (Dated: October 12, 2010) Quantum discord, as introduced by Olliver and Zurek [Phys. ...

Journal of Mathematical Physics, 2008

We settle the so-called degree conjecture for the separability of multipartite quantum states, wh... more We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. [Phys. Rev. A 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 pure multipartite quantum states, using the modified tensor product of graphs defined in [J. Phys. A: Math. Theor. 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 counterexample 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.

Quantum Information and Computation, 2008

We give a new separability criterion, a necessary condition for separability of N-partite quantum... more 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 arbitrary N-partite quantum states in mathcalH=mathcalHd1otimesmathcalHd2otimescdotsotimesmathcalHd_N.\mathcal{H}=\mathcal{H}^{d_1}\otimes \mathcal{H}^{d_2} \otimes \cdots \otimes \mathcal{H}^{d_N}.mathcalH=mathcalHd_1otimesmathcalHd2otimescdotsotimesmathcalHdN. The criterion can test whether a N-partite state is entangled and can be applied to different partitions of the NNN-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.

We investigate the quantum correlation dynamics of three independent qubits each locally interact... more 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.

We develop a geometric approach to quantify the capability of creating entanglement for a general... more 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 (Phys. Rev. A 77, 062334 (2008)). This geometric method has the distinct advantage that it gives the experimentally implementable criteria to ensure the optimal entanglement production rate without requiring a detailed knowledge of the state of the two qubit system. For the production of entanglement in practice, we need criteria for optimal entanglement production which can be checked in situ without any need to know the state, as experimentally finding out the state of a quantum system is generally a formidable task. Further, we use our method to quantify the entanglement capacity in higher level and multipartite systems. We quantify the entanglement capacity for two qutrits and find the maximal entanglement generation rate and the corresponding state for the general isotropic interaction between qutrits, using the entanglement measure of N-qudit pure states proposed by us (Phys. Rev. A 80, 042302 (2009)). Next we quantify the genuine three qubit entanglement capacity for a general interaction between qubits. We obtain the maximum entanglement generation rate and the corresponding three qubit state for a general isotropic interaction between qubits. The state maximizing the entanglement generation rate is of the GHZ class. To the best of our knowledge, the entanglement capacities for two qutrit and three qubit systems have not been reported earlier.

We show that the quantum discord in a bipartite quantum state is invariant under the action of a ... more We show that the quantum discord in a bipartite quantum state is invariant under the action of a local quantum channel if and only if the channel is invertible. In particular, quantum discord is invariant under a local unitary channel.

We settle the so-called degree conjecture for the separability of multipartite quantum states, wh... more We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. ͓Phys. Rev. A 73, 012320 ͑2006͔͒. 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 pure multipartite quantum states using the modified tensor product of graphs defined by Hassan and Joag ͓J. Phys. A 40, 10251 ͑2007͔͒, 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 counterexample 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.

We investigate how thermal quantum discord (QD) and classical correlations (CC) of a two-qubit on... more 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 (B 1 = −B 2 ), we find that QD and CC are equal for all values of (B 1 = −B 2 ) 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 B 1 and B 2 . Finally, we connect our results with the monogamy relations between the EOF, CC and the QD of two qubits and the environment.

We present a multipartite entanglement measure for N-qudit pure states, using the norm of the cor... more 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 an important class of N-qutrit pure states, namely, general GHZ states. We prove that this measure possesses all the essential and many desirable properties expected of a good entanglement measure, including monotonicity. We also discuss the feasibility of the experimental evaluation of this measure for an N-qutrit system.

Quantum discord, as introduced by Ollivier and Zurek (2001 Phys. Rev. Lett. 88 017901), is a meas... more Quantum discord, as introduced by Ollivier and Zurek (2001 Phys. Rev. Lett. 88 017901), 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 et al (2010 Phys. Rev. Lett. 105 190502) who introduced a geometric measure of quantum discord and derived an explicit formula for any two-qubit state. Recently, Luo and Fu (2010 Phys. Rev. A 82 034302) 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 an N-qubit quantum state. The exact formulas for the N-qubit quantum state can be used to get experimental estimates of the quantum discord and the total quantum correlation.

We present a multipartite entanglement measure for N-qubit pure states, using the norm of the cor... more 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 Greenberger-Horne-Zeilinger and W states and their superpositions. We compute this measure for interesting applications like the 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. of the SU͑2͒ group ͑Pauli matrices͒. These Hermitian operators form an orthogonal basis ͑under the Hilbert-Schmidt scalar product͒ of the Hilbert space of operators acting on a singlequbit state space. The N times tensor product of this basis with itself generates a product basis of the Hilbert space of operators acting on the N-qubit state space. Any N-qubit density operator can be expanded in this basis. The corre-* alisaif@physics.unipune.ernet.in † pramod@physics.unipune.ernet.in PHYSICAL REVIEW A 77, 062334 ͑2008͒

We settle the so-called degree conjecture for the separability of multipartite quantum states, wh... more We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. ͓Phys. Rev. A 73, 012320 ͑2006͔͒. 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 pure multipartite quantum states using the modified tensor product of graphs defined by Hassan and Joag ͓J. Phys. A 40, 10251 ͑2007͔͒, 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 counterexample 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.

We develop a geometric approach to quantify the capability of creating entanglement for a general... more 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 ͓Ali Saif M. Hassan and Pramod S. Joag, Phys. Rev. A 77, 062334 ͑2008͔͒. This geometric method has the distinct advantage that it gives the experimentally implementable criteria to ensure the optimal entanglement production rate without requiring a detailed knowledge of the state of the two qubit system. For the production of entanglement in practice, we need criteria for optimal entanglement production, which can be checked in situ without any need to know the state, as experimentally finding out the state of a quantum system is generally a formidable task. Further, we use our method to quantify the entanglement capacity in higher level and multipartite systems. We quantify the entanglement capacity for two qutrits and find the maximal entanglement generation rate and the corresponding state for the general isotropic interaction between qutrits, using the entanglement measure of N-qudit pure states proposed by us ͓Ali Saif M. Hassan and Pramod S. Joag, Phys. Rev. A 80, 042302 ͑2009͔͒. Next we quantify the genuine three qubit entanglement capacity for a general interaction between qubits. We obtain the maximum entanglement generation rate and the corresponding three qubit state for a general isotropic interaction between qubits. The state maximizing the entanglement generation rate is of the Greenberger-Horne-Zeilinger class. To the best of our knowledge, the entanglement capacities for two qutrit and three qubit systems have not been reported earlier.

We give a new separability criterion, a necessary condition for separability of Npartite quantum ... more We give a new separability criterion, a necessary condition for separability of Npartite 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 arbitrary N-partite quantum states in

In this paper we give a method to associate a graph with an arbitrary density matrix referred to ... more 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 related issues such as 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 (Werner 1989 Phys. Rev. A 40 4277) of a separable state can be written in terms of graphs, for the states in a real or complex Hilbert space. We generalize the separability criterion (degree criterion) due to Braunstein et al (2006 Phys. Rev. A 73 012320) 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.

http://arxiv.org/abs/quant-ph/0701040v2 by ali hassan

We settle the so-called degree conjecture for the separability of multipartite quantum states, wh... more We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. ͓Phys. Rev. A 73, 012320 ͑2006͔͒. 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 pure multipartite quantum states using the modified tensor product of graphs defined by Hassan and Joag ͓J. Phys. A 40, 10251 ͑2007͔͒, 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 counterexample 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.

Physical Review A, 2008

We present a multipartite entanglement measure for N-qudit pure states, using the norm of the cor... more 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.

Journal of Physics A: Mathematical and Theoretical, 2010

We investigate how thermal quantum discord (QD) and classical correlations (CC) of a two qubit on... more 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 temperature of the bath as well as on nonuniform external magnetic fields applied to two qubits and varied separately. We show that the behaviour of QD differs in many unexpected ways from 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 and 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, classical correlations and the quantum discord of two qubits and the environment.

Arxiv preprint arXiv:0905.0312, 2009

Arxiv preprint arXiv:1010.1920, 2010

Ali Saif M. Hassan,1, ∗ Behzad Lari,2, † and Pramod S. Joag2, ‡ 1Department of Physics, Universit... more Ali Saif M. Hassan,1, ∗ Behzad Lari,2, † and Pramod S. Joag2, ‡ 1Department of Physics, University of Amran, Amran, Yemen 2Department of Physics, University of Pune, Pune, India-411007. (Dated: October 12, 2010) Quantum discord, as introduced by Olliver and Zurek [Phys. ...

Journal of Mathematical Physics, 2008

We settle the so-called degree conjecture for the separability of multipartite quantum states, wh... more We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. [Phys. Rev. A 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 pure multipartite quantum states, using the modified tensor product of graphs defined in [J. Phys. A: Math. Theor. 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 counterexample 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.

Quantum Information and Computation, 2008

We give a new separability criterion, a necessary condition for separability of N-partite quantum... more 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 arbitrary N-partite quantum states in mathcalH=mathcalHd1otimesmathcalHd2otimescdotsotimesmathcalHd_N.\mathcal{H}=\mathcal{H}^{d_1}\otimes \mathcal{H}^{d_2} \otimes \cdots \otimes \mathcal{H}^{d_N}.mathcalH=mathcalHd_1otimesmathcalHd2otimescdotsotimesmathcalHdN. The criterion can test whether a N-partite state is entangled and can be applied to different partitions of the NNN-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.

We investigate the quantum correlation dynamics of three independent qubits each locally interact... more 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.

We develop a geometric approach to quantify the capability of creating entanglement for a general... more 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 (Phys. Rev. A 77, 062334 (2008)). This geometric method has the distinct advantage that it gives the experimentally implementable criteria to ensure the optimal entanglement production rate without requiring a detailed knowledge of the state of the two qubit system. For the production of entanglement in practice, we need criteria for optimal entanglement production which can be checked in situ without any need to know the state, as experimentally finding out the state of a quantum system is generally a formidable task. Further, we use our method to quantify the entanglement capacity in higher level and multipartite systems. We quantify the entanglement capacity for two qutrits and find the maximal entanglement generation rate and the corresponding state for the general isotropic interaction between qutrits, using the entanglement measure of N-qudit pure states proposed by us (Phys. Rev. A 80, 042302 (2009)). Next we quantify the genuine three qubit entanglement capacity for a general interaction between qubits. We obtain the maximum entanglement generation rate and the corresponding three qubit state for a general isotropic interaction between qubits. The state maximizing the entanglement generation rate is of the GHZ class. To the best of our knowledge, the entanglement capacities for two qutrit and three qubit systems have not been reported earlier.

We show that the quantum discord in a bipartite quantum state is invariant under the action of a ... more We show that the quantum discord in a bipartite quantum state is invariant under the action of a local quantum channel if and only if the channel is invertible. In particular, quantum discord is invariant under a local unitary channel.

We settle the so-called degree conjecture for the separability of multipartite quantum states, wh... more We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. ͓Phys. Rev. A 73, 012320 ͑2006͔͒. 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 pure multipartite quantum states using the modified tensor product of graphs defined by Hassan and Joag ͓J. Phys. A 40, 10251 ͑2007͔͒, 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 counterexample 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.

We investigate how thermal quantum discord (QD) and classical correlations (CC) of a two-qubit on... more 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 (B 1 = −B 2 ), we find that QD and CC are equal for all values of (B 1 = −B 2 ) 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 B 1 and B 2 . Finally, we connect our results with the monogamy relations between the EOF, CC and the QD of two qubits and the environment.

We present a multipartite entanglement measure for N-qudit pure states, using the norm of the cor... more 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 an important class of N-qutrit pure states, namely, general GHZ states. We prove that this measure possesses all the essential and many desirable properties expected of a good entanglement measure, including monotonicity. We also discuss the feasibility of the experimental evaluation of this measure for an N-qutrit system.

Quantum discord, as introduced by Ollivier and Zurek (2001 Phys. Rev. Lett. 88 017901), is a meas... more Quantum discord, as introduced by Ollivier and Zurek (2001 Phys. Rev. Lett. 88 017901), 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 et al (2010 Phys. Rev. Lett. 105 190502) who introduced a geometric measure of quantum discord and derived an explicit formula for any two-qubit state. Recently, Luo and Fu (2010 Phys. Rev. A 82 034302) 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 an N-qubit quantum state. The exact formulas for the N-qubit quantum state can be used to get experimental estimates of the quantum discord and the total quantum correlation.

We present a multipartite entanglement measure for N-qubit pure states, using the norm of the cor... more 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 Greenberger-Horne-Zeilinger and W states and their superpositions. We compute this measure for interesting applications like the 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. of the SU͑2͒ group ͑Pauli matrices͒. These Hermitian operators form an orthogonal basis ͑under the Hilbert-Schmidt scalar product͒ of the Hilbert space of operators acting on a singlequbit state space. The N times tensor product of this basis with itself generates a product basis of the Hilbert space of operators acting on the N-qubit state space. Any N-qubit density operator can be expanded in this basis. The corre-* alisaif@physics.unipune.ernet.in † pramod@physics.unipune.ernet.in PHYSICAL REVIEW A 77, 062334 ͑2008͒

We settle the so-called degree conjecture for the separability of multipartite quantum states, wh... more We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. ͓Phys. Rev. A 73, 012320 ͑2006͔͒. 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 pure multipartite quantum states using the modified tensor product of graphs defined by Hassan and Joag ͓J. Phys. A 40, 10251 ͑2007͔͒, 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 counterexample 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.

We develop a geometric approach to quantify the capability of creating entanglement for a general... more 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 ͓Ali Saif M. Hassan and Pramod S. Joag, Phys. Rev. A 77, 062334 ͑2008͔͒. This geometric method has the distinct advantage that it gives the experimentally implementable criteria to ensure the optimal entanglement production rate without requiring a detailed knowledge of the state of the two qubit system. For the production of entanglement in practice, we need criteria for optimal entanglement production, which can be checked in situ without any need to know the state, as experimentally finding out the state of a quantum system is generally a formidable task. Further, we use our method to quantify the entanglement capacity in higher level and multipartite systems. We quantify the entanglement capacity for two qutrits and find the maximal entanglement generation rate and the corresponding state for the general isotropic interaction between qutrits, using the entanglement measure of N-qudit pure states proposed by us ͓Ali Saif M. Hassan and Pramod S. Joag, Phys. Rev. A 80, 042302 ͑2009͔͒. Next we quantify the genuine three qubit entanglement capacity for a general interaction between qubits. We obtain the maximum entanglement generation rate and the corresponding three qubit state for a general isotropic interaction between qubits. The state maximizing the entanglement generation rate is of the Greenberger-Horne-Zeilinger class. To the best of our knowledge, the entanglement capacities for two qutrit and three qubit systems have not been reported earlier.

We give a new separability criterion, a necessary condition for separability of Npartite quantum ... more We give a new separability criterion, a necessary condition for separability of Npartite 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 arbitrary N-partite quantum states in

In this paper we give a method to associate a graph with an arbitrary density matrix referred to ... more 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 related issues such as 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 (Werner 1989 Phys. Rev. A 40 4277) of a separable state can be written in terms of graphs, for the states in a real or complex Hilbert space. We generalize the separability criterion (degree criterion) due to Braunstein et al (2006 Phys. Rev. A 73 012320) 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.

We settle the so-called degree conjecture for the separability of multipartite quantum states, wh... more We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. ͓Phys. Rev. A 73, 012320 ͑2006͔͒. 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 pure multipartite quantum states using the modified tensor product of graphs defined by Hassan and Joag ͓J. Phys. A 40, 10251 ͑2007͔͒, 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 counterexample 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.