In measurement-based quantum computing one starts with a large entangled resource state |Ψ RES on... more In measurement-based quantum computing one starts with a large entangled resource state |Ψ RES on n qubits. We identify a two sets of qubits, I which will represent the inputs, and O which will represent the outputs of the computation, with n ≥ |O| ≥ |I|. Generally one can consider three types of computation using this resource, one with a classical input and a classical output (let's call this CC), one with a quantum input and a classical output QC and one with a quantum input and a quantum output QQ. Clearly QQ is the most general, since one can always encode classical information onto quantum states. In this work we focus on QQ. When considering a quantum input |ψ S (on a system S of |I| qubits) the first step is to teleport the input system qubits S onto I on the resource state, by some global map on I and S. This can be done for example by entangling I with S (using, say, a control-Z gate) and performing Pauli X measurements on S then appropriate corrections (see e.g.
We exactly evaluate a number of multipartite entanglement measures for a class of graph states, i... more We exactly evaluate a number of multipartite entanglement measures for a class of graph states, including d-dimensional cluster states (d = 1, 2, 3), the Greenberger-Horne-Zeilinger states, and some related mixed states. The entanglement measures that we consider are continuous, 'distance from separable states' measures, including the relative entropy, the so-called geometric measure, and robustness of entanglement. We also show that for our class of graph states these entanglement values give an operational interpretation as the maximal number of graph states distinguishable by local operations and classical communication (LOCC), as well as supplying a tight bound on the fixed letter classical capacity under LOCC decoding.
Electronic Proceedings in Theoretical Computer Science, 2010
In this article we extend on work which establishes an analogy between one-way quantum computatio... more In this article we extend on work which establishes an analogy between one-way quantum computation and thermodynamics to see how the former can be performed on fractal lattices. We find fractals lattices of arbitrary dimension greater than one which do all act as good resources for one-way quantum computation, and sets of fractal lattices with dimension greater than one all of which do not. The difference is put down to other topological factors such as ramification and connectivity. This work adds confidence to the analogy and highlights new features to what we require for universal resources for one-way quantum computation.
Permutation-symmetric quantum states appear in a variety of physical situations, and they have be... more Permutation-symmetric quantum states appear in a variety of physical situations, and they have been proposed for quantum information tasks. This article builds upon the results of [New J. Phys. 12, 073025 (2010)], where the maximally entangled symmetric states of up to twelve qubits were explored, and their amount of geometric entanglement determined by numeric and analytic means. For this the Majorana representation, a generalization of the Bloch sphere representation, can be employed to represent symmetric n qubit states by n points on the surface of a unit sphere. Symmetries of this point distribution simplify the determination of the entanglement, and enable the study of quantum states in novel ways. Here it is shown that the duality relationship of Platonic solids has a counterpart in the Majorana representation, and that in general maximally entangled symmetric states neither correspond to anticoherent spin states nor to spherical designs. The usability of symmetric states as resources for measurement-based quantum computing is also discussed.
We trace the resistance to entanglement generation of spin coherent states when passed through a ... more We trace the resistance to entanglement generation of spin coherent states when passed through a beam splitter as we vary S through S = 1/2 → ∞. In the infinite S limit the spin coherent states are equivalent to the high-amplitude limit of the optical coherent states. These states generate no entanglement and are completely distinguishable. This transition is discussed in terms of the classicality of the states. The decline of the generated entanglement, and in this sense increase in classicality with S, is very slow and dependent on the amplitude z of the state. Surprisingly we find that, for |z| > 1, there is an initial increase in entanglement followed by an extremely gradual decline to zero. Other aspects of classicality are also discussed over the transition in S, including the distinguishability, which decreases quickly and monotonically. We illustrate the distinguishability of spin-coherent states using the representation of Majorana.
In this paper we argue that one-way quantum computation can be seen as a form of phase transition... more In this paper we argue that one-way quantum computation can be seen as a form of phase transition with the available information about the solution of the computation being the order parameter. We draw a number of striking analogies between standard thermodynamical quantities such as energy, temperature, work, and corresponding computational quantities such as the amount of entanglement, time, potential
We show that entanglement guarantees difficulty in the discrimination of orthogonal multipartite ... more We show that entanglement guarantees difficulty in the discrimination of orthogonal multipartite states locally. The number of pure states that can be discriminated by local operations and classical communication is bounded by the total dimension over the average entanglement. A similar, general condition is also shown for pure and mixed states. These results offer a rare operational interpretation for three abstractly defined distance like measures of multipartite entanglement. * Authors are listed alphabetically. Corresponding authors are Markham ([email protected]),
In this paper we study the nonlocal properties of permutation symmetric states of n-qubits. We sh... more In this paper we study the nonlocal properties of permutation symmetric states of n-qubits. We show that all these states are nonlocal, via an extended version of the Hardy paradox and associated inequalities. Natural extensions of both the paradoxes and the inequalities are developed which relate different entanglement classes to different nonlocal features. Belonging to a given entanglement class will guarantee the violation of associated Bell inequalities which see the persistence of correlations to subsets of players, whereas there are states outside that class which do not violate. PACS numbers: 03.65.Ud, 03.67.Mn
We study various distance-like entanglement measures of multipartite states under certain symmetr... more We study various distance-like entanglement measures of multipartite states under certain symmetries. Using group averaging techniques we provide conditions under which the relative entropy of entanglement, the geometric measure of entanglement and the logarithmic robustness are equivalent. We consider important classes of multiparty states, and in particular show that these measures are equivalent for all stabilizer states, symmetric basis and antisymmetric basis states. We rigorously prove a conjecture that the closest product state of permutation symmetric states can always be chosen to be permutation symmetric. This allows us to calculate the explicit values of various entanglement measures for symmetric and antisymmetric basis states, observing that antisymmetric states are generally more entangled. We use these results to obtain a variety of interesting ensembles of quantum states for which the optimal LOCC discrimination probability may be explicitly determined and achieved. We also discuss applications to the construction of optimal entanglement witnesses.
We trace the resistance to entanglement generation of spin coherent states when passed through a ... more We trace the resistance to entanglement generation of spin coherent states when passed through a beam splitter as we vary S through S = 1/2 → ∞. In the infinite S limit the spin coherent states are equivalent to the high-amplitude limit of the optical coherent states. These states generate no entanglement and are completely distinguishable. This transition is discussed in terms of the classicality of the states. The decline of the generated entanglement, and in this sense increase in classicality with S, is very slow and dependent on the amplitude z of the state. Surprisingly we find that, for |z| > 1, there is an initial increase in entanglement followed by an extremely gradual decline to zero. Other aspects of classicality are also discussed over the transition in S, including the distinguishability, which decreases quickly and monotonically. We illustrate the distinguishability of spin-coherent states using the representation of Majorana.
In this paper, building on some recent progress combined with numerical techniques, we shed some ... more In this paper, building on some recent progress combined with numerical techniques, we shed some new light on how the nonlocality of symmetric states is related to their entanglement properties and potential usefulness in quantum information processing. We use semidefinite programming techniques to devise a device independent classification of three four qubit states into two classes inequivalent under local unitaries and permutation of systems (LUP). We study nonlocal properties when the number of parties grows large for two important classes of symmetric states: the W states and the GHZ states, showing that they behave differently under the inequalities we consider. We also discuss the monogamy arising from the nonlocal correlations of symmetric states. We show that although monogamy in a strict sense is not guaranteed for all symmetric states, strict monogamy is achievable for all Dicke states when the number of parties goes to infinity.
We study the robustness of multipartite entanglement of the ground state of the one-dimensional s... more We study the robustness of multipartite entanglement of the ground state of the one-dimensional spin 1/2 XY model with a transverse magnetic field in the presence of thermal excitations, by investigating a threshold temperature, below which the thermal state is guaranteed to be entangled. We obtain the threshold temperature based on the geometric measure of entanglement of the ground state. The threshold temperature reflects three characteristic lines in the phase diagram of the correlation function. Our approach reveals a region where multipartite entanglement at zero temperature is high but is thermally fragile, and another region where multipartite entanglement at zero temperature is low but is thermally robust.
The geometric measure of entanglement is investigated for permutation symmetric pure states of mu... more The geometric measure of entanglement is investigated for permutation symmetric pure states of multipartite qubit systems, in particular the question of maximum entanglement. This is done with the help of the Majorana representation, which maps an n qubit symmetric state to n points on the unit sphere. It is shown how symmetries of the point distribution can be exploited to simplify the calculation of entanglement and also help find the maximally entangled symmetric state. Using a combination of analytical and numerical results, the most entangled symmetric states for up to 12 qubits are explored and discussed. The optimization problem on the sphere presented here is then compared with two classical optimization problems on the S 2 sphere, namely Tóth's problem and Thomson's problem, and it is observed that, in general, they are different problems.
Scalable quantum computing and communication requires the protection of quantum information from ... more Scalable quantum computing and communication requires the protection of quantum information from the detrimental effects of decoherence and noise. Previous work tackling this problem has relied on the original circuit model for quantum computing. However, recently a family of entangled resources known as graph states has emerged as a versatile alternative for protecting quantum information. Depending on the graph's structure, errors can be detected and corrected in an efficient way using measurement-based techniques. Here we report an experimental demonstration of error correction using a graph state code. We use an all-optical setup to encode quantum information into photons representing a four-qubit graph state. We are able to reliably detect errors and correct against qubit loss. The graph we realize is setup independent, thus it could be employed in other physical settings. Our results show that graph state codes are a promising approach for achieving scalable quantum information processing.
In this article we investigate the possibility of encoding classical information onto multipartit... more In this article we investigate the possibility of encoding classical information onto multipartite quantum states in the distant laboratory framework. We show that for all states generated by Clifford operations there always exists such an encoding, this includes all stabilizer states such as cluster states and all graph states. We also show local encoding for classes of symmetric states (which cannot be generated by Clifford operations). We generalise our approach using group theoretic methods introducing the unifying notion of Pseudo Clifford operations. All states generated by Pseudo Clifford operations are locally encodable (unifying all our examples), and we give a general method for generating sets of many such locally encodable states.
In this paper for a class of symmetric multiparty pure states we consider a conjecture related to... more In this paper for a class of symmetric multiparty pure states we consider a conjecture related to the geometric measure of entanglement: "for a symmetric pure state, the closest product state in terms of the fidelity can be chosen as a symmetric product state". We show that this conjecture is true for symmetric pure states whose amplitudes are all non-negative in a computational basis.
In measurement-based quantum computing one starts with a large entangled resource state |Ψ RES on... more In measurement-based quantum computing one starts with a large entangled resource state |Ψ RES on n qubits. We identify a two sets of qubits, I which will represent the inputs, and O which will represent the outputs of the computation, with n ≥ |O| ≥ |I|. Generally one can consider three types of computation using this resource, one with a classical input and a classical output (let's call this CC), one with a quantum input and a classical output QC and one with a quantum input and a quantum output QQ. Clearly QQ is the most general, since one can always encode classical information onto quantum states. In this work we focus on QQ. When considering a quantum input |ψ S (on a system S of |I| qubits) the first step is to teleport the input system qubits S onto I on the resource state, by some global map on I and S. This can be done for example by entangling I with S (using, say, a control-Z gate) and performing Pauli X measurements on S then appropriate corrections (see e.g.
We exactly evaluate a number of multipartite entanglement measures for a class of graph states, i... more We exactly evaluate a number of multipartite entanglement measures for a class of graph states, including d-dimensional cluster states (d = 1, 2, 3), the Greenberger-Horne-Zeilinger states, and some related mixed states. The entanglement measures that we consider are continuous, 'distance from separable states' measures, including the relative entropy, the so-called geometric measure, and robustness of entanglement. We also show that for our class of graph states these entanglement values give an operational interpretation as the maximal number of graph states distinguishable by local operations and classical communication (LOCC), as well as supplying a tight bound on the fixed letter classical capacity under LOCC decoding.
Electronic Proceedings in Theoretical Computer Science, 2010
In this article we extend on work which establishes an analogy between one-way quantum computatio... more In this article we extend on work which establishes an analogy between one-way quantum computation and thermodynamics to see how the former can be performed on fractal lattices. We find fractals lattices of arbitrary dimension greater than one which do all act as good resources for one-way quantum computation, and sets of fractal lattices with dimension greater than one all of which do not. The difference is put down to other topological factors such as ramification and connectivity. This work adds confidence to the analogy and highlights new features to what we require for universal resources for one-way quantum computation.
Permutation-symmetric quantum states appear in a variety of physical situations, and they have be... more Permutation-symmetric quantum states appear in a variety of physical situations, and they have been proposed for quantum information tasks. This article builds upon the results of [New J. Phys. 12, 073025 (2010)], where the maximally entangled symmetric states of up to twelve qubits were explored, and their amount of geometric entanglement determined by numeric and analytic means. For this the Majorana representation, a generalization of the Bloch sphere representation, can be employed to represent symmetric n qubit states by n points on the surface of a unit sphere. Symmetries of this point distribution simplify the determination of the entanglement, and enable the study of quantum states in novel ways. Here it is shown that the duality relationship of Platonic solids has a counterpart in the Majorana representation, and that in general maximally entangled symmetric states neither correspond to anticoherent spin states nor to spherical designs. The usability of symmetric states as resources for measurement-based quantum computing is also discussed.
We trace the resistance to entanglement generation of spin coherent states when passed through a ... more We trace the resistance to entanglement generation of spin coherent states when passed through a beam splitter as we vary S through S = 1/2 → ∞. In the infinite S limit the spin coherent states are equivalent to the high-amplitude limit of the optical coherent states. These states generate no entanglement and are completely distinguishable. This transition is discussed in terms of the classicality of the states. The decline of the generated entanglement, and in this sense increase in classicality with S, is very slow and dependent on the amplitude z of the state. Surprisingly we find that, for |z| > 1, there is an initial increase in entanglement followed by an extremely gradual decline to zero. Other aspects of classicality are also discussed over the transition in S, including the distinguishability, which decreases quickly and monotonically. We illustrate the distinguishability of spin-coherent states using the representation of Majorana.
In this paper we argue that one-way quantum computation can be seen as a form of phase transition... more In this paper we argue that one-way quantum computation can be seen as a form of phase transition with the available information about the solution of the computation being the order parameter. We draw a number of striking analogies between standard thermodynamical quantities such as energy, temperature, work, and corresponding computational quantities such as the amount of entanglement, time, potential
We show that entanglement guarantees difficulty in the discrimination of orthogonal multipartite ... more We show that entanglement guarantees difficulty in the discrimination of orthogonal multipartite states locally. The number of pure states that can be discriminated by local operations and classical communication is bounded by the total dimension over the average entanglement. A similar, general condition is also shown for pure and mixed states. These results offer a rare operational interpretation for three abstractly defined distance like measures of multipartite entanglement. * Authors are listed alphabetically. Corresponding authors are Markham ([email protected]),
In this paper we study the nonlocal properties of permutation symmetric states of n-qubits. We sh... more In this paper we study the nonlocal properties of permutation symmetric states of n-qubits. We show that all these states are nonlocal, via an extended version of the Hardy paradox and associated inequalities. Natural extensions of both the paradoxes and the inequalities are developed which relate different entanglement classes to different nonlocal features. Belonging to a given entanglement class will guarantee the violation of associated Bell inequalities which see the persistence of correlations to subsets of players, whereas there are states outside that class which do not violate. PACS numbers: 03.65.Ud, 03.67.Mn
We study various distance-like entanglement measures of multipartite states under certain symmetr... more We study various distance-like entanglement measures of multipartite states under certain symmetries. Using group averaging techniques we provide conditions under which the relative entropy of entanglement, the geometric measure of entanglement and the logarithmic robustness are equivalent. We consider important classes of multiparty states, and in particular show that these measures are equivalent for all stabilizer states, symmetric basis and antisymmetric basis states. We rigorously prove a conjecture that the closest product state of permutation symmetric states can always be chosen to be permutation symmetric. This allows us to calculate the explicit values of various entanglement measures for symmetric and antisymmetric basis states, observing that antisymmetric states are generally more entangled. We use these results to obtain a variety of interesting ensembles of quantum states for which the optimal LOCC discrimination probability may be explicitly determined and achieved. We also discuss applications to the construction of optimal entanglement witnesses.
We trace the resistance to entanglement generation of spin coherent states when passed through a ... more We trace the resistance to entanglement generation of spin coherent states when passed through a beam splitter as we vary S through S = 1/2 → ∞. In the infinite S limit the spin coherent states are equivalent to the high-amplitude limit of the optical coherent states. These states generate no entanglement and are completely distinguishable. This transition is discussed in terms of the classicality of the states. The decline of the generated entanglement, and in this sense increase in classicality with S, is very slow and dependent on the amplitude z of the state. Surprisingly we find that, for |z| > 1, there is an initial increase in entanglement followed by an extremely gradual decline to zero. Other aspects of classicality are also discussed over the transition in S, including the distinguishability, which decreases quickly and monotonically. We illustrate the distinguishability of spin-coherent states using the representation of Majorana.
In this paper, building on some recent progress combined with numerical techniques, we shed some ... more In this paper, building on some recent progress combined with numerical techniques, we shed some new light on how the nonlocality of symmetric states is related to their entanglement properties and potential usefulness in quantum information processing. We use semidefinite programming techniques to devise a device independent classification of three four qubit states into two classes inequivalent under local unitaries and permutation of systems (LUP). We study nonlocal properties when the number of parties grows large for two important classes of symmetric states: the W states and the GHZ states, showing that they behave differently under the inequalities we consider. We also discuss the monogamy arising from the nonlocal correlations of symmetric states. We show that although monogamy in a strict sense is not guaranteed for all symmetric states, strict monogamy is achievable for all Dicke states when the number of parties goes to infinity.
We study the robustness of multipartite entanglement of the ground state of the one-dimensional s... more We study the robustness of multipartite entanglement of the ground state of the one-dimensional spin 1/2 XY model with a transverse magnetic field in the presence of thermal excitations, by investigating a threshold temperature, below which the thermal state is guaranteed to be entangled. We obtain the threshold temperature based on the geometric measure of entanglement of the ground state. The threshold temperature reflects three characteristic lines in the phase diagram of the correlation function. Our approach reveals a region where multipartite entanglement at zero temperature is high but is thermally fragile, and another region where multipartite entanglement at zero temperature is low but is thermally robust.
The geometric measure of entanglement is investigated for permutation symmetric pure states of mu... more The geometric measure of entanglement is investigated for permutation symmetric pure states of multipartite qubit systems, in particular the question of maximum entanglement. This is done with the help of the Majorana representation, which maps an n qubit symmetric state to n points on the unit sphere. It is shown how symmetries of the point distribution can be exploited to simplify the calculation of entanglement and also help find the maximally entangled symmetric state. Using a combination of analytical and numerical results, the most entangled symmetric states for up to 12 qubits are explored and discussed. The optimization problem on the sphere presented here is then compared with two classical optimization problems on the S 2 sphere, namely Tóth's problem and Thomson's problem, and it is observed that, in general, they are different problems.
Scalable quantum computing and communication requires the protection of quantum information from ... more Scalable quantum computing and communication requires the protection of quantum information from the detrimental effects of decoherence and noise. Previous work tackling this problem has relied on the original circuit model for quantum computing. However, recently a family of entangled resources known as graph states has emerged as a versatile alternative for protecting quantum information. Depending on the graph's structure, errors can be detected and corrected in an efficient way using measurement-based techniques. Here we report an experimental demonstration of error correction using a graph state code. We use an all-optical setup to encode quantum information into photons representing a four-qubit graph state. We are able to reliably detect errors and correct against qubit loss. The graph we realize is setup independent, thus it could be employed in other physical settings. Our results show that graph state codes are a promising approach for achieving scalable quantum information processing.
In this article we investigate the possibility of encoding classical information onto multipartit... more In this article we investigate the possibility of encoding classical information onto multipartite quantum states in the distant laboratory framework. We show that for all states generated by Clifford operations there always exists such an encoding, this includes all stabilizer states such as cluster states and all graph states. We also show local encoding for classes of symmetric states (which cannot be generated by Clifford operations). We generalise our approach using group theoretic methods introducing the unifying notion of Pseudo Clifford operations. All states generated by Pseudo Clifford operations are locally encodable (unifying all our examples), and we give a general method for generating sets of many such locally encodable states.
In this paper for a class of symmetric multiparty pure states we consider a conjecture related to... more In this paper for a class of symmetric multiparty pure states we consider a conjecture related to the geometric measure of entanglement: "for a symmetric pure state, the closest product state in terms of the fidelity can be chosen as a symmetric product state". We show that this conjecture is true for symmetric pure states whose amplitudes are all non-negative in a computational basis.
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Papers by Damian Markham