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Surfaces of Constant Discord for Bell-Diagonal States

Matthias Lang, Carlton M. Caves

Published Jun 14, 2010• 0 citations
OpenAlex

Abstract

The set of Bell-diagonal states for two qubits can be depicted as a tetrahedron in three dimensions. We consider the level surfaces of entanglement and quantum discord for Bell-diagonal states. This provides a complete picture of the structure of entanglement and discord for this simple case and, in particular, of their nonanalytic behavior under decoherence. The pictorial approach also indicates how to show that discord is neither convex nor concave.

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The set of Bell-diagonal states for two qubits can be depicted as a tetrahedron in three dimensions. We consider the level surfaces of entanglement and quantum discord for Bell-diagonal states. This provides a complete picture of the structure of entanglement and discord for this simple case and, in particular, of their nonanalytic behavior under decoherence. The pictorial approach also indicates how to show that discord is neither convex nor concave.
The level surfaces of quantum discord for a class of two-qubit states are investigated when the Bloch vectors and are perpendicularly oriented. The geometric objects of tetrahedron T and octahedron O are deformed. The level surfaces of constant discord are formed by three interaction "tubes" along three orthogonal directions. They shrink to the center when the Bloch vectors are increased and are expanded and cut off by the state tetrahedron T when the quantum discord is increased. In the phase damping channel, the quantum discord keeps approximately a constant when the time increases.
Two-qubit Bell-diagonal states can be depicted as a tetrahedron in three dimensions. We investigate the structure of quantum resources, including coherence and quantum discord, in the tetrahedron. The ordering of different resources measures is a common problem in resource theories, and which measure should be chosen to investigate the structure of resources is still an open question. We consider the structure of quantum resources which is not affected by the choice of measure. Our work provides a complete structure of coherence and quantum discord for Bell-diagonal states. The pictorial approach also indicates how to explore the structure of resources even when we don't have consistent measure of a concrete quantum resource.
We study the level surfaces of quantum discord for a class of two-qubit states with parallel nonzero Bloch vectors. The dynamic behavior of quantum discord under decoherence is investigated. It is shown that a class of XXX states has sudden transition between classical and quantum correlations under decoherence. Our results include the ones in M. D. Lang and C. M. Caves [Phys. Rev. Lett. 105, 150501 (2010)] as a special case and show new pictures and structures of quantum discord.
We plot the geometry of several distance-based quantifiers of coherence for Bell-diagonal states. We find that along with both l1l_{1}l1​ norm and relative entropy of coherence changes continuously from zero to one, their surfaces move from the separable regions to the entangled regions. Based on this fact, it is more illuminating to use an intuitive geometry to explain quantum states with nonzero coherence can be used for entanglement creation, rather than the other way around. We find the necessary and sufficient conditions that quantum discord of Bell-diagonal states equal to its relative entropy of coherence and depict the surfaces of the equality. We give surfaces of relative entropy of coherence for XXX states. We show the surfaces of dynamics of relative entropy of coherence for Bell-diagonal states under local nondissipative channels and find that all coherence under local nondissipative channels decrease.
We evaluate analytically the quantum discord for a large family of multiqubit states. It is interesting to note that the quantum discord of three qubits and five qubits is the same, as is the quantum discord of two qubits and six qubits. We discover that the quantum discord of this family of states can be classified into three categories. The level surfaces of the quantum discord in the three categories are shown through images. Furthermore, we investigated the dynamic behavior of quantum discord under decoherence. For the odd partite systems, we prove the frozen phenomenon of quantum discord does not exist under the phase flip channel, while it can be found in the even partite systems.