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arXiv:quant-ph/0610100 · Journal of Physics A Mathematical and Theoretical

A special simplex in the state space for entangled qudits

Bernhard Baumgartner, Beatrix C. Hiesmayr, Heide Narnhofer

Published Jun 27, 2007• 95 citations
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Abstract

The focus is on two parties with Hilbert spaces of dimension d , i.e. 'qudits'. In the state space of these two possibly entangled qudits an analogue to the well-known tetrahedron with the four qubit Bell states at the vertices is presented. The simplex analogue to this magic tetrahedron includes mixed states. Each of these states appears to each of the two parties as the maximally mixed state. Some studies on these states are performed, and special elements of this set are identified. A large number of them are included in the chosen simplex which fits exactly into conditions needed for teleportation and other applications. Its rich symmetry—related to that of a classical phase space—helps to study entanglement, to construct witnesses and perform partial transpositions. This simplex has been explored in detail for d = 3. In this paper, the mathematical background and extensions to arbitrary dimensions are analysed.

DOI
10.1088/1751-8113/40/28/s03

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The focus is on two parties with Hilbert spaces of dimension d , i.e. 'qudits'. In the state space of these two possibly entangled qudits an analogue to the well-known tetrahedron with the four qubit Bell states at the vertices is presented. The simplex analogue to this magic tetrahedron includes mixed states. Each of these states appears to each of the two parties as the maximally mixed state. Some studies on these states are performed, and special elements of this set are identified. A large number of them are included in the chosen simplex which fits exactly into conditions needed for teleportation and other applications. Its rich symmetry—related to that of a classical phase space—helps to study entanglement, to construct witnesses and perform partial transpositions. This simplex has been explored in detail for d = 3. In this paper, the mathematical background and extensions to arbitrary dimensions are analysed.
Focus is on two parties with Hilbert spaces of dimension d, i.e. "qudits". In the state space of these two possibly entangled qudits an analogue to the well known tetrahedron with the four qubit Bell states at the vertices is presented. The simplex analogue to this magic tetrahedron includes mixed states. Each of these states appears to each of the two parties as the maximally mixed state. Some studies on these states are performed, and special elements of this set are identified. A large number of them is included in the chosen simplex which fits exactly into conditions needed for teleportation and other applications. Its rich symmetry - related to that of a classical phase space - helps to study entanglement, to construct witnesses and perform partial transpositions. This simplex has been explored in details for d=3. In this paper the mathematical background and extensions to arbitrary dimensions are analysed.
We investigate the state space of bipartite qutrits. For states which are locally maximally mixed we obtain an analog of the ``magic'' tetrahedron for bipartite qubits---a magic simplex W\mathcal{W}W. This is obtained via the Weyl group which is a kind of ``quantization'' of classical phase space. We analyze how this simplex W\mathcal{W}W is embedded in the whole state space of two qutrits and discuss symmetries and equivalences inside the simplex W\mathcal{W}W. Because we are explicitly able to construct optimal entanglement witnesses we obtain the border between separable and entangled states. With our method we find also the total area of bound entangled states of the parameter subspace under intervestigation. Our considerations can also be applied to higher dimensions.
We show how to detect entangled, bound entangled and separable bipartite quantum states of arbitrary dimension and mixedness using geometric entanglement witnesses. These witnesses are constructed using properties of the Hilbert–Schmidt geometry and can be shifted along parameterized lines. The involved conditions are simplified using Bloch decompositions of operators and states. As an example we determine the three different types of states for a family of two-qutrit states that is part of the 'magic simplex', i.e. the set of Bell-state mixtures of arbitrary dimension.
The characterization of entanglement properties in mixed states is important from both a theoretical and a practical point of view. While the estimation of entanglement of bipartite pure states is well established, for mixed states it is a considerably much harder task. The key elements of the mixed-state entanglement theory are given by the exact solutions which sometimes are possible for special states of high symmetry problems. In this paper, we present the exact investigation on the entanglement properties for a five-parameter family of highly symmetric two-qudit mixed states with equal but arbitrary finite local Hilbert space dimension. We achieve this by extensive analysis of various conditions of separability and the entanglement classification with respect to stochastic local operations and classical communication. Furthermore, our results can be used for an arbitrary state by proper application of the proposed twirling operator.
Quantum simulations of many-body systems using 2-qubit Yang-Baxter gates offer a benchmark for quantum hardware. This can be extended to the higher dimensional case with nnn-qubit generalisations of Yang-Baxter gates called nnn-simplex operators. Such multi-qubit gates potentially lead to shallower and more efficient quantum circuits as well. Finding them amounts to identifying unitary solutions of the nnn-simplex equations, the building blocks of higher dimensional integrable systems. These are a set of highly non-linear and over determined system of equations making it notoriously hard to solve even when the local Hilbert spaces are spanned by qubits. We systematically overcome this for higher simplex operators constructed using two methods: from Clifford algebras and by lifting Yang-Baxter operators. The n=3n=3n=3 or the tetrahedron case is analyzed in detail. For the qubit case our methods produce 13 inequivalent families of unitary tetrahedron operators. 12 of these families are obtained by appending the 5 unitary families of 4 by 4 constant Yang-Baxter operators of Dye-Hietarinta, with a single qubit operator. As applications, universal sets of single, two and three qubit gates are realized using such unitary tetrahedron operators. The ideas presented in this work can be naturally extended to the higher simplex cases.