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A special simplex in the state space for entangled qudits

Baumgartner, B, Beatrix C. Hiesmayr, Narnhofer, H

Published Oct 12, 2006• 68 citations
OpenAlex

Abstract

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.

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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.
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.
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 information processing and communication techniques rely heavily upon entangled quantum states, and this dependence motivates the development of methods and systems to generate entanglement. Much research has been dedicated to state preparation for radix-2 qubits, and due to the pursuit of entangled states, the Bell state generator and its generalized forms where the number of entangled qubits is greater than two have been defined. In this work, we move beyond radix-2 and propose techniques for quantum state entanglement in high-dimensional systems through the generalization of the binary bipartite entanglement states. These higher-radix quantum informatic systems are composed of n quantum digits, or qudits, that are each mathematically characterized as elements of an r -dimensioned Hilbert vector space where r > 2. Consequently, the wave function is a time-dependent state vector of dimension r n . The generalization of the binary controlled-NOT to the controlled-modulo-addition gate, the concept of partial versus maximal entanglement, and architectures for generating higher-radix entangled states for the partial and maximal case are all presented.