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arXiv:quant-ph/0307124 · Physical Review A

Entanglement, mixedness, and spin-flip symmetry in multiple-qubit systems

Gregg S. Jaeger, Alexander V. Sergienko, Bahaa E. A. Saleh, Malvin Carl Teich

Published Aug 27, 2003• 59 citations
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Abstract

A relationship between a recently introduced multipartite entanglement measure, state mixedness, and spin-flip symmetry is established for any finite number of qubits. It is also shown that, for those classes of states invariant under the spin-flip transformation, there is a complementarity relation between multipartite entanglement and mixedness. A number of example classes of multiple-qubit systems are studied in light of this relationship.

DOI
10.1103/physreva.68.022318

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A relationship between a recently introduced multipartite entanglement measure, state mixedness, and spin-flip symmetry is established for any finite number of qubits. It is also shown that, for those classes of states invariant under the spin-flip transformation, there is a complementarity relation between multipartite entanglement and mixedness. A number of example classes of multiple-qubit systems are studied in light of this relationship.
The behavior of quantum states under local unitary transformations (LUTs) and stochastic local quantum operations and classical communication (SLOCC) has proven central to the understanding of entanglement in multipartite quantum systems. In particular, invariants under these operations have provided insight into quantum entanglement in multiple‐qubit states. Relationships between entanglement, mixedness and spin symmetry in multiple‐qubit quantum states can be found by exploiting these properties. For example, concurrence and n‐tangle are naturally expressed in terms of spin‐flip transformations. In the case of specialized entanglement measures and/or special families of states, complementarity relations involving these lengths and state transformations are derivable. Here, the role of geometry in such investigations is explored. Minkowskian geometry is seen to provide elegant representations of entanglement in multiple‐qubit systems, particularly the language of twistors.
Via a multidimensional complementarity relation we derive an operational entanglement measure for any discrete quantum system, i.e., for any multidimensional and multipartite system. This measure admits a separation into different classes of entanglement obtained by using a flip operator 2,3,…,n2,3,\dots{},n2,3,…,n times, defining a mmm-flip concurrence. This operator sum has the practical feature to allow one to calculate for mixed states bounds on this mmm-flip concurrence. Moreover, the information content of an nnn-partite multidimensional system admits a simple and intuitive interpretation in terms of single particle obtainable information, entanglement, and information due to lack of classical knowledge of the quantum state under investigation. Explicitly, the three qubit system is analyzed and, e.g., the physical difference in entanglement of the W state, the Greenberger-Horne-Zeilinger (GHZ) state, or a biseparable state is revealed.
We derive two complementarity relations that constrain the individual and bipartite properties that may simultaneously exist in a multi-qubit system. The first expression, valid for an arbitrary pure state of n qubits, demonstrates that the degree to which single particle properties are possessed by an individual member of the system is limited by the bipartite entanglement that exists between that qubit and the remainder of the system. This result implies that the phenomenon of entanglement sharing is one specific consequence of complementarity. The second expression, which holds for an arbitrary state of two qubits, pure or mixed, quantifies a tradeoff between the amounts of entanglement, separable uncertainty, and single particle properties that are encoded in the quantum state. The separable uncertainty is a natural measure of our ignorance about the properties possessed by individual subsystems, and may be used to completely characterize the relationship between entanglement and mixedness in two-qubit systems. The two-qubit complementarity relation yields a useful geometric picture in which the root mean square values of local subsystem properties act like coordinates in the space of density matrices, and suggests possible insights into the problem of interpreting quantum mechanics.
In this work, superposition of spin coherent states composed of two qubits is formed and by calculating the geometric measure of mixing of the system introduced in this work, the degree of mixedness of the studied state is investigated. In the following, the changes in the entanglement of the introduced state are studied by using the negativity measure. Finally, by drawing the appropriate plots, according to the influential parameters in this system, we examine the maximum and the minimum of the changes in the mixedness and entanglement and also their relationship to the mixed system consisting of two-qubit spin coherent states. It was observed that for maximizing α, when the changes of parameters indicate the purity of the state, the entanglement of the system reaches its maximum value for two-qubit systems. Also, for all values of coherence parameter, the entanglement decreases when mixedness of the system increases.
The mixedness of the N -qubit quantum states with exchange symmetry has been studied, and the results show that the linear entropy of the single qubit reduced density matrix (RDM), which can describe the mixedness, is completely determined by the expectation values ( S z ) and ( S ± ) for both the pure and the mixed states. The mixedness of the pure states can be used to describe the bipartite entanglement, as an example we have calculated the mixedness of the Dicke state and the spin squeezed Kitagawa–Ueda state. For the mixed states, we determine the mixedness properties of both the ground states and the thermal states in mean-field clusters of spin-1/2 particles interacting via the anisotropy Heisenberg XXZ interaction, and found for the ferromagnetic case ( J < 0), the mixedness will approximate to the pairwise entanglement when the anisotropic parameter Δ > Δ c .
A general method in constructing a complete set of wave functions for multipartite identical qubits is presented based on the irreducible representations of the permutation group and the nth rank tensors. Particular examples for n =2, 3, and 4 are derived and the entanglement behavior for each state is examined from several criteria. It is found that the states so constructed are all bound entangled states. For the case of even n, all the states are found to have maximum "n-tangle". The symmetry in spin space is found to increase the n-tangle in general. The "n-tangle" for n = 4 is found not always representing 4-way entanglement. It measures the degree of spin-space symmetry instead. A useful relationship in the classification between systems containing different number of qubits is given in terms of the Young's Tableaux based on our analysis.