Extremal entanglement for triqubit pure states

Abstract

A complete analysis of entangled triqubit pure states is carried out based on a new simple entanglement measure. An analysis of all possible extremally entangled pure triqubit states with up to eight terms is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis presented are most helpful for finding different entanglement types in multipartite pure state systems.Entanglement is a fundamental concept that underpins quantum information and computation. [1−3] As a consequence, the quantification of entanglement emerges as a central challenge. Many authors have contributed to this topic, [4−23] among which the basic requirements for entanglement measures proposed in [4] provide with guidelines for its definition. In [6], Bennett et al defined stochastic local operations and classical communication (SLOCC) based on the concept of local operations assisted with classical communication (LOCC). Dür et al applied such an operation to a triqubit pure state system and found that triqubit states can be entangled at least in two inequivalent ways, [7] namely, in the GHZ form [24] or the W form. [7] In this Letter, we will use a recently proposed entanglement measure [23] for N -qubit pure states to analyze all extremally entangled triqubit pure states with the constrained maximization, and to see whether there are other inequivalent types of entanglement.In the following, based on the method used in [7], only entanglement properties of a single copy of a state will be considered. Therefore, asymptotic properties will not be discussed. At single copy level, it is well known that two pure states can always be transformed with certainty from each other by means of LOCC if and only if they are related by Local Unitary transformation (LU). [6] However, even in the simplest bipartite cases, entangled states are not always related by LU, and continuous parameters are needed to label all equivalence classes. Hence, it seems that one needs to deal with infinitely many kinds of entanglement. Fortunately, such arbitrariness has been overcome with the help of SLOCC. [7] According to [23], for a genuine entangled N -qubit pure state Ψ, the measure can be defined by

Entropy product measure for multipartite pure states

“…Now we proceed to discuss separately the various types of states according to the classification given in ref. [13].…”

“…[12] in which the conditions of triqubit pure states [13] and bipartite qutrit pure states [14] were discussed and a significant result was found. Some of us found that this measure is equivalent to the scheme of the distance from the maximal entangled state 1) .…”

Entanglement measure for pure six-qubit quantum states

“…Only permutations of the subsystems ABC and, separately, within DEF, do not change π 6 . For example, π 6 (|G H Z 3 …”

“…The complete understanding of the role played by the entanglement in quantum algorithms and protocols pass by its quantification and classification, thus, a good (and, preferable, easy to calculate) entanglement measure for multipartite system is important. There are a number of entanglement measures for multipartite (n > 2) quantum states [1][2][3][4][5][6]. Here, we are concerned with those based on negativity [7].…”

Effect of Substrate Temperature on Epitaxial Orientation of Rh Thin Films Sputtered on A-Plane Sapphire

“…The four-fold degenerate states at t = 0 come from the double occurrence of S = 1/2, while two states from S = −S 0 = −k + N/2 and another two from S = −S 0 = −(k + 1) + N/2 form the corresponding four-fold degeneracy for 0 < t 2/3. However, it has been proved, at least for small-N cases, that GHZ-and Wtype states are inequivalent under the stochastic local operations and classical communication (SLOCC) transformations [25][26][27]. Therefore, the ground state should be classified into three phases in the thermodynamic limit for the N even case under SLOCC.…”