Bounds on probability of transformations between multi-partite pure states

Abstract

For a tripartite pure state of three qubits, it is well known that there are two inequivalent classes of genuine tripartite entanglement, namely the GHZ-class and the W-class. Any two states within the same class can be transformed into each other with stochastic local operations and classical communication (SLOCC) with a non-zero probability. The optimal conversion probability, however, is only known for special cases. Here, we derive new lower and upper bounds for the optimal probability of transformation from a GHZ-state to other states of the GHZ-class. A key idea in the derivation of the upper bounds is to consider the action of the LOCC protocol on a different input state, namely $1/\sqrt{2}[|000⟩-|111⟩]$, and demand that the probability of an outcome remains bounded by 1. We also find an upper bound for more general cases by using the constraints of the so-called interference term and 3-tangle. Moreover, we generalize some of our results to the case where each party holds a higher-dimensional system. In particular, we found that the GHZ state generalized to three qutrits, i.e., $|{\mathrm{GHZ}}_3⟩=1/\sqrt{3}[|000⟩+|111⟩+|222⟩]$, shared among three parties can be transformed to {\it any} tripartite 3-qubit pure state with probability 1 via LOCC. Some of our results can also be generalized to the case of a multipartite state shared by more than three parties.

16 pages, 11 figures

References in corpus (5)

• Four qubits can be entangled in nine different ways
• Weak measurements are universal
• Tripartite entanglement transformations and tensor rank
• Binary search trees for generalized measurement
• Conditions for entanglement transformation between a class of multipartite pure states with generalized Schmidt decompositions

Cited by in corpus (9)

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• Necessary and sufficient conditions for local manipulation of multipartite pure quantum states
• Transformations of $W$-Type Entangled States
• Randomly distilling W-class states into general configurations of two-party entanglement
• Deterministic LOCC transformation of three-qubit pure states and entanglement transfer
• Deterministic transformations of three-qubit entangled pure states
• Canonical operators and the optimal concentration of three-qubit Greenberger-Horne-Zeilinger states
• Optimal Local Transformations of Flip and Exchange Symmetric Entangled States
• Approximate and ensemble local entanglement transformations for multipartite states