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Journal of Physics A Mathematical and General

Studying entanglement-assisted entanglement transformation

Li‐Yi Hsu

Published Nov 11, 2004• 1 citation
Publisher

Abstract

In this paper, we study catalysis of entanglement transformations for n -level pure entangled states. We propose an algorithm of finding the required catalystic entanglement. We introduce several examples by way of demonstration. We evaluate the lower and upper bound of the required inequalities for deciding whether there are m -level appropriate catalyst states for entanglement transformations for two n -level pure entangled states.

DOI
10.1088/0305-4470/37/47/012

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In this paper, we study catalysis of entanglement transformations for n -level pure entangled states. We propose an algorithm of finding the required catalystic entanglement. We introduce several examples by way of demonstration. We evaluate the lower and upper bound of the required inequalities for deciding whether there are m -level appropriate catalyst states for entanglement transformations for two n -level pure entangled states.
We introduce the general catalysts for pure entanglement transformations under local operations and classical communications in such a way that we disregard the profit and loss of entanglement of the catalysts per se. As such, the possibilities of pure entanglement transformations are greatly expanded. Remarkably, we find an interesting phenomenon that, in some situations, incomparable pairs ${| ψ> ,| ϕ> \} $ and ${| χ> ,| χ^{\prime}> \} $ can assist each other mutually so as to realize the transformation ∣ψ>∣χ>→∣ϕ>∣χ′>| ψ> | χ> \to| ϕ> | χ^{\prime}>∣ψ>∣χ>→∣ϕ>∣χ′>. We also design an efficient algorithm to detect whether a k×kk\times kk×k general catalyst exists for a given entanglement transformation. This algorithm can as well be exploited to witness the existence of standard catalysts.
Given a pure state transformation $\ensuremath{\psi}\ensuremath{\mapsto}\ensuremath{\phi}$ restricted to entanglement-assisted local operations with classical communication, we determine a lower bound for the dimension of a catalyst allowing that transformation. Our bound is stated in terms of the generalized concurrence monotones (the usual concurrence of two qubits is one such monotone). We further provide tools for deriving further conditions upon catalysts of pure state transformations.
Entanglement catalysis is a phenomenon that usually enhances the conversion probability in the transformation of entangled states by the temporary involvement of another entangled state (so-called catalyst), where after the process is completed the catalyst is returned to the same state. For some pairs of bipartite pure entangled states, catalysis enables a transformation with unit probability of success, in which case the respective Schmidt coefficients of the states are said to satisfy the trumping relation, a mathematical relation which is an extension of the majorization relation. This paper provides all necessary and sufficient conditions for the trumping and two other associated relations. Using these conditions, the least upper bound of conversion probabilities using catalysis is also obtained. Moreover, best conversion ratios achievable with catalysis are found for transformations involving many copies of states.
Abstract — We examine catalyst-assisted probabilistic entanglement transformations. A sufficient and necessary condition of when some appropriate catalyst exists that can increase the maximal transformation probability is presented. We also design an algorithm which leads to an efficient method to find the most economical partial catalyst with minimal dimension. The mathematical structure of catalyst-assisted probabilistic transformation is carefully examined. Index Terms — Probabilistic entanglement transformation, catalyst-assisted transformation, partial catalyst.
Given a finite-dimensional pure state transformation restricted by entanglement-assisted local operations and classical communication (ELOCC), we derive minimum and maximum bounds on the entanglement of an ancillary catalyst that allows that transformation. These bounds are nontrivial even when the Schmidt number of both the original and ancillary states becomes large. We identify a lower bound for the dimension of a catalyst allowing a particular ELOCC transformation. Along with these bounds, we present further constraints on ELOCC transformations by identifying restrictions on the Schmidt coefficients of the target state. In addition, an example showing the existence of qubit ELOCC transformations with multiple ranges of potential ancillary states is provided. This example reveals some additional difficulty in finding strict bounds on ELOCC transformations, even in the qubit case. Finally, a comparison of the bounds in this paper with previously discovered bounds is presented.
Catalysis refers to the possibility of performing an otherwise impossible local state transformation by sharing an additional state, i.e. a catalyst, which is returned at the end of the protocol. There is a stronger version, known as supercatalysis, in which the borrowed catalyst is returned in an enhanced form, i.e. more entangled. However, this phenomenon has remained little explored. In this work we introduce the supercatalytic entanglement gain as a figure of merit taking values in [0,1] that quantifies the performance of the protocol (with 0 corresponding to the standard case of catalysis and 1 representing the maximal possible gain) and we study in which cases it can be greater than zero and which strategies can maximize it. While it turns out that every catalytic transformation can be implemented in a supercatalytic fashion with entanglement gain equal to 1 if the state that is borrowed is chosen appropriately, other choices can make the gain strictly less than 1 and even 0. In fact, we prove that a large class of catalytic transformations are not fully supercatalyzable, i.e. there is at least one choice of catalyst for which the entanglement gain vanishes. On the other hand, the construction that shows that supercatalysis is always possible with maximal gain uses artificially highly entangled catalysts. For this reason, we also study minimal supercatalysis, where the entanglement content of the borrowed state is constrained in a precise and natural way. While we consider a scenario where we prove it is impossible to have entanglement gain equal to 1 in this case, we show that there exist minimal supercatalytic transformations with gain as close to 1 as desired. We also explore several examples and observe that, although choosing a catalyst with the least possible entanglement is often an optimal strategy for minimal supercatalysis, this is not necessarily always the case.