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arXiv:1205.3970 · Physical Review A

Bound on remote preparation of entanglement from isotropic states

Soojoon Lee

Published May 17, 2012• 3 citations
arXivHTMLPDF

Abstract

Using the negativity as an entanglement measure, we investigate the possible amount of remotely prepared entanglement. For two identical isotropic states on two-qudit systems 12 and 34, we calculate the average amount of entanglement remotely distributed on the system 13 by joint measurement on the system 24, and show that the remote preparation of entanglement by the generalized Bell measurement is optimal among rank-one measurements if the isotropic states have a certain fidelity with a maximally entangled state in higher dimensional quantum systems, or if the fidelity of the isotropic states is greater than a certain value depending on the dimension. In addition, we construct a measurement better than the generalized Bell measurement with respect to the remote preparation of entanglement when the isotropic states have small fidelity.

DOI
10.1103/physreva.85.052311

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Using the negativity as an entanglement measure, we investigate the possible amount of remotely prepared entanglement. For two identical isotropic states on two-qudit systems 12 and 34, we calculate the average amount of entanglement remotely distributed on the system 13 by joint measurement on the system 24, and show that the remote preparation of entanglement by the generalized Bell measurement is optimal among rank-one measurements if the isotropic states have a certain fidelity with a maximally entangled state in higher dimensional quantum systems, or if the fidelity of the isotropic states is greater than a certain value depending on the dimension. In addition, we construct a measurement better than the generalized Bell measurement with respect to the remote preparation of entanglement when the isotropic states have small fidelity.
Using the convex-roof extended negativity and the negativity of assistance as quantifications of bipartite entanglement, we consider the possible remotely-distributed entanglement. For two pure states ∣ϕ⟩AB\ketϕ_{AB}∣ϕ⟩AB​ and ∣ψ⟩CD\ketψ_{CD}∣ψ⟩CD​ on bipartite systems ABABAB and CDCDCD, we first show that the possible amount of entanglement remotely distributed on the system ACACAC by joint measurement on the system BDBDBD is not less than the product of two amounts of entanglement for the states ∣ϕ⟩AB\ketϕ_{AB}∣ϕ⟩AB​ and ∣ψ⟩CD\ketψ_{CD}∣ψ⟩CD​ in two-qubit and two-qutrit systems. We also provide some sufficient conditions, for which the result can be generalized into higher-dimensional quantum systems.
Entanglement cost is the asymptotic rate of Bell pairs required to prepare a quantum state. Its regularized definition has made exact evaluation difficult, even for isotropic states (i.e., depolarized maximally entangled states). In this work, we determine the entanglement cost of every qubit isotropic state and, more generally, every two-qubit Bell-diagonal state. Our proof uses a family of suitably tuned qubit semigroups to transform a general log-Sobolev entropy bound into supporting lines of Wootters' function. A recent exact tensorization theorem of Dong et al. [arXiv:2606.17729] extends these bounds to arbitrarily many copies, yielding a general lower bound on entanglement cost. Matching this bound with the entanglement of formation also determines the exact cost for a broader class of two-qubit states. For qudit isotropic states, we derive another general lower bound from the full depolarizing 2→32\to 32→3 norm. This bound substantially improves on the PPT-relative entropy of entanglement and nearly matches the entanglement of formation, leaving a gap of at most 2%2\%2% of log⁡d\log dlogd that vanishes as ddd grows. Finally, the exact qubit results and qudit bounds extend to the entanglement cost of preparing quantum depolarizing channels under both parallel and sequential strategies.
Remote state preparation (RSP) is the act of preparing a quantum state at a remote location without actually transmitting the state itself. Using at most two classical bits and a single shared maximally entangled state, one can in theory remotely prepare any qubit state with certainty and with perfect fidelity. However, in any experimental implementation the average fidelity between the target and output states cannot be perfect. In order for an RSP experiment to demonstrate genuine quantum advantages, it must surpass the optimal threshold of a comparable classical protocol. Here we study the fidelity achievable by RSP protocols lacking shared entanglement and determine the optimal value for the average fidelity in several different cases. We implement an experimental scheme for deterministic remote preparation of arbitrary photon polarization qubits, preparing 178178178 different pure and mixed qubit states with an average fidelity of 0.9950.9950.995. Our experimentally achieved average fidelities surpass our derived classical thresholds whenever the classical protocol does not trivially allow for perfect RSP.
We present a scheme for remotely preparing an arbitrary two-qubit pure state by using two bipartite partially entangled states as the quantum channel, the scheme is then generalized to the arbitrary three-qubit case. For the two cases of remote state preparation, we construct two different efficient projective measurement bases at sender's side and calculate the corresponding success probabilities. It is shown that remote preparation of the two-qubit or three-qubit state can be probabilistically achieved with unity fidelity. Moreover, for some special ensembles of the initial states, we find that the success probability of preparation can be increased to four times for two-qubit states and eight times for three-qubit states, and is equal to one in the case of the maximal entanglement resources.
In a process of remote state preparation, the universality of quantum channel is an essential ingredient. That is, one quantum channel should be feasible to remotely prepare any given qubit state. This problem appears in a process where one uses non-maximally entangled state as the passage. We present a scheme in which any given qubit |ϕ⟩ = cos θ|0⟩+sin θ e i φ |1⟩ could be remotely prepared by using minimum classical bits and the previously shared non-maximally entangled state with a high fidelity, under the condition that the receiver holds the knowledge of θ. This condition is helpful to reduce the necessary amount of quantum channels, which is proven to be a low quantity to realize the universality. We also give several methods to investigate the trade-off between this amount and the achievable fidelity of the protocol.
We study the connection between the Hilbert-Schmidt measure of entanglement (that is the minimal distance of an entangled state to the set of separable states) and entanglement witness in terms of a generalized Bell inequality which distinguishes between entangled and separable states. A method for checking the nearest separable state to a given entangled one is presented. We illustrate the general results by considering isotropic states, in particular two-qubit and two-qutrit states---and their generalizations to arbitrary dimensions---where we calculate the optimal entanglement witnesses explicitly.