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Optics and Spectroscopy

Asymmetric universal entangling machine for qubits

D. B. Horoshko, Mikhail I. Kolobov, S. Ya. Kilin

Published Sep 1, 2005• 0 citations
Publisher

Abstract

We give a definition of a universal entangling machine that entangles a system in an unknown state to a specially prepared ancilla. We describe explicitly such a machine for one qubit; i.e., we show how a qubit in an a priori unknown state can be entangled to a four-level ancilla by means of a unitary transformation. As a result, a fixed state-independent amount of entanglement is generated in exchange for a fixed degradation of qubit state fidelity.

DOI
10.1134/1.2055929

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We give a definition of a universal entangling machine that entangles a system in an unknown state to a specially prepared ancilla. We describe explicitly such a machine for one qubit; i.e., we show how a qubit in an a priori unknown state can be entangled to a four-level ancilla by means of a unitary transformation. As a result, a fixed state-independent amount of entanglement is generated in exchange for a fixed degradation of qubit state fidelity.
We give a definition of an asymmetric universal entangling machine which entangles a system in an unknown state to a specially prepared ancilla. The machine produces a fixed state-independent amount of entanglement in exchange for a fixed degradation of the system state fidelity. We describe explicitly such a machine for any quantum system having d levels and prove its optimality. We show that a d 2-dimensional ancilla is sufficient for reaching optimality. The introduced machine is a generalization to a number of widely investigated universal quantum devices such as the symmetric and asymmetric quantum cloners, the symmetric and antisymmetric quantum entanglers, the quantum information distributor and the universal-NOT gate.
It is not possible to disentangle a qubit in an unknown state $|\ensuremath{\psi}〉$ from a set of $N\ensuremath{-}1$ ancilla qubits prepared in a specific reference state ∣0〉.|0〉.∣0〉. That is, it is not possible to perfectly perform the transformation $(|\ensuremath{\psi},0,\dots{},0〉+|0,\ensuremath{\psi},\dots{},0〉+\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}+|0,0,\dots{},\ensuremath{\psi}〉)\ensuremath{\rightarrow}|0,\dots{},0〉\ensuremath{\bigotimes}|\ensuremath{\psi}〉.$ The question is then how well we can do? We consider a number of different methods of extracting an unknown state from an entangled state formed from that qubit and a set of ancilla qubits in a known state. Measuring the whole system is, as expected, the least effective method. We present various quantum ``devices'' which disentangle the unknown qubit from the set of ancilla qubits. In particular, we present the optimal universal disentangler which disentangles the unknown qubit with a fidelity that does not depend on the state of the qubit, and a probabilistic disentangler which performs the perfect disentangling transformation, but with a probability less than 1.
We analyze various scenarios for entangling two initially unentangled qubits. In particular, we propose an optimal universal entangler that entangles a qubit in unknown state $|\ensuremath{\Psi}〉$ with a qubit in a reference (known) state ∣0〉.|0〉.∣0〉. That is, our entangler generates the output state that is as close as possible to the pure (symmetrized) state $(|\ensuremath{\Psi}〉|0〉+|0〉|\ensuremath{\Psi}〉).$ The most attractive feature of this entangling machine, is that the fidelity of its performance (i.e., the distance between the output and the ideally entangled---symmetrized state) does not depend on the input and takes the constant value $\mathcal{F}=(9+3\sqrt{2})/14\ensuremath{\simeq}0.946.$ We also analyze how to optimally generate from a single qubit initially prepared in an unknown state $|\ensuremath{\Psi}〉$ a two qubit entangled system, which is as close as possible to a Bell state $(|\ensuremath{\Psi}〉|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉+|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉|\ensuremath{\Psi}〉),$ where $〈\ensuremath{\Psi}|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉=0.$
We present a unified universal quantum cloning machine, which combines several different existing universal cloning machines together, including the asymmetric case. In this unified framework, the identical pure states are projected equally into each copy initially constituted by input and one half of the maximally entangled states. We show explicitly that the output states of those universal cloning machines are the same. One importance of this unified cloning machine is that the cloning procession is always the symmetric projection, which reduces dramatically the difficulties for implementation. Also, it is found that this unified cloning machine can be directly modified to the general asymmetric case. Besides the global fidelity and the single-copy fidelity, we also present all possible arbitrary-copy fidelities.