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arXiv:quant-ph/0006045 · Physical Review A

Optimal manipulations with qubits: Universal quantum entanglers

Vladimír Bužek, Mark S. Hillery

Published Jul 12, 2000• 31 citations
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Abstract

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 $|0〉.$ 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 $〈\ensuremath{\Psi}|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉=0.$

DOI
10.1103/physreva.62.022303

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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 ∣\ensuremathΨ〉|\ensuremath{\Psi}〉∣\ensuremathΨ〉 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 (∣\ensuremathΨ〉∣0〉+∣0〉∣\ensuremathΨ〉).(|\ensuremath{\Psi}〉|0〉+|0〉|\ensuremath{\Psi}〉).(∣\ensuremathΨ〉∣0〉+∣0〉∣\ensuremathΨ〉). 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 F=(9+32)/14\ensuremath≃0.946.\mathcal{F}=(9+3\sqrt{2})/14\ensuremath{\simeq}0.946.F=(9+32​)/14\ensuremath≃0.946. We also analyze how to optimally generate from a single qubit initially prepared in an unknown state ∣\ensuremathΨ〉|\ensuremath{\Psi}〉∣\ensuremathΨ〉 a two qubit entangled system, which is as close as possible to a Bell state (∣\ensuremathΨ〉∣\ensuremathΨ\ensuremath⊥〉+∣\ensuremathΨ\ensuremath⊥〉∣\ensuremathΨ〉),(|\ensuremath{\Psi}〉|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉+|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉|\ensuremath{\Psi}〉),(∣\ensuremathΨ〉∣\ensuremathΨ\ensuremath⊥〉+∣\ensuremathΨ\ensuremath⊥〉∣\ensuremathΨ〉), where 〈\ensuremathΨ∣\ensuremathΨ\ensuremath⊥〉=0.〈\ensuremath{\Psi}|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉=0.〈\ensuremathΨ∣\ensuremathΨ\ensuremath⊥〉=0.
Quantum information processing is usually associated with a specific computational basis. Nevertheless, for a deeper understanding of the fundamental principles of quantum computing, it is essential to investigate what is the fidelity of universal operations that are basis independent; i.e., the task is to perform optimally a specific operation on a qubit or a quantum register that is in an unknown state. In this paper we introduce the universal-NOT gate that takes as an input a qubit in an arbitrary state $|\ensuremath{\Psi}〉$ and generates an output that is as close as possible to the orthogonal state $|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉.$
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.
It is not a problem to complement a classical bit, i.e. to change the value of a bit, a 0 to a 1 and vice versa. This is accomplished by a NOT gate. Complementing a qubit in an unknown state, however, is another matter. We show that this operation cannot be done perfectly. We define the Universal-NOT (U-NOT) gate which out of N identically prepared pure input qubits generates M output qubits in a state which is as close as possible to the perfect complement. This gate can be realized by classical estimation and subsequent re-preparation of complements of the estimated state. Its fidelity is therefore equal to the fidelity F= (N+1)/(N+2) of optimal estimation, and does not depend on the required number of outputs. We also show that when some additional a priori information about the state of input qubit is available, than the fidelity of the quantum NOT gate can be much better than the fidelity of estimation.
Given two arbitrary pure states $ |\\phi>$ and $ |\\psi>$ of qubits or higher level states, we provide arguments in favor of states of the form $ \\frac{1}{\\sqrt{2}}(|\\psi> |\\phi> + i |\\phi> |\\psi>) $ instead of symmetric or anti-symmetric states, as natural candidates for optimally entangled states constructed from these states. We show that such states firstly have on the average a high value of concurrence, secondly can be constructed by a universal unitary operator independent of the input states. We also show that these states are the only ones which can be produced with perfect fidelity by any quantum operation designed for intertwining two pure states with a relative phase. A probabilistic method is proposed for producing any pre-determined relative phase into the combination of any two arbitrary states.
Beyond the no-cloning theorem, the universal symmetric quantum cloning machine was first addressed by Bu\ifmmode \check{z}\else \v{z}\fi{}ek and Hillery. Here, we realized the $\stackrel{\ensuremath{\rightarrow}}{1}$2 qubit Bu\ifmmode \check{z}\else \v{z}\fi{}ek-Hillery cloning machine with linear optical devices. This method relies on the representation of several qubits by a single photon. We showed that the fidelities between the two output qubits and the original qubit are both 56\frac{5}{6}65​ (which proved to be the optimal fidelity of $\stackrel{\ensuremath{\rightarrow}}{1}$2 qubit universal cloner) for arbitrary input pure states.
We analyze how information encoded in quantum systems can be optimally processed. In particular, we investigate copying (cloning) of quantum information (represented as states of 2-level quantum systems--qubits). We present unitary transformations which describe the optimal universal cloning of a qubit. Universality of the transformation guarantees that the fidelity of the cloning does not depend on the input state of the qubit, i.e. all states are cloned equally well. We present network for the optimal universal quantum cloning `machine' (transformation) which produces N + 1 copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copes, N. We also present the `machine' which universally and optimally clones states of quantum objects in arbitrary- dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers. In addition to cloning of qubits we analyzed another universal operation-- the Universal NOT. We present the optimal transformation and the corresponding logical network which optimally complements an arbitrary input state of a qubit. We show that the fidelity of the performance of the Universal NOT operation increases as a function of the number of input qubits prepared in the same state.