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Physical Review A

Optimal manipulations with qubits: Universal-NOT gate

Vladimír Bužek, Mark S. Hillery, Reinhard F. Werner

Published Oct 1, 1999• 235 citations
Publisher

Abstract

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 $|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉.$

DOI
10.1103/physreva.60.r2626

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Fortschritte der Physik•Apr 24, 2003•2 citations
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 ∣\ensuremathΨ〉|\ensuremath{\Psi}〉∣\ensuremathΨ〉 and generates an output that is as close as possible to the orthogonal state ∣\ensuremathΨ\ensuremath⊥〉.|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉.∣\ensuremathΨ\ensuremath⊥〉.
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.
The action of a NOT gate on a classical bit results in a change of its value from a 0 to a 1 and vice versa. The action of the classical NOT gate is in principle perfect because with fidelity equal to unity it complements the value of a bit. The action of the quantum NOT gate in a computational basis |0⟩ and |1⟩ is very similar to the action of the classical NOT gate. However, a more general quantum mechanical operation which corresponds to a classical NOT gate would take a qubit in an arbitrary state |Ψ⟩ and produce a qubit in the state |Ψ⊥⟩ orthogonal to |Ψ⟩. This operation is anti-unitary and therefore, cannot be realized exactly. So how well we can do? We find a unitary transformation acting on an input qubit and some auxiliary qubits, which represent degrees of freedom of the quantum NOT gate itself, which approximately realizes the NOT operation on the state of the original qubit. We call this ‘device’ a universal-NOT gate because the size of the error it produces is independent of the input state. We show that an optimal U-NOT gate which has as its input N identical qubits and produces M outputs achieves a fidelity of F = (N + 1)/ (N + 2), which is equal to the fidelity of estimation of the input qubits. We also show that when a priori information about the state of the input qubit is available, the fidelity of a quantum NOT gate can be much better than the fidelity of estimation.
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 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.
A quantum analog of the fundamental classical not gate is a quantum gate that would transform any input qubit state onto an orthogonal state. Intriguingly, this universal NOT gate is forbidden by the laws of quantum physics. This striking phenomenon has far-reaching implications concerning quantum information processing and encoding information about directions and reference frames into quantum states. It also triggers the question of under what conditions the preparation of quantum states orthogonal to input states becomes possible. Here we report on experimental demonstration of orthogonalization of partly unknown single- and two-qubit quantum states. A state orthogonal to an input state is conditionally prepared by quantum filtering, and the only required information about the input state is a mean value of a single arbitrary operator. We show that perfect orthogonalization of partly unknown two-qubit entangled states can be performed by applying the quantum filter to one of the qubits only.
Abstract An arbitrary quantum state cannot be “cloned” perfectly, i.e. reproduced with “fidelity” F = 1 into M > 1 states identical to the original by any conceivable physical device. The main root of this impossibility resides in the linearity of quantum mechanics. A second “quantum impossibility” process, based on the complete positivity character of any quantum operation and conceivably subtly related to the first one, forbids the realization of a universal NOT gate i.e. one that flips exactly any input qubit into an orthogonal one. A detailed investigation of these results, representing the most fundamental difference between classical and quantum information, indeed can reveal the detailed structure of the latter one. We report an experimental demonstration of the process of cloning of N = 1 input qubit into M = 2 output qubits by a quantum‐injected optical parametric amplifier OPA. By the same apparatus the realization of a universal NOT gate is also demonstrated. The two processes will be found to be “universal” and optimal, i.e. the measured fidelity of both processes F < 1 will be found close to the theoretical values.