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arXiv:quant-ph/0111153 · Physical Review A

General impossible operations in quantum information

Arun Kumar Pati

Published Dec 30, 2002• 50 citations
arXivHTMLPDF

Abstract

We prove a general limitation in quantum information that unifies the impossibility principles such as no-cloning and no-anticloning. Further, we show that for an unknown qubit one cannot design a universal Hadamard gate for creating equal superposition of the original and its complement state. Surprisingly, we find that Hadamard transformations exist for an unknown qubit chosen either from the polar or equatorial great circles. Also, we show that for an unknown qubit one cannot design a universal unitary gate for creating unequal superpositions of the original and its complement state. We discuss why it is impossible to design a controlled-NOT gate for two unknown qubits and discuss the implications of these limitations. 03.67.Hk, 03.65.Ta

DOI
10.1103/physreva.66.062319

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We prove a general limitation in quantum information that unifies the impossibility principles such as no-cloning and no-anticloning. Further, we show that for an unknown qubit one cannot design a universal Hadamard gate for creating equal superposition of the original and its complement state. Surprisingly, we find that Hadamard transformations exist for an unknown qubit chosen either from the polar or equatorial great circles. Also, we show that for an unknown qubit one cannot design a universal unitary gate for creating unequal superpositions of the original and its complement state. We discuss why it is impossible to design a controlled-NOT gate for two unknown qubits and discuss the implications of these limitations. 03.67.Hk, 03.65.Ta
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.
We establish the non-existence of a universal Hadamard gate for an arbitrary qubit, by considering two different principles; namely, no-superluminal signalling and non-increase of entanglement under LOCC. It is also shown that these principles are not violated if and only if the qubit states belong to the special ensemble obtained recently. We then extend the non-existence of the Hadamard operation to a multi-qubit system. In higher dimensions, the analog of the Hadamard gate is the quantum Fourier transform. We show that it is not possible to design this gate for an arbitrary qudit.
The no-cloning theorem, the no-deleting theorem, and other no-go theorems play a crucial role in quantum mechanics and quantum information. In this paper, we show that it is impossible to conjugate or reflect an unknown quantum state or conjugate or reflect the bit-flip (the phase-flip) of an unknown state via a universally unitary evolution. We also indicate that the partial conjugating operations are not anti-unitary and it is impossible to universally partially conjugate an unknown state via a unitary evolution. We compute the fidelities of the optimal physical approximations to conjugating and conjugating the bit-flip and phase-flip operations.
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.
In this work we show that the most general class of anti-unitary operators are nonphysical in nature through the existence of incomparable pure bipartite entangled states. It is also shown that a large class of inner-product-preserving operations defined only on the three qubits having spin-directions along x,y and z are impossible. If we perform such an operation locally on a particular pure bipartite state then it will exactly transform to another pure bipartite state that is incomparable with the original one. As subcases of the above results we find the nonphysical nature of universal exact flipping operation and existence of universal Hadamard gate. Beyond the information conservation in terms of entanglement, this work shows how an impossible local operation evolve with the joint system in a nonphysical way.