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Fortschritte der Physik

Universal Transformations for Finite Dimensional Quantum Systems

Chiara Macchiavello

Published May 1, 2000• 0 citations
Publisher

Abstract

We describe in a unified way universal operations for quantum systems with arbitrary finite dimension and show some links among them. In particular, we discuss a connection between optimal universal cloning and optimal state estimation. We then describe the operation of the depolarising quantum channel, its connection to quantum cloning and ways to contrast the undesired effects of its action.

DOI
10.1002/(sici)1521-3978(200005)48:5/7<545::aid-prop545>3.0.co;2-g

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We describe in a unified way universal operations for quantum systems with arbitrary finite dimension and show some links among them. In particular, we discuss a connection between optimal universal cloning and optimal state estimation. We then describe the operation of the depolarising quantum channel, its connection to quantum cloning and ways to contrast the undesired effects of its action.
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 present the universal cloning transformation of states in arbitrary-dimensional Hilbert spaces. This unitary transformation attains the optimal fidelity of cloning as specified by Werner [Phys. Rev. A 58, 1827 (1998)]. With this cloning transformation, pure as well as impure states can be optimally copied, and the quality of the copies does not depend on the state being copied. We discuss the properties of quantum clones. In particular, we show that in the limit of high dimension the fidelity of clones does not converge to zero but attains the limit 1/21/21/2. We also show that our cloning transformation is most suitable for cloning of entanglement.
We review our recent work on the universal (i.e. input state independent) optimal quantum copying (cloning) of qubits. We present unitary transformations which describe the optimal cloning of a qubit and we present the corresponding quantum logical network. We also present network for an optimal quantum copying ``machine'' (transformation) which produces N+1 identical 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 copies, N. In addition, we present the machine which universaly and optimally clones states of quantum objects in arbitrary-dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers.
We derive a tight upper bound for the fidelity of a universal $N\ensuremath{\rightarrow}M$ qubit cloner, valid for any $M\ensuremath{\ge}N$, where the output of the cloner is required to be supported on the symmetric subspace. Our proof is based on the concatenation of two cloners and the connection between quantum cloning and quantum state estimation. We generalize the operation of a quantum cloner to mixed and/or entangled input qubits described by a density matrix supported on the symmetric subspace of the constituent qubits. We also extend the validity of optimal state estimation methods to inputs of this kind.
We present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system. The algorithm specifies a physically realizable (i.e., finite) positive operator valued measurement on a finite number of identically prepared systems. We illustrate the general formalism by applying it to different scenarios of the state estimation of NNN independent and identically prepared two-level systems (qubits).