Quantum cloning of orthogonal qubits

Abstract

An optimal universal cloning transformation is derived that produces M copies of an unknown qubit from a pair of orthogonal qubits. For M > 6, the corresponding cloning fidelity is higher than that of the optimal copying of a pair of identical qubits. It is shown that this cloning transformation can be implemented probabilistically via parametric down-conversion by feeding the signal and idler modes of a nonlinear crystal with orthogonally polarized photons.In contrast to classical information, quantum information cannot be copied. This so-called no-cloning theorem [1], which is a direct consequence of the linearity of quantum theory, states that it is impossible to prepare several exact copies (or clones) of an unknown quantum state |ψ . Although exact cloning is forbidden, one can design various quantum cloning machines which produce approximate clones. In particular, much attention has been devoted to the optimal universal cloning machines for qubits, which prepare M identical approximate clones out of N replicas of an unknown qubit, and such that the fidelity of the clones is state-independent [2]. Cloning machines for states in a d-dimensional Hilbert space (qudits) were also investigated [3], as well as continuousvariable cloning machines for coherent states [4].In the limit of an infinite number of clones, the optimal cloning reduces to the optimal quantum measurement. In this context, a very interesting observation has been made by Gisin and Popescu [5] who noted that the information about a direction in space is better encoded into two orthogonal qubits than in two identical ones. If we possess a two-qubit state |ψ, ψ ⊥ with ψ|ψ ⊥ = 0, then we can estimate |ψ with a fidelity 789 [5, 6]. This slightly exceeds the fidelity of the optimal measurement on a qubit pair |ψ, ψ , F || = 3/4. A similar situation occurs for continuous quantum variables. Suppose we want to encode a (randomly chosen) position in phase space. A possible strategy would be to prepare a pair of coherent states |α, α , where the real and imaginary parts of the complex number α represent the phase-space coordinates. However, it is actually better to supply a state |α, α * from which α can be inferred via optimal measurement with a lower error [7]. It can also be shown that the state |α, α * gives an advantage when cloning coherent states: M identical approximate clones of a coherent state |α can be prepared with a higher fidelity from the state |α, α * than from |α, α [8].Motivated by this result, we were led to ask whether a similar situation might also occur for qubits. Can M clones of qubit |ψ be produced from an orthogonal qubit pair |ψ, ψ ⊥ with a higher fidelity than from an identical pair |ψ, ψ ? In this Letter, we answer this question by an affirmative. We present a universal cloning machine acting on an orthogonal qubit pair that approximately implements the transformation |ψ |ψ ⊥ → |ψ ⊗M with the optimal fidelity. Then, we show that this cloning transformation can be implemented probabilistically in quantum optics by use of par...

Optimal discrimination of mixed quantum states involving inconclusive results

“…Here we will not investigate this issue in detail and we refer the reader to the papers [34,40,41] where the formulation of optimal quantum-state discrimination as a semidefinite program is described in detail. Note also that the semidefinite programming has recently found its applications in several branches of quantum information theory such as the optimization of completely positive maps [42,43,44], the analysis of the distillation protocols that preserve the positive partial transposition [45], or the tests of separability of quantum states [46].…”

Conditional generation of arbitrary multimode entangled states of light with linear optics

“…It was shown recently that these states may be conditionally generated with the help of linear optics, single photon sources and photodetectors [14,15,16,17]. We then combine the modes 1 and 2 on a balanced beam splitter and measure the number of photons in output modes 1 and 2 by means of two photodetectors P D 1 and P D 2 with single-photon resolution.…”

“…Schemes for probabilistic preparation of Fock states [9], arbitrary superpositions of Fock states of single-mode field [10,11], and superpositions of classically distinguishable states (Schrödinger-cat-like states) [12,13], have been found. Several setups for the generation of twomode N -photon path-entangled states have been suggested [14,15,16,17,18]. Recently, the experimental conditional preparation of a single-photon Fock state with negative Wigner function has been reported [19].…”

Optimal copying of entangled two-qubit states

“…As arbitrary quantum states cannot be copied perfectly, the interesting question arises as to which extent a quantum process can perform this task in an optimal way. Recently, much work has been devoted to the copying of pure quantum states [2,3,4,5,6,7,8,9,10]. These investigations demonstrate that the maximal fidelity which can be achieved in an optimal copying process depends on characteristic properties of the set of states which are to be copied.…”