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International Journal of Quantum Information

OPTIMAL STATE ESTIMATION AND CLONING FOR EQUATORIAL QUANTUM SYSTEMS WITH ARBITRARY DIMENSION

Chiara Macchiavello

Published Dec 17, 2004• 0 citations
Publisher

Abstract

We review the problem of optimal estimation of multiple phases for systems with arbitrary finite dimension and derive the optimal estimation fidelity for equatorial states. We present optimal phase covariant cloning transformations for d-dimensional systems, relating these results to the optimal estimation procedure.
DOI
10.1142/s0219749905000554

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We describe a unified framework of phase-covariant multiuser quantum transformations for d-dimensional quantum systems. We derive the optimal phase-covariant cloning and transposition transformations for multiphase states. We show that for some particular relations between the input and output number of copies, they correspond to economical transformations, which can be achieved without the need of auxiliary systems. We prove a relation between the optimal phase-covariant cloning and transposition maps and optimal estimation of multiple phases for equatorial states.
Since the initial discovery of the Wootters-Zurek no-cloning theorem, a wide variety of quantum cloning machines have been proposed aiming at imperfect but optimal cloning of quantum states within its own context. Remarkably, most previous studies have employed the Bures fidelity or the Hilbert-Schmidt norm as the figure of merit to characterize the quality of the corresponding cloning scenarios. However, in many situations, what we truly care about is the relevant information about certain parameters encoded in quantum states. In this work, we investigate the multiple phase estimation problem in the framework of quantum cloning machines, from the perspective of quantum Fisher information matrix (QFIM). Focusing on the generalized ddd-dimensional equatorial states, we obtain the analytical formulas of QFIM for both universal quantum cloning machine (UQCM) and phase-covariant quantum cloning machine (PQCM), and prove that PQCM indeed performs better than UQCM in terms of QFIM. We highlight that our method can be generalized to arbitrary cloning schemes where the fidelity between the single-copy input and output states is input-state independent. Furthermore, the attainability of the quantum Cram\'er-Rao bound is also explicitly discussed.
We derive the optimal $N\ensuremath{\rightarrow}M$ phase-covariant quantum cloning for equatorial states in dimension ddd with M=kd+NM=kd+NM=kd+N, kkk integer. The cloning maps are optimal for both global and single-qudit fidelity. The map is achieved by an ``economical'' cloning machine, which works without ancilla.
We consider cloning transformations of equatorial qubits $|{\ensuremath{\psi}}_{\ensuremath{\varphi}}〉=1/\sqrt{2}(|0〉{+e}^{i\ensuremath{\varphi}}|1〉)$ and qutrits $|{\ensuremath{\psi}}_{\ensuremath{\varphi},\ensuremath{\theta}}〉=1/\sqrt{3}(|0〉{+e}^{i\ensuremath{\varphi}}|1〉{+e}^{i\ensuremath{\theta}}|2〉),$ with the transformation covariant for rotation of the phases $\ensuremath{\varphi}$ and $\ensuremath{\theta}.$ The optimal cloning maps are derived without simplifying assumptions from first principles, for any number of input and output qubits, and for a single-input qutrit and any number of output qutrits. We also compare the cloning maps for global and single-particle fidelities, and we show that the two criteria lead to different optimal maps.
We consider cloning transformations of d-dimensional states of the form eiϕ0∣0>+eiϕ1∣1>+...+eiϕd−1∣d−1>e^{i\phi_0}|0> + e^{i\phi_1}|1> +...+ e^{i\phi_{d-1}}|d-1>eiϕ0​∣0>+eiϕ1​∣1>+...+eiϕd−1​∣d−1> that are covariant with respect to rotations of the phases ϕi\phi_iϕi​'s. The optimal cloning maps are easily obtained using a well-defined general characterization of state-dependent 1→21 \rightarrow 21→2 cloning transformations in arbitrary dimensions. Our results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can be analyzed.
It is known that in phase-covariant quantum cloning, the equatorial states on the Bloch sphere can be cloned with a fidelity higher than the optimal bound established for universal quantum cloning. We generalize this concept to include other states on the Bloch sphere with a definite z component of spin. It is shown that once we know the z component, we can always clone a state with a fidelity higher than the universal value and that of equatorial states. We also make a detailed study of the entanglement properties of the output copies and show that the equatorial states are the only states that give rise to a separable density matrix for the outputs.