Massimiliano F. Sacchi

Maximum-likelihood estimation of the density matrix

We present a universal technique for quantum-state estimation based on the maximum-likelihood method. This approach provides a positive-definite estimate for the density matrix from a sequence of measurements performed on identically prepared copies of the system. The method is versatile and can be applied to multimode radiation fields as well as to spin systems. The incorporation of physical constraints, which is natural in the maximum-likelihood strategy, leads to a substantial reduction of statistical errors. Numerical implementation of the method is based on a particular form of the Gauss decomposition for positive-definite Hermitian matrices. PACS number~s!: 03.67.2a, 03.65.Bz In quantum mechanics, the achievable information on a physical system is encoded into the density matrix % ˆ , which allows one to evaluate all possible expectation values through the Born statistical rule ^O ˆ &5Tr(% ˆO ˆ). In order to obtain full information on a quantum system we need to estimate its density matrix. In principle, this can be accomplished by successive measurements on repeated identical preparations of the same system. With a proper choice of the measurements, and after collecting a suitably large number of data, we can arrive at a reliable knowledge of the quantum state of the system.

Universal Teleportation with a Twist

We give a transfer theorem for teleportation based on twisting the entanglement measurement. This allows one to say what local unitary operation must be performed to complete the teleportation in any situation, generalizing the scheme to include overcomplete measurements, non-Abelian groups of local unitary operations (e.g., angular momentum teleportation), and the effect of nonmaximally entangled resources.

Efficient Use of Quantum Resources for the Transmission of a Reference Frame

We propose a covariant protocol for transmitting reference frames encoded on N spins, achieving sensitivity N-2 without the need of a preestablished reference frame and without using entanglement between sender and receiver. The protocol exploits the use of equivalent representations that were overlooked in the previous literature.

Bell measurements and observables

A general matrix approach to study entangled states is presented, based on operator completeness relations. Bases of unitary operators are considered, with focus on irreducible representations of groups. Bell measurements for teleportation are considered, and robustness of teleportation to various kinds of non idealities is shown. q 2000 Elsevier Science B.V. All rights reserved.

Informationally complete measurements and group representation

Informationally complete measurements on a quantum system allow one to estimate the expectation value of any arbitrary operator by just averaging functions of the experimental outcomes. We show that such kinds of measurement can be achieved through positive-operator valued measures (POVMs) related to unitary irreducible representations of a group on the Hilbert space of the system. With the help of frame theory we provide a constructive way to evaluate the data-processing function for arbitrary operators.

Optimal discrimination of quantum operations

We address the problem of discriminating with minimal error probability two given quantum operations. We show that the use of entangled input states generally improves the discrimination. For Pauli channels we provide a complete comparison of the optimal strategies where either entangled or unentangled input states are used.

Optical Fock-state synthesizer

We suggest a tunable optical device to synthesize Fock states and their superpositions starting from a coherent source. The scheme involves an avalanche-triggering photodetector and a ring cavity coupled to a traveling wave through a cross-Kerr medium. Low quantum efficiency at the photodetector improves the synthesizing quality at the expense of reducing the synthesizing rate.

Optimal estimation of group transformations using entanglement

We derive the optimal input states and the optimal quantum measurements for estimating the unitary action of a given symmetry group, showing how the optimal performance is obtained with a suitable use of entanglement. Optimality is defined in a Bayesian sense, as minimization of the average value of a given cost function. We introduce a class of cost functions that generalizes the Holevo class for phase estimation, and show that for states of the optimal form all functions in such a class lead to the same optimal measurement. As a first application of the main result is the complete proof of the optimal efficiency in the transmission of a Cartesian reference frame. As a second application, we derive the optimal estimation of a completely unknown two-qubit maximally entangled state, provided that N copies of the state are available. In the limit of large N, the fidelity of the optimal estimation is shown to be 1-3/(4N)

Approximate quantum cloning and the impossibility of superluminal information transfer

We show that nonlocality of quantum mechanics cannot lead to superluminal transmission of information, even if most general local operations are allowed, as long as they are linear and trace preserving. In particular, any quantum mechanical approximate cloning transformation does not allow signalling. On the other hand, the no-signalling constraint on its own is not sufficient to prevent a transformation from surpassing the known cloning bounds. We illustrate these concepts on the basis of some examples.

Parameter estimation in quantum optics

We address several estimation problems in quantum optics by means of the maximum-likelihood principle. We consider Gaussian state estimation and the determination of the coupling parameters of quadratic Hamiltonians. Moreover, we analyze different schemes of phase-shift estimation. Finally, the absolute estimation of the quantum efficiency of both linear and avalanche photodetectors is studied. In all the considered applications, the Gaussian bound on statistical errors is attained with a few thousand data.