Group Theoretical Quantum Tomography
The paper is devoted to the mathematical foundation of the quantum tomography using the theory of square-integrable representations of unimodular Lie groups.
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arXiv subjects
G. M. D'Ariano
Publications and source records attributed to G. M. D'Ariano.
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.
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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.
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Phase covariant quantum cloning
We consider an N -> M quantum cloning transformation acting on pure two-level states lying on the equator of the Bloch sphere. An upper bound for its fidelity is presented, by establishing a connection between optimal phase covariant cloning and phase estimation. We give the explicit form of a cloning transformation that achieves the bound for the case N=1, M=2, and find a link between this case and optimal eavesdropping in the quantum cryptographic scheme BB84.
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Feedback-assisted measurement of the free mass position under the Standard Quantum Limit
The Standard Quantum Limit (SQL) for the measurement of a free mass position is illustrated, along with two necessary conditions for breaching it. A measurement scheme that overcomes the SQL is engineered. It can be achieved in three-steps: i) a pre-squeezing stage; ii) a standard von Neumann measurement with momentum-position object-probe interaction and iii) a feedback. Advantages and limitations of this scheme are discussed. It is shown that all of the three steps are needed in order to overcome the SQL. In particular, the von Neumann interaction is crucial in getting the right state reduction, whereas other experimentally achievable Hamiltonians, as, for example, the radiation-pressure interaction, lead to state reductions that on the average cannot overcome the SQL.
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Tomographic test of Bell's inequality
We present a homodyne detection scheme to verify Bell's inequality on correlated optical beams at the output of a nondegenerate parametric amplifier. Our approach is based on tomographic measurement of the joint detection probabilities, which allows high quantum efficiency at detectors. A self-homodyne scheme is suggested to simplify the experimental set-up.
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On the correspondence between classical and quantum measurements on a bosonic field
We study the correspondence between classical and quantum measurements on a harmonic oscillator that describes a one-mode bosonic field. We connect the quantum measurement of an observable of the field with the possibility of amplifying the observable ideally through a quantum amplifier. The ``classical'' measurement corresponds to the joint measurement of the position $q$ and momentum $p$ of the harmonic oscillator, with following evaluation of a function $f$ of the outcome $α=q+ip$. For the electromagnetic field the joint measurement is achieved by a heterodyne detector. The quantum measurement of an observable $\hat O$ is obtained by preamplifying the heterodyne detector through an ideal amplifier of $\hat O$, and rescaling the outcome by the gain $g$. We give a general criterion which states when this preamplified heterodyne detection scheme approaches the ideal quantum measurement of $\hat O$ in the limit of infinite gain. We show that this criterion is satisfied and the ideal measurement is achieved for the case of the photon number operator and for the quadrature. For both operators the method is robust to nonunit quantum efficiency of the heterodyne detector. On the other hand, we show that the preamplified heterodyne detection scheme does not work for arbitrary observable of the field. As a counterexample, we prove that the simple quadratic function of the field $\hat K=i(a^{†2}-a^2)/2$ has no corresponding polynomial function $f(α,\bar α)$---including the obvious choice $f=\hbox{Im}(α^2)$---that allows the measurement of $\hat K$ through the preamplified heterodyne measurement scheme.
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State preparation by photon filtering
We propose a setup capable of generating Fock states of a single mode radiation field. The scheme is based on coupling the signal field to a ring cavity through cross-Kerr phase modulation, and on conditional ON-OFF photodetection at the output cavity mode. The same setup allows to prepare selected superpositions of Fock states and entangled two-mode states. Remarkably, the detector's quantum efficiency does not affect the reliability of the state synthesis.
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Quantum tomography of the GHZ state
We present a method of generation of the Greenberger-Horne-Zeilinger state involving type II and type I parametric downconversion, and triggering photodetectors. The state generated by the proposed experimental set-up can be reconstructed through multi-mode quantum homodyne tomography. The feasibility of the measurement is studied on the basis of Monte-Carlo simulations.
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Generation and Measurement of Nonclassical States by Quantum Fock Filter
We study a novel optical setup which is able to select a specific Fock component from a generic input state. The device allows to synthesize number states and superpositions of few number states, and to measure the photon distribution and the density matrix of a generic input signal.
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On the parametric approximation in quantum optics
We perform the exact numerical diagonalization of the Hamiltonians that describe both degenerate and nondegenerate parametric amplifiers, by exploiting the conservation laws pertaining each device. We clarify the conditions under which the parametric approximation holds, showing that the most relevant requirement is the coherence of the pump after the interaction, rather than its undepletion.
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Tomographic measurement of nonclassical radiation states
We propose to experimentally test the nonclassicality of quantum states through homodyne tomography. For single-mode states we check violations of inequalities involving the photon-number probability. For two-mode states we test the nonclassicality by reconstructing some suitable number-operator functions. The test can be performed with available quantum efficiency of homodyne detection, by measuring the pertaining quantities on the corresponding noisy states.
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On the general problem of quantum phase estimation
The problem of estimating a generic phase-shift experienced by a quantum state is addressed for a generally degenerate phase shift operator. The optimal positive operator-valued measure is derived along with the optimal input state. Two relevant examples are analyzed: i) a multi-mode phase shift operator for multipath interferometry; ii) the two mode heterodyne phase detection.
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Measuring quantum optical Hamiltonians
We show how recent state-reconstruction techniques can be used to determine the Hamiltonian of an optical device that evolves the quantum state of radiation. A simple experimental setup is proposed for measuring the Liouvillian of phase-insensitive devices. The feasibility of the method with current technology is demonstrated on the basis of Monte Carlo simulated experiments.
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Quantum tomography of mesoscopic superpositions of radiation states
We show the feasibility of a tomographic reconstruction of Schrödinger cat states generated according to the scheme proposed by S. Song, C.M. Caves and B. Yurke [Phys. Rev. A 41, 5261 (1990)]. We present a technique that tolerates realistic values for quantum efficiency at photodetectors. The measurement can be achieved by a standard experimental setup.
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Noise, errors and information in quantum amplification
We analyze and compare the characterization of a quantum device in terms of noise, transmitted bit-error-rate (BER) and mutual information, showing how the noise description is meaningful only for Gaussian channels. After reviewing the description of a quantum communication channel, we study the insertion of an amplifier. We focus attention on the case of direct detection, where the linear amplifier has a 3 decibels noise figure, which is usually considered an unsurpassable limit, referred to as the standard quantum limit (SQL). Both noise and BER could be reduced using an ideal amplifier, which is feasible in principle. However, just a reduction of noise beyond the SQL does not generally correspond to an improvement of the BER or of the mutual information. This is the case of a laser amplifier, where saturation can greatly reduce the noise figure, although there is no corresponding improvement of the BER. Such mechanism is illustrated on the basis of Monte Carlo simulations.
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Homodyning as universal detection
Homodyne tomography - i. e. homodyning while scanning the local oscillator phase - is now a well assessed method for ``measuring'' the quantum state. In this paper I will show how it can be used as a kind of universal detection, for measuring generic field operators, however at expense of some additional noise. The general class of field operators that can be measured in this way is presented, and includes also operators that are inaccessible to heterodyne detection. The noise from tomographical homodyning is compared to that from heterodyning, for those operators that can be measured in both ways. It turns out that for some operators homodyning is better than heterodyning when the mean photon number is sufficiently small. Finally, the robustness of the method to additive phase-insensitive noise is analyzed. It is shown that just half photon of thermal noise would spoil the measurement completely.
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Comment on "Loss-error compensation in quantum-state measurements"
In the two papers [T. Kiss, U. Herzog, and U. Leonhardt, Phys. Rev. A 52, 2433 (1995); U. Herzog, Phys. Rev. A 53, 1245 (1996)] with titles similar to the one given above, the authors assert that in some cases it is possible to compensate a quantum efficiency $η\leq 1/2$ in quantum-state measurements, violating the lower bound 1/2 proved in a preceding paper [G. M. D'Ariano, U. Leonhardt and H. Paul, Phys. Rev. A 52, R1801 (1995)]. Here we re-establish the bound as unsurpassable for homodyning any quantum state, and show how the proposed loss-compensation method would always fail in a real measurement outside the allowed $η>1/2$ region.
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