The modern tools of quantum mechanics (A tutorial on quantum states, measurements, and operations)

This tutorial is devoted to review the modern tools of quantum mechanics, which are suitable to describe states, measurements, and operations of realistic, not isolated, systems in interaction with their environment, and with any kind of measuring and processing devices. We underline the central role of the Born rule and and illustrate how the notion of density operator naturally emerges, together the concept of purification of a mixed state. In reexamining the postulates of standard quantum measurement theory, we investigate how they may formally generalized, going beyond the description in terms of selfadjoint operators and projective measurements, and how this leads to the introduction of generalized measurements, probability operator-valued measures (POVM) and detection operators. We then state and prove the Naimark theorem, which elucidates the connections between generalized and standard measurements and illustrates how a generalized measurement may be physically implemented. The "impossibility" of a joint measurement of two non commuting observables is revisited and its canonical implementations as a generalized measurement is described in some details. Finally, we address the basic properties, usually captured by the request of unitarity, that a map transforming quantum states into quantum states should satisfy to be physically admissible, and introduce the notion of complete positivity (CP). We then state and prove the Stinespring/Kraus-Choi-Sudarshan dilation theorem and elucidate the connections between the CP-maps description of quantum operations, together with their operator-sum representation, and the customary unitary description of quantum evolution. We also address transposition as an example of positive map which is not completely positive, and provide some examples of generalized measurements and quantum operations.Comment: Tutorial. 26 pages, 1 figure. Published in a special issue of EPJ - ST devoted to the memory of Federico Casagrand

Simultaneous minimum-uncertainty measurement of discrete-valued complementary observables

We have made the first experimental demonstration of the simultaneous minimum uncertainty product between two complementary observables for a two-state system (a qubit). A partially entangled two-photon state was used to perform such measurements. Each of the photons carries (partial) information of the initial state thus leaving a room for measurements of two complementary observables on every member in an ensemble.Comment: 4 pages, 4 figures, REVTeX, submitted to PR

Decoherence in a double-slit quantum eraser

We study and experimentally implement a double-slit quantum eraser in the presence of a controlled decoherence mechanism. A two-photon state, produced in a spontaneous parametric down conversion process, is prepared in a maximally entangled polarization state. A birefringent double-slit is illuminated by one of the down-converted photons, and it acts as a single-photon two-qubits controlled not gate that couples the polarization with the transversal momentum of these photons. The other photon, that acts as a which-path marker, is sent through a Mach-Zehnder-like interferometer. When the interferometer is partially unbalanced, it behaves as a controlled source of decoherence for polarization states of down-converted photons. We show the transition from wave-like to particle-like behavior of the signal photons crossing the double-slit as a function of the decoherence parameter, which depends on the length path difference at the interferometer.Comment: Accepted in Physical Review

Witnessing effective entanglement in a continuous variable prepare&measure setup and application to a QKD scheme using postselection

We report an experimental demonstration of effective entanglement in a prepare&measure type of quantum key distribution protocol. Coherent polarization states and heterodyne measurement to characterize the transmitted quantum states are used, thus enabling us to reconstruct directly their Q-function. By evaluating the excess noise of the states, we experimentally demonstrate that they fulfill a non-separability criterion previously presented by Rigas et al. [J. Rigas, O. G\"uhne, N. L\"utkenhaus, Phys. Rev. A 73, 012341 (2006)]. For a restricted eavesdropping scenario we predict key rates using postselection of the heterodyne measurement results.Comment: 12 pages, 12 figures, 2 table

Approximate joint measurement of qubit observables through an Arthur-Kelly type model

We consider joint measurement of two and three unsharp qubit observables through an Arthur-Kelly type joint measurement model for qubits. We investigate the effect of initial state of the detectors on the unsharpness of the measurement as well as the post-measurement state of the system. Particular emphasis is given on a physical understanding of the POVM to PVM transition in the model and entanglement between system and detectors.Two approaches for characterizing the unsharpness of the measurement and the resulting measurement uncertainty relations are considered.The corresponding measures of unsharpness are connected for the case where both the measurements are equally unsharp. The connection between the POVM elements and symmetries of the underlying Hamiltonian of the measurement interaction is made explicit and used to perform joint measurement in arbitrary directions. Finally in the case of three observables we derive a necessary condition for the approximate joint measurement and use it show the relative freedom available when the observables are non-orthogonal.Comment: 22 pages; Late

Complementarity and the uncertainty relations

We formulate a general complementarity relation starting from any Hermitian operator with discrete non-degenerate eigenvalues. We then elucidate the relationship between quantum complementarity and the Heisenberg-Robertson's uncertainty relation. We show that they are intimately connected. Finally we exemplify the general theory with some specific suggested experiments.Comment: 9 pages, 4 figures, REVTeX, uses epsf.sty and multicol.st

Joint measurements via quantum cloning

We explore the possibility of achieving optimal joint measurements of noncommuting observables on a single quantum system by performing conventional measurements of commuting self adjoint operators on optimal clones of the original quantum system. We consider the case of both finite dimensional and infinite dimensional Hilbert spaces. In the former we study the joint measurement of three orthogonal components of a spin 1/2, in the latter we consider the case of the joint measurements of any pair of noncommuting quadratures of one mode of the electromagnetic field. We show that universally covariant cloning is not ideal for joint measurements, and a suitable non universally covariant cloning is needed.Comment: 8 page

Interpretation for a positive P representation

We show that a "canonical" form of the positive P representation has a simple interpretation as the statistics of four detectors, two of which make redundant position measurements, while the other two simultaneously make redundant momentum measurements. This interpretation allows us to understand the additional degrees of freedom for the canonical positive P representation

Processing and Transmission of Information

Contains research objectives and reports on four research projects.Lincoln Laboratory, Purchase Order DDL-B222Department of the ArmyDepartment of the NavyDepartment of the Air Force under Contract AF19(122)-45

Control of quantum interference in the quantum eraser

We have implemented an optical quantum eraser with the aim of studying this phenomenon in the context of state discrimination. An interfering single photon is entangled with another one serving as a which-path marker. As a consequence, the visibility of the interference as well as the which-path information are constrained by the overlap (measured by the inner product) between the which-path marker states, which in a more general situation are non-orthogonal. In order to perform which-path or quantum eraser measurements while analyzing non-orthogonal states, we resort to a probabilistic method for the unambiguous modification of the inner product between the two states of the which-path marker in a discrimination-like process.Comment: Submitted to New Journal of Physics, March 200