Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups.

We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument's evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.

Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group.

The canonical commutation relation, [Q,P]=iℏ, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following. The simultaneous P and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg (IWH) group. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of the IWH group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument.

Fisher-Symmetric Informationally Complete Measurements for Pure States.

We introduce a new kind of quantum measurement that is defined to be symmetric in the sense of uniform Fisher information across a set of parameters that uniquely represent pure quantum states in the neighborhood of a fiducial pure state. The measurement is locally informationally complete-i.e., it uniquely determines these parameters, as opposed to distinguishing two arbitrary quantum states-and it is maximal in the sense of a multiparameter quantum Cramér-Rao bound. For a d-dimensional quantum system, requiring only local informational completeness allows us to reduce the number of outcomes of the measurement from a minimum close to but below 4d-3, for the usual notion of global pure-state informational completeness, to 2d-1.

Optimal quantum-enhanced interferometry using a laser power source.

We consider an interferometer powered by laser light (a coherent state) into one input port and ask the following question: what is the best state to inject into the second input port, given a constraint on the mean number of photons this state can carry, in order to optimize the interferometer's phase sensitivity? This question is the practical question for high-sensitivity interferometry. We answer the question by considering the quantum Cramér-Rao bound for such a setup. The answer is squeezed vacuum.

Fundamental quantum limit to waveform estimation.

We derive a quantum Cramér-Rao bound (QCRB) on the error of estimating a time-changing signal. The QCRB provides a fundamental limit to the performance of general quantum sensors, such as gravitational-wave detectors, force sensors, and atomic magnetometers. We apply the QCRB to the problem of force estimation via continuous monitoring of the position of a harmonic oscillator, in which case the QCRB takes the form of a spectral uncertainty principle. The bound on the force-estimation error can be achieved by implementing quantum noise cancellation in the experimental setup and applying smoothing to the observations.

Coherent quantum-noise cancellation for optomechanical sensors.

Using a flow chart representation of quantum optomechanical dynamics, we design coherent quantum-noise-cancellation schemes that can eliminate the backaction noise induced by radiation pressure at all frequencies and thus overcome the standard quantum limit of force sensing. The proposed schemes can be regarded as novel examples of coherent feedforward quantum control.

Quantum discord and the geometry of Bell-diagonal states.

The set of Bell-diagonal states for two qubits can be depicted as a tetrahedron in three dimensions. We consider the level surfaces of entanglement and quantum discord for Bell-diagonal states. This provides a complete picture of the structure of entanglement and discord for this simple case and, in particular, of their nonanalytic behavior under decoherence. The pictorial approach also indicates how to show that discord is neither convex nor concave.

Quantum metrology: dynamics versus entanglement.

A parameter whose coupling to a quantum probe of n constituents includes all two-body interactions between the constituents can be measured with an uncertainty that scales as 1/n3/2, even when the constituents are initially unentangled. We devise a protocol that achieves the 1/n3/2 scaling without generating any entanglement among the constituents, and we suggest that the protocol might be implemented in a two-component Bose-Einstein condensate.

Quantum discord and the power of one qubit.

We use quantum discord to characterize the correlations present in the model called deterministic quantum computation with one quantum bit (DQC1), introduced by Knill and Laflamme [Phys. Rev. Lett. 81, 5672 (1998)10.1103/PhysRevLett.81.5672]. The model involves a collection of qubits in the completely mixed state coupled to a single control qubit that has nonzero purity. The initial state, operations, and measurements in the model all point to a natural bipartite split between the control qubit and the mixed ones. Although there is no entanglement between these two parts, we show that the quantum discord across this split is nonzero for typical instances of the DQC1 ciruit. Nonzero values of discord indicate the presence of nonclassical correlations. We propose quantum discord as figure of merit for characterizing the resources present in this computational model.

Generalized limits for single-parameter quantum estimation.

We develop generalized bounds for quantum single-parameter estimation problems for which the coupling to the parameter is described by intrinsic multisystem interactions. For a Hamiltonian with k-system parameter-sensitive terms, the quantum limit scales as 1/Nk, where N is the number of systems. These quantum limits remain valid when the Hamiltonian is augmented by any parameter-independent interaction among the systems and when adaptive measurements via parameter-independent coupling to ancillas are allowed.

Classical phase-space descriptions of continuous-variable teleportation.

The non-negative Wigner function of all quantum states involved in teleportation of Gaussian states using the standard continuous-variable teleportation protocol means that there is a local realistic phase-space description of the process. This includes the coherent states teleported up to now in experiments. We extend the phase-space description to teleportation of non-Gaussian states using the standard protocol and conclude that teleportation of non-Gaussian pure states with a fidelity of 2/3 is a "gold standard" for this kind of teleportation.

Local realistic model for the dynamics of bulk-ensemble NMR information processing.

We construct a local realistic hidden-variable model that describes the states and dynamics of bulk-ensemble NMR information processing up to about 12 nuclear spins. The existence of such a model rules out violation of any Bell inequality, temporal or otherwise, in present high-temperature, liquid-state NMR experiments. The model does not provide an efficient description in that the number of hidden variables grows exponentially with the number of nuclear spins.