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Measurements play a crucial role in doing physics: Their results provide the basis on which we adopt or reject physical theories. In this note, we examine the effect of subjecting measurements themselves to our experience. We require that our contact with the world is empirically warranted. Therefore, we study theories that satisfy the following assumption: Interactions are accounted for so that they are empirically traceable, and observations necessarily go with such an interaction with the observed system. Examining, with regard to these assumptions, an abstract representation of measurements with tools from quantum logic leads us to contextual theories. Contextuality becomes a means to render interactions, thus also measurements, empirically tangible. The measurement becomes problematic---also beyond quantum mechanics---if one tries to commensurate the assumption of tangible interactions with the notion of a spectator theory, i.e., with the idea that measurement results are read off without effect. The problem, thus, presents itself as the collision of different epistemological stances with repercussions beyond quantum mechanics.
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Time plays a fundamental role in our ability to make sense of the physical laws in the world around us. The nature of time has puzzled people -- from the ancient Greeks to the present day -- resulting in a long running debate between philosophers and physicists alike to whether time needs change to exist (the so-called relatival theory), or whether time flows regardless of change (the so-called substantival theory). One way to decide between the two is to attempt to measure the flow of time with a stationary clock, since if time were substantival, the flow of time would manifest itself in the experiment. Alas, conventional wisdom suggests that in order for a clock to function, it cannot be a static object, thus rendering this experiment seemingly impossible. We show, counter-intuitively, that a quantum clock can measure the passage of time even while being switched off, lending support for the substantival theory of time.
Measurement connects the world of quantum phenomena to the world of classical events. It plays both a passive role, observing quantum systems, and an active one, preparing quantum states and controlling them. Surprisingly - in the light of the central status of measurement in quantum mechanics - there is no general recipe for designing a detector that measures a given observable. Compounding this, the characterization of existing detectors is typically based on partial calibrations or elaborate models. Thus, experimental specification (i.e. tomography) of a detector is of fundamental and practical importance. Here, we present the realization of quantum detector tomography: we identify the optimal positive-operator-valued measure describing the detector, with no ancillary assumptions. This result completes the triad, state, process, and detector tomography, required to fully specify an experiment. We characterize an avalanche photodiode and a photon number resolving detector capable of detecting up to eight photons. This creates a new set of tools for accurately detecting and preparing non-classical light.
Measurement is integral to quantum information processing and communication; it is how information encoded in the state of a system is transformed into classical signals for further use. In quantum optics, measurements are typically destructive, so that the state is not available afterwards for further steps - crucial for sequential measurement schemes. The development of practical methods for non-destructive measurements on optical fields is therefore an important topic for future practical quantum information processing systems. Here we show how to measure the presence or absence of the vacuum in a quantum optical field without destroying the state, implementing the ideal projections onto the respective subspaces. This not only enables sequential measurements, useful for quantum communication, but it can also be adapted to create novel states of light via bare raising and lowering operators.
The entropy of a quantum system is a measure of its randomness, and has applications in measuring quantum entanglement. We study the problem of measuring the von Neumann entropy, $S(rho)$, and Renyi entropy, $S_alpha(rho)$ of an unknown mixed quantum state $rho$ in $d$ dimensions, given access to independent copies of $rho$. We provide an algorithm with copy complexity $O(d^{2/alpha})$ for estimating $S_alpha(rho)$ for $alpha<1$, and copy complexity $O(d^{2})$ for estimating $S(rho)$, and $S_alpha(rho)$ for non-integral $alpha>1$. These bounds are at least quadratic in $d$, which is the order dependence on the number of copies required for learning the entire state $rho$. For integral $alpha>1$, on the other hand, we provide an algorithm for estimating $S_alpha(rho)$ with a sub-quadratic copy complexity of $O(d^{2-2/alpha})$. We characterize the copy complexity for integral $alpha>1$ up to constant factors by providing matching lower bounds. For other values of $alpha$, and the von Neumann entropy, we show lower bounds on the algorithm that achieves the upper bound. This shows that we either need new algorithms for better upper bounds, or better lower bounds to tighten the results. For non-integral $alpha$, and the von Neumann entropy, we consider the well known Empirical Young Diagram (EYD) algorithm, which is the analogue of empirical plug-in estimator in classical distribution estimation. As a corollary, we strengthen a lower bound on the copy complexity of the EYD algorithm for learning the maximally mixed state by showing that the lower bound holds with exponential probability (which was previously known to hold with a constant probability). For integral $alpha>1$, we provide new concentration results of certain polynomials that arise in Kerov algebra of Young diagrams.
We study local-realistic inequalities, Bell-type inequalities, for bipartite pure states of finite dimensional quantum systems -- qudits. There are a number of proposed Bell-type inequalities for such systems. Our interest is in relating the value of Bell-type inequality function with a measure of entanglement. Interestingly, we find that one of these inequalities, the Son-Lee-Kim inequality, can be used to measure entanglement of a pure bipartite qudit state and a class of mixed two-qudit states. Unlike the majority of earlier schemes in this direction, where number of observables needed to characterize the entanglement increases with the dimension of the subsystems, this method needs only four observables. We also discuss the experimental feasibility of this scheme. It turns out that current experimental set ups can be used to measure the entanglement using our scheme.
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