Quantum mechanical measurement of non-orthogonal states and a test of non-locality (original) (raw)
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Quantum Theory Allows Measurement of Non-Hermitian Operators
In quantum theory, a physical observable is represented by a Hermitian operator as it admits real eigenvalues. This stems from the fact that any measuring apparatus that is supposed to measure a physical observable will always yield a real number. However, reality of eigenvalue of some operator does not mean that it is necessarily Hermitian. There are examples of non-Hermitian operators which may admit real eigenvalues under some symmetry conditions. However, in general, given a non-Hermitian operator, its average value in a quantum state is a complex number and there are only very limited methods available to measure it. Following standard quantum mechanics, we provide an experimentally feasible protocol to measure the expectation value of any non-Hermitian operator via weak measurements. The average of a non-Hermitian operator in a pure state is a complex multiple of the weak value of the positive semi-definite part of the non-Hermitian operator. We also prove a new uncertainty relation for any two non-Hermitian operators and show that the fidelity of a quantum state under quantum channel can be measured using the average of the corresponding Kraus operators. The importance of our method is shown in testing the stronger uncertainty relation, verifying the Ramanujan formula and in measuring the product of non commuting projectors.
Experimental nonlocal measurement of a product observable
Optica, 2019
Nonlocal measurement, or instantaneous measurement of nonlocal observables, is a considerably difficult task even for a simple form of product observable since relativistic causality prohibits interaction between spacelike separate subsystems. Following a recent proposal for effectively creating the von Neumann measurement Hamiltonian of nonlocal observables [Brodutch and Cohen, Phys. Rev. Lett. 116, 070404 (2016)], here we report a proof-of-principle demonstration of nonlocally measuring a product observable using linear optics without the violation of relativistic causality. Our scheme provides a feasible approach to perform nonlocal measurements via quantum erasure with linear optics.
Foundations of Physics, 1990
A partial ordering in the class of observables (∼ positive operator-valued measures, introduced by Davies and by Ludwig) is explored. The ordering is interpreted as a form of nonideality, and it allows one to compare ideal and nonideal versions of the same observable. Optimality is defined as maximality in the sense of the ordering. The framework gives a generalization of the usual (implicit) definition of self-adjoint operators as optimal observables (von Neumann), but it can, in contrast to this latter definition, be justified operationally. The nonideality notion is compared to other quantum estimation theoretic methods. Measures for the amount of nonideality are derived from information theory.
Local and nonlocal observables in quantum optics
New Journal of Physics, 2014
It is pointed out that there exists an unambiguous definition of locality that enables one to distinguish local and nonlocal quantities. Observables of both types coexist in quantum optics but one must be very careful when attempting to measure them. A nonlocal observable which formally depends on the spatial position r cannot be locally measured without disturbing the measurements of this observable at all other positions.
Measurement As Spontaneous Symmetry Breaking, Non-locality and Non-Boolean Holism
It is shown that having degenerate ground states over the domain of the wavefunction of a system is a sufficient condition for a quantum system to act as a measuring apparatus for the system. Measurements are then instances of spontaneous symmetry breaking to one of these ground states, induced by environmental perturbations. Together with non-Boolean holism this constitutes an optimal formulation of quantum mechanics that does not imply non-locality.
Quantum mechanics and the local observer
International Journal of Theoretical Physics, 1983
A notion of local observer inspired by the work of Segal is introduced here in the Hilbert space theory of quantum mechanics. The local observer finds a mathematical place in the Hilbert space through local negation or complementation. A logicomathematical theory of local negation is presented and its implications for quantum logic and the problem of measurement are discussed. The setting is constructivist mathematics and the main result of the paper states that the introduction of a local observer implies the nonorthocomplementability of the whole Hilbert space even in the finite-dimensional case. Making a mathematical place for the observer (the “projector”) thus modifies the structure of the observables or the system of the projections, in accordance with a nonclassical theory of quantum-mechanical measurement.
Unambiguous quantum measurement of nonorthogonal states
Physical Review A, 1996
Generalized quantum measurements can be used to separate deterministically two nonorthogonal quantum states. However, such measurements also lead to inconclusive results, where the initial state remains unknown. We introduce a particular type of generalized quantum measurement, which we term loss induced generalized ͑LIGe͒ quantum measurement, and present an experimental realization. This LIGe measurement achieves optimal deterministic separation of two nonorthogonally polarized single photons.
Orthogonalization of partly unknown quantum states
Physical Review A, 2014
A quantum analog of the fundamental classical NOT gate is a quantum gate that would transform any input qubit state onto an orthogonal state. Intriguingly, this universal NOT gate is forbidden by the laws of quantum physics. This striking phenomenon has far-reaching implications concerning quantum information processing and encoding information about directions and reference frames into quantum states. It also triggers the question of under what conditions the preparation of quantum states orthogonal to input states becomes possible. Here we report on experimental demonstration of orthogonalization of partly unknown single-and two-qubit quantum states. A state orthogonal to an input state is conditionally prepared by quantum filtering, and the only required information about the input state is a mean value of a single arbitrary operator. We show that perfect orthogonalization of partly unknown two-qubit entangled states can be performed by applying the quantum filter to one of the qubits only.
Entanglement, detection, and geometry of non-classical States
2010
Non-classical states that are characterized by their non-positive quasi-probabilities in phase space are known to be the basis for various quantum effects. In this work, we investigate the interrelation between the non-classicality and entanglement, and then characterize the non-classicality that precisely corresponds to entanglement. The results naturally follow from two findings: one is the general structure among non-classical, entangled, separable, and classical states over Hermitian operators, and the other a general scheme to detect non-classical states.
Demonstration of Quantum Nonlocality in the Presence of Measurement Dependence
Physical Review Letters, 2015
Quantum nonlocality stands as a resource for Device Independent Quantum Information Processing (DIQIP), as, for instance, Device Independent Quantum Key Distribution. We investigate experimentally the assumption of limited Measurement Dependence, i.e., that the measurement settings used in Bell inequality tests or DIQIP are partially influenced by the source of entangled particle and/or by an adversary. Using a recently derived Bell-like inequality [Phys. Rev. Lett. 113 190402 ] and a 99% fidelity source of partially entangled polarization photonic qubits, we obtain a clear violation of the inequality, excluding a much larger range of measurement dependent local models than would be possible with an adapted Clauser, Horne, Shimony and Holt (CHSH) inequality. It is therefore shown that the Measurement Independence assumption can be widely relaxed while still demonstrating quantum nonlocality.