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We derive and implement a general method to characterize the nonclassicality in compound discrete- and continuous-variable systems. For this purpose, we introduce the operational notion of conditional hybrid nonclassicality which relates to the ability to produce a nonclassical continuous-variable state by projecting onto a general superposition of discrete-variable subsystem. We discuss the importance of this form of quantumness in connection with interfaces for quantum communication. To verify the conditional hybrid nonclassicality, a matrix version of a nonclassicality quasiprobability is derived and its sampling approach is formulated. We experimentally generate an entangled, hybrid Schrodinger cat state, using a coherent photon-addition process acting on two temporal modes, and we directly sample its nonclassicality quasiprobability matrix. The introduced conditional quantum effects are certified with high statistical significance.
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We examine weak measurements of arbitrary observables where the object is prepared in a mixed state and on which measurements with imperfect detectors are made. The weak value of an observable can be expressed as a conditional expectation value over an infinite class of different generalized Kirkwood quasi-probability distributions. Strange weak values for which the real part exceeds the eigenvalue spectrum of the observable can only be found if the Terletsky-Margenau-Hill distribution is negative, or, equivalently, if the real part of the weak value of the density operator is negative. We find that a classical model of a weak measurement exists whenever the Terletsky-Margenau-Hill representation of the observable equals the classical representation of the observable and the Terletsky-Margenau-Hill distribution is nonnegative. Strange weak values alone are not sufficient to obtain a contradiction with classical models. We propose feasible weak measurements of photon number of the radiation field. Negative weak values of energy contradicts all classical stochastic models, whereas negative weak values of photon number contradict all classical stochastic models where the energy is bounded from below by the zero-point energy. We examine coherent states in particular, and find negative weak values with probabilities of 16% for kinetic energy (or squared field quadrature), 8% for harmonic oscillator energy and 50% for photon number. These experiments are robust against detector inefficiency and thermal noise.
It is demonstrated that thermal radiation of small occupation number is strongly nonclassical. This includes most forms of naturally occurring radiation. Nonclassicality can be observed as a negative weak value of a positive observable. It is related to negative values of the Margenau-Hill quasi-probability distribution.
We investigate the nonclassicality of an open quantum system using Leggett-Garg inequality (LGI) which test the correlations of a single system measured at different times. Violation of LGI implies nonclassical behavior of the open system. We investigate the violation of the Leggett-Garg inequality for a two level system (qubit) spontaneously decaying under a general non-Markovian dissipative environment. Our results are exact as we have calculated the two-time correlation functions exactly for a wide range of system-environment parameters beyond Born-Markov regime.
We present a set of practical benchmarks for $N$-qubit arrays that economically test the fidelity of achieving multi-qubit nonclassicality. The benchmarks are measurable correlators similar to 2-qubit Bell correlators, and are derived from a particular set of geometric structures from the $N$-qubit Pauli group. These structures prove the Greenberger-Horne-Zeilinger (GHZ) theorem, while the derived correlators witness genuine $N$-partite entanglement and establish a tight lower bound on the fidelity of particular stabilizer state preparations. The correlators need only $M leq N+1$ distinct measurement settings, as opposed to the $2^{2N}-1$ settings that would normally be required to tomographically verify their associated stabilizer states. We optimize the measurements of these correlators for a physical array of qubits that can be nearest-neighbor-coupled using a circuit of controlled-$Z$ gates with constant gate depth to form $N$-qubit linear cluster states. We numerically simulate the provided circuits for a realistic scenario with $N=3,...,9$ qubits, using ranges of $T_1$ energy relaxation times, $T_2$ dephasing times, and controlled-$Z$ gate-fidelities consistent with Googles 9-qubit superconducting chip. The simulations verify the tightness of the fidelity bounds and witness nonclassicality for all nine qubits, while also showing ample room for improvement in chip performance.
One of the central problems in quantum theory is to characterize, detect, and quantify quantumness in terms of classical strategies. Dephasing processes, caused by non-dissipative information exchange between quantum systems and environments, provides a natural platform for this purpose, as they control the quantum-to-classical transition. Recently, it has been shown that dephasing dynamics itself can exhibit (non)classical traits, depending on the nature of the system-environment correlations and the related (im)possibility to simulate these dynamics with Hamiltonian ensembles---the classical strategy. Here we establish the framework of detecting and quantifying the nonclassicality for pure dephasing dynamics. The uniqueness of the canonical representation of Hamiltonian ensembles is shown, and a constructive method to determine the latter is presented. We illustrate our method for qubit, qutrit, and qubit-pair pure dephasing and describe how to implement our approach with quantum process tomography experiments. Our work is readily applicable to present-day quantum experiments.
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