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Register a new userErgodicity properties of quantum expectation values in entangled states
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Using a model Hamiltonian for a single-mode electromagnetic field interacting with a nonlinear medium, we show that quantum expectation values of subsystem observables can exhibit remarkably diverse ergodic properties even when the dynamics of the total system is regular. The time series of the mean photon number is studied over a range of values of the ratio of the strength $gamma$ of the nonlinearity to that of the inter-mode coupling $g$. We obtain the power spectrum, estimate the embedding dimension of the reconstructed phase space and the maximal Liapunov exponent $lambda_{rm max}$, and find the recurrence-time distribution of the coarse-grained dynamics. The dynamical behavior ranges from quasiperiodicity (for $gamma/g ll 1$) to chaos as characterized by $lambda_{rm max} > 0$ (for $gamma/g gtrsim 1$), and is interpreted.
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In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in spherical polar coordinate comes. We have tried to point out most of the elementary quantum mechanical results, related to the momentum operator, which has coordinate dependence. We explicitly calculate the momentum expectation values in various bound states and show that the expectation value really turns out to be zero, a consequence of the fact that the momentum expectation value is real. We comment briefly on the status of the angular variables in quantum mechanics and the problems related in interpreting them as dynamical variables. At the end, we calculate the Heisenbergs equation of motion for the radial component of the momentum for the Hydrogen atom.
The quantum fluctuations of a physical property can be observed in the measurement statistics of any measurement that is at least partially sensitive to that physical property. Quantum theory indicates that the effective distribution of values taken by the physical property depends on the specific measurement context based on which these values are determined and weak values have been identified as the contextual values describing this dependence of quantum fluctuations on the measurement context. Here, the relation between classical statistics and quantum contextuality is explored by considering systems entangled with a quantum reference. The quantum fluctuations of the system can then be steered by precise projective measurements of the reference, resulting in different contextual values of the quantum fluctuations depending on the effective state preparation context determined by the measurement of the reference. The results show that mixed state statistics are consistent with a wide range of potential contexts, indicating that the precise definition of a context requires maximal quantum coherence in both state preparation and measurement.
The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical operators are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfests theorem is shown to be Lie-Poisson for a semidirect-product Lie group, named the `Ehrenfest group. The underlying Poisson structure produces classical and quantum mechanics as special limit cases. In addition, quantum dynamics is expressed in the frame of the expectation values, in which the latter undergo canonical Hamiltonian motion. In the case of Gaussian states, expectation values dynamics couples to second-order moments, which also enjoy a momentum map structure. Eventually, Gaussian states are shown to possess a Lie-Poisson structure associated to another semidirect-product group, which is called the Jacobi group. This structure produces the energy-conserving variant of a class of Gaussian moment models previously appeared in the chemical physics literature.
Measurements on current quantum-hardware are subject to hardware imperfections that lead to readout-errors. These errors manifest themselves as a bias in quantum expectation values. Here, we propose a method to remove this bias from the expectation values of Pauli observables. No specific form of the noise is assumed, other than requiring that it is `weak. We apply a method that forces the bias in the expectation value to appear as a multiplicative factor irrespective of the actual noise process. This factor can be measured directly and removed, at the cost of an increase in the sampling complexity for the observable. A bound relating the error in the expectation value to the sample complexity is derived.
Noise in existing quantum processors only enables an approximation to ideal quantum computation. However, these approximations can be vastly improved by error mitigation, for the computation of expectation values, as shown by small-scale experimental demonstrations. However, the practical scaling of these methods to larger system sizes remains unknown. Here, we demonstrate the utility of zero-noise extrapolation for relevant quantum circuits using up to 26 qubits, circuit depths of 60, and 1080 CNOT gates. We study the scaling of the method for canonical examples of product states and entangling Clifford circuits of increasing size, and extend it to the quench dynamics of 2-D Ising spin lattices with varying couplings. We show that the efficacy of the error mitigation is greatly enhanced by additional error suppression techniques and native gate decomposition that reduce the circuit time. By combining these methods, we demonstrate an accuracy in the approximate quantum simulation of the quench dynamics that surpasses the classical approximations obtained from a state-of-the-art 2-D tensor network method. These results reveal a path to a relevant quantum advantage with noisy, digital, quantum processors.
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