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arXiv:1909.08931 · Physical Review Research

Characterizing coherence with quantum observables

Suman Mandal, Marek Narożniak, Chandrashekar Radhakrishnan, Zhi-Qiang Jiao, Xian-Min Jin, Tim Byrnes

Published Feb 13, 2020• 12 citations
arXivHTMLPDF

Abstract

We introduce a procedure based on quantum expectation values of measurement observables to characterize quantum coherence.Our measure allows one to quantify coherence without having to perform tomography of the quantum state and can be directly calculated from measurement expectation values.This definition of coherence allows the decomposition into contributions corresponding to the nonclassical correlations between the subsystems and localized on each subsystem.The method can also be applied to cases where the full set of measurement operators is unavailable.An estimator using the truncated measurement operators can be used to obtain lower bound to the genuine value of coherence.We illustrate the method for several bipartite systems and show the singular behavior of the coherence measure in a spin-1 chain, characteristic of a quantum phase transition.

DOI
10.1103/physrevresearch.2.013157

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We introduce a procedure based on quantum expectation values of measurement observables to characterize quantum coherence.Our measure allows one to quantify coherence without having to perform tomography of the quantum state and can be directly calculated from measurement expectation values.This definition of coherence allows the decomposition into contributions corresponding to the nonclassical correlations between the subsystems and localized on each subsystem.The method can also be applied to cases where the full set of measurement operators is unavailable.An estimator using the truncated measurement operators can be used to obtain lower bound to the genuine value of coherence.We illustrate the method for several bipartite systems and show the singular behavior of the coherence measure in a spin-1 chain, characteristic of a quantum phase transition.
We introduce a procedure based on quantum expectation values of measurement observables to characterize quantum coherence. In a bipartite system, the two-point correlation function is used in combination with the Schatten-1 norm to estimate the quantum coherence. This definition of coherence allows the decomposition into contributions corresponding to the non-classical correlations between the subsystems and localized on each subsystem. We illustrate the method for several bipartite systems, showing that an incomplete basis still gives a qualitative measure of the coherence.
We introduce a measure Q of bipartite quantum correlations for arbitrary two-qubit states, expressed as a state-independent function of the density matrix elements. The amount of quantum correlations can be quantified experimentally by measuring the expectation value of a small set of observables on up to four copies of the state, without the need for a full tomography. We extend the measure to 2×d systems, providing its explicit form in terms of observables and applying it to the relevant class of multiqubit states employed in the deterministic quantum computation with one quantum bit model. The number of required measurements to determine Q in our scheme does not increase with d. Our results provide an experimentally friendly framework to estimate quantitatively the degree of general quantum correlations in composite systems.
Quantum coherence quantifies the amount of superposition a quantum state can have in a given basis. Since there is a difference in the structure of eigenstates of the ergodic and many-body localized systems, we expect them also to differ in terms of their coherences in a given basis. Here, we numerically calculate different measures of quantum coherence in the excited eigenstates of an interacting disordered Hamiltonian as a function of the disorder. We show that quantum coherence can be used as an order parameter to detect the well-studied ergodic to many-body localized phase transition. We also perform quantum quench studies to distinguish the behavior of coherence in thermalized and localized phases. We then present a protocol to calculate measurement-based localizable coherence to investigate the thermal and many-body localized phases. The protocol allows one to investigate quantum correlations experimentally in a nondestructive way, in contrast to measures that require tracing out a subsystem, which always destroys coherence and correlation.
Quantum correlations in a physical system are usually studied with respect to a unique and fixed decomposition of the system into subsystems, without fully exploiting the rich structure of the state space. Here, we show several examples in which the consideration of different ways to decompose a physical system enhances the quantum resources and accounts for a more flexible definition of quantumness measures. Furthermore, we give a different perspective regarding how to reassess the fact that local operations play a key role in general quantumness measures that go beyond entanglement---as discordlike ones. We propose a family of measures to quantify the maximum quantumness of a given state. For the discord-based case, we present some analytical results for $2\ifmmode\times\else\texttimes\fi{}d$-dimensional states. Applying our definition to low-dimensional bipartite states, we show that different behaviors can be reported for separable and entangled states vis-\`a-vis those corresponding to the usual measures of quantum correlations. We show that there is a close link between our proposal and the criterion to witness quantum correlations based on the rank of the correlation matrix, proposed by Daki\ifmmode \acute{c}\else \'{c}\fi{}, Vedral, and Brukner [Phys. Rev. Lett. 105, 190502 (2010)].
Abstract The geometry of quantum states provides a unifying framework for estimation processes based on quantum probes, and it establishes the ultimate bounds of the achievable precision. We show a relation between the statistical distance between infinitesimally close quantum states and the second order variation of the coherence of the optimal measurement basis with respect to the state of the probe. In quantum phase estimation protocols, this leads to propose coherence as the relevant resource that one has to engineer and control to optimize the estimation precision. Furthermore, the main object of the theory i.e. the symmetric logarithmic derivative, in many cases allows one to identify a proper factorization of the whole Hilbert space in two subsystems. The factorization allows one to discuss the role of coherence versus correlations in estimation protocols; to show how certain estimation processes can be completely or effectively described within a single-qubit subsystem; and to derive lower bounds for the scaling of the estimation precision with the number of probes used. We illustrate how the framework works for both noiseless and noisy estimation procedures, in particular those based on multi-qubit GHZ-states. Finally we succinctly analyze estimation protocols based on zero-temperature critical behavior. We identify the coherence that is at the heart of their efficiency, and we show how it exhibits the non-analyticities and scaling behavior proper of a large class of quantum phase transitions.
Entanglement measures quantify nonclassical correlations present in a quantum system, but can be extremely difficult to calculate, even more so, when information on its state is limited. Here, we consider broad families of entanglement criteria that are based on variances of arbitrary operators and analytically derive the lower bounds these criteria provide for two relevant entanglement measures: the best separable approximation and the generalized robustness. This yields a practical method for quantifying entanglement in realistic experimental situations, in particular, when only few measurements of simple observables are available. As a concrete application of this method, we quantify bipartite and multipartite entanglement in spin-squeezed Bose-Einstein condensates of ∼500 atoms, by lower bounding the best separable approximation and the generalized robustness only from measurements of first and second moments of the collective spin operator.