Quantum Physics

The Leggett-Garg (LG) inequalities were introduced, as a temporal parallel of the Bell inequalities, to test macroscopic realism -- the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The talk will begin with a review of the LG framework. Unlike the Bell inequalities, the original LG inequalities are only a necessary condition for macrorealism, and are therefore not a decisive test. I argue, for the case of measurements of a single dichotomic variable Q, that when the original four three-time LG inequalities are augmented with a set of twelve two-time inequalities also of the LG form, Fine's theorem applies and these augmented conditions are then both necessary and sufficient [1]. A comparison is carried out with the alternative necessary and sufficient conditions for macrorealism  based on no-signaling in time conditions which ensure that all probabilities for Q at one and two times are independent of whether earlier or intermediate measurements are made. I argue that the two tests differ in their implementation of the key requirement of non-invasive measurability so are testing different notions of macrorealism, and these notions are elucidated.  I also describe some alternative protocols which achieve non-invasiveness, one involving continuous measurement of the velocity conjugate to Q [2], which was recently implemented in an experiment at IQC, the other involving a modification of the standard ideal negative measurement protocol [3].

[1] J.J.Halliwell, Phys Rev A96, 012121 (2017); A93, 022123 (2016); arxiv:1811.10408.

[2]  J.J.Halliwell, Phys. Rev. A94, 052114 (2016).

[3] J.J.Halliwell, Phys. Rev. A99, 022119 (2019).

The theory of statistical comparison was formulated (chiefly by David Blackwell in the 1950s) in order to extend the theory of majorization to objects beyond probability distributions, like multivariate statistical models and stochastic transitions, and has played an important role in mathematical statistics ever since. The central concept in statistical comparison is the so-called "information ordering," according to which information need not always be a totally ordered quantity, but often takes on a multi-faceted form whose content may vary depending on its use. In this talk, after reviewing the basic ideas of statistical comparison with an emphasis on their operational character, I will discuss various generalizations to quantum theory (and beyond). I will then argue that quantum statistical comparison provides a natural framework, somehow complementary to semi-definite programming, to study quantum resource theories, with explicit examples given by the resource theories of quantum nonlocality, quantum communication, and quantum thermodynamics.

We study correlations in fermionic systems with long-range interactions in thermal equilibrium. We prove an upper-bound on the correlation decay between anti-commut-ing operators based on long-range Lieb-Robinson type bounds. Our result shows that correlations between such operators in fermionic long-range systems of spatial dimension $D$ with at most two-site interactions decaying algebraically with the distance with an exponent $\alpha \geq 2\,D$, decay at least algebraically with an exponent arbitrarily close to $\alpha$. Our bound is asymptotically tight, which we demonstrate by numerically analysing density-density correlations in a 1D quadratic (free, exactly solvable) model, the Kitaev chain with long-range interactions. Away from the quantum critical point correlations in this model are found to decay asymptotically as slowly as our bound permits.