Proud to support the new journal! Here is my first submission: note on Leggett-Garg 4. *Abstract*: In this paper we first analyse Leggett and Garg’s argument to the effect that macroscopic realism contradicts quantum mechanics. After making explicit all the assumptions in Leggett and Garg’s reasoning, we argue against the plausibility of their auxiliary assumption of non-invasive measurability, using Bell’s construction of stochastic pilot-wave theories as a counterexample. Violations of the Leggett-Garg inequality thus do not provide a good argument against macrorealism. We then apply Dzhafarov and Kujala’s analysis of contextuality in the presence of signalling to the case of the Leggett-Garg inequalities, with rather surprising results. An analogy with pilot-wave theory again helps to clarify the situation. *Acknowledgements*: This paper was prompted by the talks of and discussions with Ehtibar Dzhafarov and Acacio de Barros at the conference on ‘Quantum Theory: from Problems to Advances’ at Linnaeus University, Växjö, Sweden, 9-12 June 2014. Many thanks to Ehti and especially to Acacio for detailed subsequent correspondence and comments on previous drafts of this paper.

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This paper has been sent out to review.

Johannes Kofler has kindly alerted me to the fact that he and Caslav Brukner also have a paper on the Leggett-Garg inequalities which somewhat overlaps with mine. Please see http://journals.aps.org/pra/abstract/10.1103/PhysRevA.87.052115 .

Hi Guido, Many thanks for the useful information. I will look at the paper. Best, Shan

Referee Report

This is a review of Bacciagaluppi paper entitled “Leggett–Garg Inequalities, Pilot Waves and Contextuality” and submitted to International Journal of Quantum Foundations. This paper can almost be split into two parts. First, the author analyses the claims of Leggett-Garg (LG) about macroscopic realism by making their (very obscure) assumptions more explicit. In particular, the author shows that a realistic pilot-wave theory provides a counter-example for LG’s claims, by present a macroscopically realistic model that violates LG’s assumption of non-invasive measurability.

In the second part, an analysis of the violations of the LG inequalities is done in terms of the inequalities of Dzhafarov and Kujala, which take into account not only the pairwise joint expectations, but also possible violations of the nosignalling

condition. The fact that the LG setup, as often discussed, does not lead to a violation of contextuality, but instead to a violation of no-signaling is, in itself, a very interesting result. Compounding this, there is the even more surprising result that contextuality, without signalling, can be violated by a particular mixed state.

Though the paper is, by and large, well written, my main concern is about the second part. In it, the author uses eq. (25) as a modified version of Suppes and Zanotti that takes into account signalling. To show that this expression is not violated, equation (29) and (30) are used. However, these are not the same as (2) and (3). Why are they different equations, since in both we have measurements in ti and tj . In fact, this also brings a puzzle as to why, using different inequalities but yet the same measurements, would a pure versus a mixed state matter for such violations. I would be important for the author to clarify those issues, which leave the reader puzzled.

So, in short, this is a very interesting paper, but has some unclear parts that need to be fixed. Therefore, I would not recommend it for publication in its current form, but I believe that changes to Section 4 can be made that would result in a suitable paper.

Author’s Reply

I wish to thank the referee for the positive comments on my paper. As regards the suggestions for improvement, I have now made clear what is needed in the respective evaluation of equations (15) and (26) (the original Leggett–Garg inequality and the modified Suppes–Zanotti inequality). I have also phrased more carefully various passages in the last section (in which I am commenting on the differences between the case of pure and mixed initial states).

I have taken the opportunity to make further minor amendments in some of the notation, and to duly mention a couple of further references, in particular the recent paper by Kofler and Brukner and the one in preparation by Maroney and Timpson.

Revised paper

Dear Guido,

I appreciate your article and agree with your conclusion regarding the implausibility of the Leggett–Garg (LG) conditions. I would just like to add, for the interested reader and in support of this general thrust, that the LG assumptions are so strong that the can be used to produce a (quite readily violated) _equality_ resembling the CHSH Bell-type _inequality_ (as I along with Sahotra Sarkar should in our pubication “Bell-type equalities for SQUIDs on the assumption of macroscopic realism and non-invasive measurability,” Gregg Jaeger, Chris Viger and Sahotra Sarkar, Physics Letters A, 210: 5 -10 (1996).)

p.s. Here is a link to said article: http://math.bu.edu/people/jaeger/JS%20PLA210%2096.pdf