In Phys. Rev. A 101 (2020) 022117 it was argued that Bell inequalities are based on classical, not quantum, physics, and hence their violation in experiments provides no support for the claimed existence of peculiar nonlocal and superluminal influenc
es in the real (quantum) world. Following a brief review of some aspects of the Consistent Histories approach used in that work, the objections raised in Lambares Comment, arXiv:2102.075243v3, are examined and shown to rest on serious misunderstandings, and as a result fail to identify any errors in, or problems with, the work being criticized.
The positivity and nonadditivity of the one-letter quantum capacity (maximum coherent information) $Q^{(1)}$ is studied for two simple examples of complementary quantum channel pairs $(B,C)$. They are produced by a process, we call it gluing, for com
bining two or more channels to form a composite. (We discuss various other forms of gluing, some of which may be of interest for applications outside those considered in this paper.) An amplitude-damping qubit channel with damping probability $0leq p leq 1$ glued to a perfect channel is an example of what we call a generalized erasure channel characterized by an erasure probability $lambda$ along with $p$. A second example, using a phase-damping rather than amplitude-damping qubit channel, results in the dephrasure channel of Ledtizky et al. [Phys. Rev. Lett. 121, 160501 (2018)]. In both cases we find the global maximum and minimum of the entropy bias or coherent information, which determine $Q^{(1)}(B_g)$ and $Q^{(1)}(C_g)$, respectively, and the ranges in the $(p,lambda)$ parameter space where these capacities are positive or zero, confirming previous results for the dephrasure channel. The nonadditivity of $Q^{(1)}(B_g)$ for two channels in parallel occurs in a well defined region of the $(p,lambda)$ plane for the amplitude-damping case, whereas for the dephrasure case we extend previous results to additional values of $p$ and $lambda$ at which nonadditivity occurs. For both cases, $Q^{(1)}(C_g)$ shows a peculiar behavior: When $p=0$, $C_g$ is an erasure channel with erasure probability $1-lambda$, so $Q^{(1)}(C_g)$ is zero for $lambda leq 1/2$. However, for any $p>0$, no matter how small, $Q^{(1)}(C_g)$ is positive, though it may be extremely small, for all $lambda >0$. Despite the simplicity of these models we still lack an intuitive understanding of the nonadditivity of $Q^{(1)}(B_g)$ and the positivity of $Q^{(1)}(C_g)$.
In quantum physics the term `contextual can be used in more than one way. One usage, here called `Bell contextual since the idea goes back to Bell, is that if $A$, $B$ and $C$ are three quantum observables, with $A$ compatible (i.e., commuting) with
$B$ and also with $C$, whereas $B$ and $C$ are incompatible, a measurement of $A$ might yield a different result (indicating that quantum mechanics is contextual) depending upon whether $A$ is measured along with $B$ (the ${A,B}$ context) or with $C$ (the ${A,C}$ context). An analysis of what projective quantum measurements measure shows that quantum theory is Bell noncontextual: the outcome of a particular $A$ measurement when $A$ is measured along with $B$ would have been exactly the same if $A$ had, instead, been measured along with $C$. A different definition, here called `globally (non)contextual refers to whether or not there is (noncontextual) or is not (contextual) a single joint probability distribution that simultaneously assigns probabilities in a consistent manner to the outcomes of measurements of a certain collection of observables, not all of which are compatible. A simple example shows that such a joint probability distribution can exist even in a situation where the measurement probabilities cannot refer to properties of a quantum system, and hence lack physical significance, even though mathematically well-defined. It is noted that the quantum sample space, a projective decomposition of the identity, required for interpreting measurements of incompatible properties in different runs of an experiment using different types of apparatus has a tensor product structure, a fact sometimes overlooked.
It is shown that when properly analyzed using principles consistent with the use of a Hilbert space to describe microscopic properties, quantum mechanics is a local theory: one system cannot influence another system with which it does not interact. C
laims to the contrary based on quantum violations of Bell inequalities are shown to be incorrect. A specific example traces a violation of the CHSH Bell inequality in the case of a spin-3/2 particle to the noncommutation of certain quantum operators in a situation where (non)locality is not an issue. A consistent histories analysis of what quantum measurements measure, in terms of quantum properties, is used to identify the basic problem with derivations of Bell inequalities: the use of classical concepts (hidden variables) rather than a probabilistic structure appropriate to the quantum domain. A difficulty with the original Einstein-Podolsky-Rosen (EPR) argument for the incompleteness of quantum mechanics is the use of a counterfactual argument which is not valid if one assumes that Hilbert-space quantum mechanics is complete; locality is not an issue. The quantum correlations that violate Bell inequalities can be understood using local quantum common causes. Wavefunction collapse and Schrodinger steering are calculational procedures, not physical processes. A general Principle of Einstein Locality rules out nonlocal influences between noninteracting quantum systems. Some suggestions are made for changes in terminology that could clarify discussions of quantum foundations and be less confusing to students.
While much of the technical analysis in the preceding Comment [1] is correct, in the end it confirms the conclusion reached in my previous work [2]: a consistent histories analysis provides no support for the claim of counterfactual quantum communication put forward in [3]
This paper answers Bells question: What does quantum information refer to? It is about quantum properties represented by subspaces of the quantum Hilbert space, or their projectors, to which standard (Kolmogorov) probabilities can be assigned by usin
g a projective decomposition of the identity (PDI or framework) as a quantum sample space. The single framework rule of consistent histories prevents paradoxes or contradictions. When only one framework is employed, classical (Shannon) information theory can be imported unchanged into the quantum domain. A particular case is the macroscopic world of classical physics whose quantum description needs only a single quasiclassical framework. Nontrivial issues unique to quantum information, those with no classical analog, arise when aspects of two or more incompatible frameworks are compared.
A solution to the second measurement problem, determining what prior microscopic properties can be inferred from measurement outcomes (pointer positions), is worked out for projective and generalized (POVM) measurements, using consistent histories. T
he result supports the idea that equipment properly designed and calibrated reveals the properties it was designed to measure. Applications include Einsteins hemisphere and Wheelers delayed choice paradoxes, and a method for analyzing weak measurements without recourse to weak values. Quantum measurements are noncontextual in the original sense employed by Bell and Mermin: if $[A,B]=[A,C]=0,, [B,C] eq 0$, the outcome of an $A$ measurement does not depend on whether it is measured with $B$ or with $C$. An application to Bohms model of the Einstein-Podolsky-Rosen situation suggests that a faulty understanding of quantum measurements is at the root of this paradox.
The correctness of the consistent histories analysis of weakly interacting probes, related to the path of a particle, is maintained against the criticisms in the Comment, and against the alternative approach described there, which receives no support from standard (textbook) quantum mechanics.
Possible paths of a photon passing through a nested Mach-Zehnder interferometer on its way to a detector are analyzed using the consistent histories formulation of quantum mechanics, and confirmed using a set of weak measurements (but not weak values
). The results disagree with an analysis by Vaidman [ Phys. Rev. A 87 (2013) 052104 ], and agree with a conclusion reached by Li et al. [ Phys. Rev. A 88 (2013) 046102 ]. However, the analysis casts serious doubt on the claim of Salih et al. (whose authorship includes Li et al.) [ Phys. Rev. Lett. 110 (2013) 170502 ] to have constructed a protocol for counterfactual communication: a channel which can transmit information even though it contains a negligible number of photons.
A recent experiment yielding results in agreement with quantum theory and violating Bell inequalities was interpreted [Nature 526 (29 Octobert 2015) p. 682 and p. 649] as ruling out any local realistic theory of nature. But quantum theory itself is b
oth local and realistic when properly interpreted using a quantum Hilbert space rather than the classical hidden variables used to derive Bell inequalities. There is no spooky action at a distance in the real world we live in if it is governed by the laws of quantum mechanics.
