The violation of a Leggett–Garg inequality confirms the incompatibility between quantum mechanics and the combined premises (called macro-realism) of macroscopic realism (MR) and noninvasive measurability (NIM). Arguments can be given that the incompatibility arises because MR fails for systems in a superposition of macroscopically distinct states—or else, that NIM fails. In this paper, we consider a strong negation of macro-realism, involving superpositions of coherent states, where the NIM premise is replaced by Bell’s locality premise. We follow recent work and propose the validity of a subset of Einstein–Podolsky–Rosen (EPR) and Leggett–Garg premises, referred to as weak macroscopic realism (wMR). In finding consistency with wMR, we identify that the Leggett–Garg inequalities are violated because of failure of both MR and NIM, but also that both are valid in a weaker (less restrictive) sense. Weak MR is distinguished from deterministic macroscopic realism (dMR) by recognizing that a measurement involves a reversible unitary interaction that establishes the measurement setting. Weak MR posits that a predetermined value for the outcome of a measurement can be attributed to the system after the interaction, when the measurement setting is experimentally specified. An extended definition of wMR considers the “element of reality” defined by EPR for system A, where one can predict with certainty the outcome of a measurement on A by performing a measurement on system B. Weak MR posits that this element of reality exists once the unitary interaction determining the measurement setting at B has occurred. We demonstrate compatibility of systems violating Leggett–Garg inequalities with wMR but point out that dMR has been shown to be falsifiable. Other tests of wMR are proposed, the predictions of wMR agreeing with quantum mechanics. Finally, we compare wMR with macro-realism models discussed elsewhere. An argument in favour of wMR is presented: wMR resolves a potential contradiction pointed out by Leggett and Garg between failure of macro-realism and assumptions intrinsic to quantum measurement theory.
The violation of a Leggett–Garg inequality confirms the incompatibility between quantum mechanics and the combined premises (called macro-realism) of macroscopic realism (MR) and noninvasive measurability (NIM). Arguments can be given that the incompatibility arises because MR fails for systems in a superposition of macroscopically distinct states—or else, that NIM fails. In this paper, we consider a strong negation of macro-realism, involving superpositions of coherent states, where the NIM premise is replaced by Bell’s locality premise. We follow recent work and propose the validity of a subset of Einstein–Podolsky–Rosen (EPR) and Leggett–Garg premises, referred to as weak macroscopic realism (wMR). In finding consistency with wMR, we identify that the Leggett–Garg inequalities are violated because of failure of both MR and NIM, but also that both are valid in a weaker (less restrictive) sense. Weak MR is distinguished from deterministic macroscopic realism (dMR) by recognizing that a measurement involves a reversible unitary interaction that establishes the measurement setting. Weak MR posits that a predetermined value for the outcome of a measurement can be attributed to the system after the interaction, when the measurement setting is experimentally specified. An extended definition of wMR considers the “element of reality” defined by EPR for system A, where one can predict with certainty the outcome of a measurement on A by performing a measurement on system B. Weak MR posits that this element of reality exists once the unitary interaction determining the measurement setting at B has occurred. We demonstrate compatibility of systems violating Leggett–Garg inequalities with wMR but point out that dMR has been shown to be falsifiable. Other tests of wMR are proposed, the predictions of wMR agreeing with quantum mechanics. Finally, we compare wMR with macro-realism models discussed elsewhere. An argument in favour of wMR is presented: wMR resolves a potential contradiction pointed out by Leggett and Garg between failure of macro-realism and assumptions intrinsic to quantum measurement theory.
The Einstein-Podolsky-Rosen (EPR) paradox gives an argument for the incompleteness of quantum mechanics based on a plausible criterion for reality, as well as the assumption of locality. A general view is that the argument is compromised, because EPR's premises are falsified by Greenberger-Horne-Zeilinger (GHZ) and Bell experiments. In this paper, we present an EPR argument based on premises not falsified by these experiments. First, we consider macroscopic EPR-Bohm, Bell, and GHZ experiments using spins ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}_{\ensuremath{\theta}}$ defined by two macroscopically distinct states. The analyzers that realize the unitary operations ${U}_{\ensuremath{\theta}}$ determining the measurement settings $\ensuremath{\theta}$ are nonlinear devices creating macroscopic superposition states. We note two definitions of macroscopic realism (MR). For a system with two macroscopically distinct states available to it, MR posits a predetermined outcome for the measurement ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}_{\ensuremath{\theta}}$ distinguishing between the states. Deterministic macroscopic realism assumes MR for the system defined prior to the interaction ${U}_{\ensuremath{\theta}}$ being carried out but is falsifiable by the macroscopic Bell and GHZ proposals. Motivated by arguments to uphold MR (without resorting to decoherence), as well as no-signaling and the properties of a meter, we define a set of premises referred to as weak macroscopic realism (wMR). Weak macroscopic realism (wMR) posits MR for the system after ${U}_{\ensuremath{\theta}}$, at the time tf---when the system is prepared with respect to the measurement basis, ready for a ``pointer'' measurement and readout. For this system, wMR posits that the outcome of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}_{\ensuremath{\theta}}$ is determined and not changed by interactions that might subsequently occur at a remote system B. The premise wMR also posits a weaker version of EPR's criterion for reality. Importantly, we show that the GHZ and Bell predictions are consistent with wMR. Yet, an EPR paradox arises for a macroscopic EPR-Bohm state, based on wMR. As considered by Schr\"odinger, it is possible to measure two complementary spins of system A simultaneously, ``one by direct, the other by indirect measurement'': If we assume wMR, then once both settings are fixed, the outcomes of the two spins are both determined. We revisit the original EPR paradox and find a similar result: An EPR argument can be based on a set of premises, weak local realism, which we show are not falsifiable by GHZ or Bell experiments.
The Einstein-Podolsky-Rosen (EPR) paradox gives an argument for the incompleteness of quantum mechanics based on the premises of local realism. A general view is that the argument is compromised, because EPR's premises are falsified by Greenberger-Horne-Zeilinger (GHZ) and Bell experiments. In this paper, we present an EPR argument based on premises not falsifiable by these experiments. We propose macroscopic EPR and GHZ experiments using spins Sθ defined by two macroscopically distinct states. The analyzers that realize the unitary operations Uθ determining the measurement settings θ are devices that create macroscopic superposition states. For a system with two macroscopically distinct states available, macroscopic realism (MR) posits a predetermined outcome for a measurement Sθ distinguishing between the states. Deterministic macroscopic realism (dMR) posits MR for the system prior to the interaction Uθ. Weak macroscopic realism (wMR) posits MR for the system after Uθ, at the time tf (when the system is prepared for a final "pointer" measurement), the outcome of Sθ not being changed by interactions that might occur at a remote system B. The premise also posits that if the outcome for SθA of a system A can be predicted by a pointer measurement on a system B defined after the interaction fixing the setting at B, then the outcome for SθA is determined at this time. The GHZ predictions negate dMR but are consistent with wMR. Yet, an EPR paradox arises based on wMR for the set-up proposed by Schrödinger, where one measures two complementary spins simultaneously, "one by direct, the other by indirect" measurement. We revisit the original EPR paradox and find similarly that an EPR argument can be based on a weak form of local realism not falsifiable by GHZ or Bell tests.
We consider tests of Leggett-Garg's macrorealism and of macroscopic local realism, where for spacelike separated measurements the assumption of macroscopic noninvasive measurability is justified by that of macroscopic Bell locality. We give a mapping between the Bell and Leggett-Garg experiments for microscopic qubits based on spin-1/2 eigenstates and gedanken experiments for macroscopic qubits based on two macroscopically distinct coherent states $|\ensuremath{\alpha}\ensuremath{\rangle}$ and $|\ensuremath{-}\ensuremath{\alpha}\ensuremath{\rangle}$ (as $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}$). In this mapping, the unitary rotation ${U}_{\ensuremath{\theta}}$ of the Stern-Gerlach analyzer or polarizing beam splitter is realized by a local interaction $H=\mathrm{\ensuremath{\Omega}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}^{4}$ where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}$ is the number of quanta. By adjusting the time of interaction, one alters the measurement setting $\ensuremath{\theta}$. We thus predict violations of Leggett-Garg and Bell inequalities in a macroscopic regime where coarse-grained measurements $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ need only discriminate between two macroscopically distinct coherent states. To interpret the violations, we distinguish between different definitions of macroscopic realism. Deterministic macroscopic local realism (dMR) assumes the system is in a state with a definite outcome ${\ensuremath{\lambda}}_{\ensuremath{\theta}}$ for the measurement $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ prior to the unitary rotation ${U}_{\ensuremath{\theta}}$, and is negated by the violations. Weak macroscopic realism (wMR) assumes a definite outcome for systems prepared in a superposition ${\ensuremath{\psi}}_{\text{pointer}}$ of two macroscopically distinct eigenstates of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$, after the unitary rotation ${U}_{\ensuremath{\theta}}$. We find that wMR can be viewed consistently with the violations and with the predictions of quantum mechanics for two tests of wMR. A model is presented in which wMR holds and the macroscopic violations emerge over the course of the unitary dynamics occurring at both sites. Implications for the realism of the micro-macro state generated during a quantum measurement are discussed. Finally, we point out an EPR-type paradox, which demonstrates inconsistency between wMR and the completeness of quantum mechanics at a microscopic level.
Macroscopic Realism (MR) says that a macroscopic system is always determinately in one or other of the macroscopically distinguishable states available to it. The Leggett-Garg (LG) inequality was derived to allow experimental test of whether or not this doctrine is true; it is also often thought of as a temporal version of a Bell-inequality. Despite recent interest in the inequality, controversy remains regarding what would be shown by its violation. Here we resolve this controversy, which arises due to an insufficiently general and model-independent approach to the question so far. We argue that LG's initial characterisation of MR does not pick out a particularly natural realist position, so we articulate an operationally well-defined and well-motivated position in its place. We show that much weaker conditions than LG's are sufficient to derive the inequality: in the first instance, its violation only demonstrates that certain measurements fail to be non-disturbing at the operational level. We articulate three distinct species of MR-ist position, and argue that it is only the first of these which can be refuted by LG inequality violation. This first position is an attractive one, so ruling it out remains of interest, however. A crucial role is played in LG's argument by the assumption of noninvasive measurability. We show that this notion is ambiguous between the weaker notion of disturbance at the operational level, and the stronger notion of invasiveness at the ontic level of properties of the system. Ontic noninvasiveness would be required to rule out MR per se but this property is not entailed by MR, and its presence cannot be established in a model-independent way. It follows that despite the formal parallels, Bell's and LG's inequalities are not methodologically on a par. We close with some reflections on the implications of our analysis for the pedagogy of quantum superposition.
The Einstein-Podolsky-Rosen (EPR) paradox was presented as an argument that quantum mechanics is an incomplete description of physical reality. However, the premises on which the argument is based are falsifiable by Bell experiments. In this paper, we examine the EPR paradox from the perspective of Schrodinger's reply to EPR. Schrodinger pointed out that the correlated states of the paradox enable the simultaneous measurement of x^ and p^, one by direct, the other by indirect measurement. Schrodinger's analysis takes on a timely importance because a recent experiment realizes these correlations for macroscopic atomic systems. Different to the original argument, Schrodinger's analysis applies to the experiment at the time when the measurement settings have been fixed. In this context, a subset of local realistic assumptions (not negated by Bell's theorem) implies that x and p are simultaneously precisely defined. Hence, an alternative EPR argument can be presented that quantum mechanics is incomplete, based on a set of (arguably) nonfalsifiable premises. As systems are amplified, macroscopic realism can be invoked, and the premises are referred to as weak macroscopic realism (wMR). In this paper, we propose a realization of Schrodinger's gedanken experiment where field quadrature phase amplitudes X^ and P^ replace position and momentum. Assuming wMR, we derive a criterion for the incompleteness of quantum mechanics, showing that the criterion is feasible for current experiments. Questions raised by Schrodinger are resolved. By performing simulations based on an objective-field (Q-based) model for quantum mechanics, we illustrate the emergence on amplification of simultaneous predetermined values for X^ and P^. The values can be regarded as weak elements of reality, along the lines of Bell's macroscopic beables.
The quantum three-box paradox considers a ball prepared in a superposition of being in any one of three boxes. Bob makes measurements by opening either box 1 or box 2. After performing some unitary operations (shuffling), Alice can infer with certainty that the ball was detected by Bob, regardless of which box he opened, if she detects the ball after opening box 3. The paradox is that the ball would have been found with certainty by Bob in either box if that box had been opened. Resolutions of the paradox include that Bob’s measurement cannot be made non-invasively or else that realism cannot be assumed at the quantum level. Here, we strengthen the case for the former argument by constructing macroscopic versions of the paradox. Macroscopic realism implies that the ball is in one of the boxes prior to Bob or Alice opening any boxes. We demonstrate the consistency of the paradox with macroscopic realism, if carefully defined (as weak macroscopic realism, wMR) to apply to the system at the times prior to Alice or Bob opening any boxes but after the unitary operations associated with preparation or shuffling. By solving for the dynamics of the unitary operations and comparing with mixed states, we demonstrate agreement between the predictions of wMR and quantum mechanics: the paradox only manifests if Alice’s shuffling combines both local operations (on box 3) and nonlocal operations, on the other boxes. Following previous work, the macroscopic paradox is shown to correspond to a violation of a Leggett–Garg inequality, which implies failure of non-invasive measurability if wMR holds.