Paths, negative "probabilities", and the Leggett-Garg inequalities

TL;DR

This paper analyzes how measurement choices influence future outcomes in quantum systems, showing that negative probabilities are more reliable indicators of signaling in time than Leggett-Garg inequalities.

Contribution

It introduces a path-based analysis of signaling in time and compares the effectiveness of negative probabilities versus Leggett-Garg inequalities as witnesses.

Findings

Signaling in time occurs when measurements destroy interference between paths.

Negative probabilities serve as a more reliable witness of signaling than Leggett-Garg inequalities.

Both methods can fail under certain conditions.

Abstract

We present a path analysis of the condition under which the outcomes of previous observation affect the results of the measurements yet to be made. It is shown that this effect, also known as "signalling in time", occurs whenever the earlier measurements are set to destroy interference between two or more virtual paths. We also demonstrate that Feynman's negative "probabilities" provide for a more reliable witness of "signalling in time", than the Leggett-Garg inequalities, while both methods are frequently subject to failure

Figures

b into the one shown in Fig. 2c, the number of real paths increases to four, and odds for arriving in the final states at $t=T$, are clearly not what they were before. On the horizontal and diagonal dark lines, the system is *”classical stochastic”*. It has only two non-interfering paths [CONS ](#bib.bib16), leading to different final destinations [see insets a) and c) in Fig. 3]. There is no interference to destroy and, as in classical statistics, we can monitor the system’s progress, without disturbing it, Finally, at the intersection of any two lines, we may call the system’s behaviour *”cl

Equations

$$\{1\}\equiv \{1,1,1\},$$

$$\{2\}\equiv \{1,-1,1\},$$

$$\{3\}\equiv \{-1,1,1\},$$

$$\{4\}\equiv \{-1,-1,1\},$$

$$A[1]\equiv ⟨+|{\hat{U}}_{T-\tau }|+⟩⟨+|{\hat{U}}_{\tau }|+⟩=\cos (T-\tau )\cos \tau ,$$

$$A[2]\equiv ⟨+|{\hat{U}}_{T-\tau }|-⟩⟨-|{\hat{U}}_{\tau }|+⟩=-\sin (T-\tau )\sin \tau ,$$

$$A[3]\equiv ⟨-|{\hat{U}}_{T-\tau }|+⟩⟨+|{\hat{U}}_{\tau }|+⟩=-i\sin (T-\tau )\cos \tau ,$$

$$A[4]\equiv ⟨-|{\hat{U}}_{T-\tau }|-⟩⟨-|{\hat{U}}_{\tau }|+⟩=-i\cos (T-\tau )\sin \tau ,$$

$$\hat{U}(t)=\cos t\hat{I}-i\sin t{\hat{\sigma }}_x,$$

$$P[I]=|A[1]+A[2]{|}^2={\cos }^2(T),$$

$$P[II]=|A[3]+A[4]{|}^2={\sin }^2(T).$$

$$P[1]={\cos }^2(T-\tau ){\cos }^2(\tau ),$$

$$P[2]={\sin }^2(T-\tau ){\sin }^2(\tau ),$$

$$P[3]={\sin }^2(T-\tau ){\cos }^2(\tau ),$$

$$P[4]={\cos }^2(T-\tau ){\sin }^2(\tau ).$$

$$P^{\prime }[I]=P[1]+P[3]={\cos }^2(\tau ),$$

$$P^{\prime }[II]=P[2]+P[4]={\sin }^2(\tau ).$$

$$Prob(Q_3=1)=P[1]+P[2]=co{s}^2(T)+\frac{1}{2}\sin (2\tau )\sin (2(T-\tau ))$$

$$Pro{b}_0(Q_3=1)+\delta P(Q_3=1).$$

$$p[1]=\int p_3(1|1,1,\lambda )p_2(1|1,\lambda )w(\lambda )d\lambda ,$$

$$p[2]=\int p_3(1|-1,1,\lambda )p_2(-1|1,\lambda )w(\lambda )d\lambda ,$$

$$p[3]=\int p_3(-1|1,1,\lambda )p_2(1|1,\lambda )w(\lambda )d\lambda ,$$

$$p[4]=\int p_3(-1|-1,1\lambda )p_2(-1,1|\lambda )w(\lambda )d\lambda .$$

$$p[i]=P[i],i=1,2,3,4.$$

$$Prob(Q_3=1)=P[I]=p[1]+p[2],$$

$$Prob(Q_3=-1)=P[II]=p[3]+p[4],$$

$$Prob(Q_2=1)=P^{\prime }[I]=p[1]+p[3],$$

$$Prob(Q_2=-1)=P^{\prime }[II]=p[2]+p[4],$$

$$⟨Q_1{Q}_2{⟩}_{cl}=p[1]-p[2]+p[3]-p[4],$$

$$⟨Q_1{Q}_3{⟩}_{cl}=p[1]+p[2]-p[3]-p[4],$$

$$⟨Q_2{Q}_3{⟩}_{cl}=p[1]-p[2]-p[3]+p[4].$$

$$⟨Q_1{Q}_2⟩\equiv P^{\prime }[I]-P^{\prime }[II]=\cos (2\tau )\equiv \alpha ,$$

$$⟨Q_1{Q}_3⟩\equiv P[I]-P[II]=\cos (2T)\equiv \beta ,$$

$$⟨Q_2{Q}_3⟩\equiv P[1]-P[2]-P[3]+P[4]=$$

$$\cos (2(T-\tau ))\equiv \gamma ,$$

$$p[1]+p[2]+p[3]+p[4]=1,$$

$$p[1]=(\alpha +\beta +\gamma +1)/4,$$

$$p[2]=(-\alpha +\beta -\gamma +1)/4,$$

$$p[3]=(\alpha -\beta -\gamma +1)/4,$$

$$p[4]=(-\alpha -\beta +\gamma +1)/4.$$

$$\delta p(\tau ,T)\equiv \sum _{i=1}^4|p[i]|-1,$$

$$⟨L⟩\equiv ⟨Q_1{Q}_2⟩+⟨Q_1{Q}_3⟩+⟨Q_2{Q}_3⟩\ge -1,$$

$$\delta L(\tau ,T)\equiv ⟨L⟩+1=\alpha +\beta +\gamma +1,$$

Full text

Recently, the authors of nat have shown that superconducting flux qubits possess, despite their macroscopic nature, such quantum properties, as the ability to exist in a superposition of distinct states. After reviewing an approach based on the so-called Leggett-Garg inequalities (LGI), which may or may not be satisfied by certain quantum mechanical averages LGI , they chose to employ a simpler experimental protocol. The method used in nat was similar to the one proposed by Koffler an Bruckner Koff , who suggested that the relevant evidence can be obtained more efficiently by analysing corresponding probability distributions, and coined a term ”signalling in time”.

Both the LGI, and the notion of ”signalling”, are closely related to a different problem, the so-called Bell test Bell , in which Alice an Bob are given two spins in the zero total spin state. Alice’s measurement along a chosen axis immediately aligns Bob’s spin in the opposite direction, and the study of inequalities, formally similar to the LGI, allowed Bell to show that the phenomenon cannot be explained by the existence of certain classical-like hidden variables. There is, however, no ”signalling” in the Bell’s experiment, in the sense that Bob is unable to recognise the choice of the axis made by Alice. Reduced density matrix of the Bob’s spin is not affected by Alice’s decision, and the no-cloning theorem noclo , constraints his ability to reconstruct the spin’s state.

Feynman’s approach to Bell’s problem has been more direct. In the Feyn he demonstrated that, in order to reproduce quantum results for an entangled Bell’s state with hidden variables, some the probabilities would inevitably turn negative. In a recent essay on the relationship between Feynman and Bell Whit , Whitaker notes that ”what Feynman describes is indeed Bell’s Theorem”. A similar, yet somewhat different approach to the ”signalling in time” problem was recently proposed in Hall , where negative values taken by quasi probabilities, defined in terms of quantum projection operators, were related with violations of the LGI.

Several authors Koff ,Hall ,Em1 , emphasise the difference between the Bell’s case, and the problem, to which the LGI is usually applied. Indeed, here one makes several consecutive measurements on the same quantum system, and asks whether the outcomes of previous observations can influence the results of the measurements yet to be made. Interaction with a measurement device at some $t_{1}$ can scatter the system, at $t_{2}>t_{1}$ into a state it would not have visited otherwise, or visit it with a different frequency. Should this happen, ”signalling in time” is said to have occurred Foot . No ”signalling” means that a measurement does not change the outcome statistics of later measurements Koff .

The literature on the Leggett-Garg inequalities is extensive, and we refer the reader to a recent review Em2 for relevant references, covering different aspects of the problem. The scope of this paper is much narrower. First, we analyse ”signalling in time” in terms of the virtual (Feynman) paths, and illustrate the analysis on the simple example of a qubit undergoing Rabi oscillations. Having done so, we compare the Feynman’s direct ”negative probability” test, and the violation of the LGI, as possible indicators of the ”signalling” phenomenon.

I Path analysis of ”signalling in time”.

Consider a sequence of accurate measurements of quantities {$\hat{Q}_{1}$,$\hat{Q}_{2}$, …,$\hat{Q}_{K}$ } which could, in principle made on a quantum system in a Hilbert space of a dimension $N$ at different times, {$t_{1}$,$t_{2}$, …,$t_{K}$ }. Let us call a path a sequence of possible measurement outcomes (numbers), {$Q_{1}$,$Q_{2}$, …,$Q_{K}$ }, where $Q_{i}$, $1\leq i\leq K$ is one of the eigenvalues of the operator $\hat{Q}_{i}$. (The simplest paths would connect just two outcomes, e.g., $Q_{2}\leftarrow Q_{1}$). For every path quantum mechanics provides a complex valued probability amplitude, $A(Q_{1},Q_{2},...,Q_{K})$. If all (or possibly some) of the measurements are actually made, quantum mechanics provides also the probabilities Feynl , e.g., $P(Q_{1},Q_{2},...,Q_{K})=|A(Q_{1},Q_{2},...,Q_{K}|^{2}$, if all of the measurements are realised. The probabilities are related to the frequencies, with which a given sequence will be observed, and the system will be seen to ”travel” the corresponding path. While paths endowed only with probability amplitudes are usually called virtual, it seems reasonable to describe the paths, to which both the amplitude and the probability, as real DS1 . We will call the set of all relevant real paths, together with the corresponding probabilities, a statistical ensemble. The probabilities for real paths can be obtained by adding the probability amplitudes of the virtual paths, and taking the absolute square, as appropriate Feynl . Importantly, choosing to make different measurements from the set {$\hat{Q}_{1}$,$\hat{Q}_{2}$, …,$\hat{Q}_{K}$ } may lead to essentially different statistical ensembles DS1 , DS2 . Now the problem of ”signalling in time” can be seen as follows. Two measurements at $t_{1}<t_{3}$ produce and ensemble with $N^{2}$ real paths. Adding a third measurement at $t_{1}<t_{2}<t_{3}$ yields another ensemble, with $N^{3}$ real paths. This ensemble is different, i.e. incompatible, with the first one, in the sense that ignoring the outcomes at $t_{2}$ (non-selective measurement), and adding the corresponding probabilities, does not recover the ensemble, obtained with the measurements made at $t_{1}$ and $t_{3}$ only. Incompatibility of different ensembles comes as a natural consequence of the fact that one inevitably perturbs an accurately measured quantum system.

II Virtual and real paths for a qubit

As an example, consider a two-level system, such as spin-$1/2$ (a qubit), and three consecutive measurements, made at $t=0$, $t=\tau$, and $t=T$, of the same quantity $\hat{Q}=|+{\rangle}{\langle}+|-|-{\rangle}{\langle}-|$, which can take the values of $\pm 1$. For simplicity, we will assume that at $t=0$ the system is in an eigenstate of $\hat{Q}$, so that the result of the first measurement is always $Q_{1}=+1$. Now there are four virtual paths shown in Fig.1,

[TABLE]

endowed with the probability amplitudes

[TABLE]

Here we have let the system performs Rabi oscillations of a unit frequency, $\omega=1$, between the states $|+{\rangle}$ and $|-{\rangle}$, so that its evolution operator in Eq.(2) can be written as

[TABLE]

with $\hat{I}$ and $\hat{\sigma}_{x}$ denoting the unity, and Pauli $x$-matrix, respectively.

We will consider three sets of measurements, (i) made at $t_{1}=0$ and $t_{3}=\tau$, thus yielding the values $Q_{1}$ and $Q_{2}$,

(ii) made at $t_{1}=0$ and $t_{3}=T$, yielding $Q_{1}$ and $Q_{3}$, and

(iii) made at $t_{1}=0$, $t_{2}={\tau}$ and $t_{3}T$, yielding $Q_{1}$, $Q_{2}$ and $Q_{3}$. The corresponding statistical ensembles are shown in Fig.2. In the case (ii) there are just two real paths, $\{I\}=\{1,1\}$ and $\{II\}=\{-1,1\}$, given by the superpositions of the paths of $\{1\}$ and $\{2\}$, and of $\{3\}$ and $\{4\}$, respectively (see Fig. 2b). The corresponding probabilities, therefore, are

[TABLE]

In the case (iii), all four paths in Eqs.(1) become real (see Fig. 2c), and are travelled with the probabilities

[TABLE]

Finally, since future measurements do not affect current results, the probabilities of the two real paths in the case (i), $\{I\}$ and $\{II\}$ in Fig. 2a, can be found by summing over the outcomes at $t=T$, shown in Fig. 2c,

[TABLE]

Our aim is to identify the conditions under which the ensembles in Figs. 2 are found to be essentially different. In other words, we want to know LGI -Koff , Hall -Em1 , Em2 when making the measurement of $Q_{2}$ at $t={\tau}$ affects the distribution of the values $Q_{3}$ at $t=T$. It is sufficient to look at the probability to have $Q_{3}=1$ in the case c) shown in Fig.2,

[TABLE]

Now $Prob_{0}(Q_{3}=1)=P[I]$ is just the probability with no measurement made at $\tau$, and $\delta P(Q_{3}=1)$ is the change brought about by the disturbance produced at $t=\tau$. We note that $\delta P(Q_{3}=1)$ vanishes when ${\tau}$ or $T-{\tau}$ equals $k\pi/2$, $k=0,1,2...$ FOOT2 , and plot its as a function of $\tau$ and $T$, $\tau\leq T$ in Fig.3. In the light-coloured regions, behaviour of the ensemble can be described as ”quantum stochastic”. A measurement, added at $t=\tau$ changes the ensemble in Fig. 2b into the one shown in Fig. 2c, the number of real paths increases to four, and odds for arriving in the final states at $t=T$, are clearly not what they were before. On the horizontal and diagonal dark lines, the system is ”classical stochastic”. It has only two non-interfering paths CONS , leading to different final destinations [see insets a) and c) in Fig. 3]. There is no interference to destroy and, as in classical statistics, we can monitor the system’s progress, without disturbing it, Finally, at the intersection of any two lines, we may call the system’s behaviour ”classical deterministic”. There a single path [see inset b) in Fig.3], which leads to a unique final state, and is travelled every time the experiment is repeated, regardless of whether the measurement at $t=\tau$ is made, or not. Note that this classification refers to the present choice of measurements, and choosing a different measured operator, or a different initial state, would result in a picture, different from the one shown in Fig. 3.

In general, we note that there can be no pre-determined values (or average values) of $Q_{3}$, independent of what being done at ${\tau}$. Rather, we must conclude that different sets of measurements may ”fabricate” completely different statistical ensembles from the same quantum system DS2 .

III No ”pre-existing” path probabilities

Next we expand on the last statement of the previous Section. Let us assume (incorrectly) that there are probabilities to have classical-like pre-determined values of $Q_{i}=\pm 1$ at $t_{i}$, possibly depending on some unknown random classical parameter $\lambda$, [as in Bell we will allow multiples $\lambda=(\lambda_{1},\lambda_{2}...\lambda_{M}$), in which case $d\lambda$ will mean $d\lambda_{1}d\lambda_{2}...d\lambda_{M}$]. With $\lambda$ distributed according to some $w(\lambda)\geq 0$, $\int w(\lambda)d\lambda=1$, we can evaluate the probabilities for the sequences of outcomes $Q^{i_{3}}\leftarrow Q^{i_{2}}\leftarrow Q^{1}=1$,

[TABLE]

In Eq.(8) $p_{2}(Q_{2}|Q_{1}=1,\lambda)$ stands for the probability to have an outcome $Q_{2}$, given a previous outcome $Q_{1}$, and $p_{3}(Q_{3}|Q_{2},Q_{1}=1,\lambda)$ yields the odds for having a value $Q-3$, given the previous values of $Q_{2}$ and $Q_{1}$. (Recall that $Q_{1}$ is always $1$, since the system is prepared in $|+{\rangle}$.) We expect to have no access to the actual value(s) of the ”hidden variable(s)” $\lambda$. We assume, however, that each time the system is set to evolve from its initial state particular path probabilities $p_{3}(Q_{3}|Q_{2},Q_{1},\lambda)p_{2}(Q_{2}|Q_{1},\lambda)$ exist, even if no measurements are made.

For our assumption to be correct, we need to demonstrate that the classical probabilities (small $p$’s in Eqs.(8) are the same as the correct quantum results (capital $P$’s) of the previous Section. Firstly, we must have

[TABLE]

Secondly, summing the $p[i]$’s over the outcomes at $t=\tau$ we should obtain the probabilities $P(I)$ and $P(II)$ in Eqs.(4), i.e.,

[TABLE]

and, similarly,

[TABLE]

However, Eq.(7) states that, in general, $P[1]+P[2]\neq P(I)$, so that Eqs.(9) and (10) cannot always hold.

This, in turn, demonstrates, that the path probabilities cannot ”pre-exist” a set of consecutive measurements, just as the result of an individual measurement cannot pre-exist the measurement Merm . Different measurements may produce statistical ensembles with distributions as different as the distributions of heads and tails for differently skewed coins. This will happen whenever an additional earlier measurement destroys interference between virtual paths leading to later outcomes.

IV The negative probability test

We could look for other proofs of the same point, e.g., by following Feynman’s example, described in Feyn . We will not rely on a particular type of quasi-probabilities, as was done, for example, in Hall , but rather assume that the classical-like path probabilities $p[i]$, similar to those in (8), can somehow be defined. We will then look for the values they must take in order to reproduce the correct quantum mechanical results. With the help of Eqs.(8) it is easy to express the average values of the products, ${\langle}Q_{i}Q_{j}{\rangle}_{cl}$, in terms of the $p[i]$’s,

[TABLE]

Using the path probabilities in Eqs. (5)-(7) yields the correct quantum value for the same quantities

[TABLE]

If our assumption is correct, results (12) and (13) will agree. Equating ${\langle}Q_{i}Q_{j}{\rangle}_{cl}={\langle}Q_{i}Q_{j}{\rangle}$, and adding a condition

[TABLE]

yields four linear equations the probabilities $p[i]$ must satisfy. Their solutions are

[TABLE]

Our assumption will be proven wrong, if at least one of the $p[i]$’s turned out to be negative. Thus, we evaluate

[TABLE]

which is zero if, and only if, all $p[i]$’s are non-negative, and map it on the $({\tau},T)$ plane in Fig. 4. As in Fig.3, $\delta p({\tau},T)\neq 0$ in the light-coloured regions, and vanishes on the horizontal and diagonal lines where, as we already know from Sect.III, the classical-like probabilities can be defined. We note that this negative probability test is also passed also on the vertical lines $T=k\pi/2$, $k=1,2,...$. An inspection of Eqs.(15) shows that for any $\tau$, and $T=(2k+1/2)\pi$, or $T=k\pi$ there exists a suitable classical ensemble. Such ensembles, with only two paths leading to the same destination, are shown in the insets if Fig. 4. Just because such ensembles can be found in principle, does not, of course, mean that they correspond to what actually happens. To warn the reader about the misrepresentation, we crossed the insets in Fig. 4 with red lines.

Clearly, the appearance of negative ”probabilities” is a sufficient, yet not necessary condition for the classical-like reasoning, based on Eqs.(12), to fail. Such is the price of relying on average values, instead of the probability distributions, which contain full information about a statistical ensemble. Relying on the properties of sums of averages, rather than on the averages themselves, would be an even less precise tool, as we will discuss next.

V The Leggett-Garg inequalities

Alternatively, we might note that the existence of the classical-like non-negative path probabilities (8) imposes certain restrictions on the sums of the averages (12) Following LGI one notes that in all of the four possible sequences, $Q_{3}\leftarrow Q_{2}\leftarrow Q_{1}$ the sum of products $L=Q_{1}Q_{2}+Q_{1}Q_{3}+Q_{2}Q_{3}$ equals $3$, if all the $Q$’s have the same sign, and takes the value of $-1$ otherwise. It is readily seen that if the sequences occur with the probabilities in Eq.(8), also the sum of the averages (12) cannot be smaller that $-1$,

[TABLE]

since a chance to add the $1\leftarrow 1\leftarrow 1$, sequence would only increase the value of $L$. The LGI test consist in inserting the correct quantum values (13) into (18) and looking for the values of ${\tau}$ and $T$, such that the inequality does not hold. Thus, we will look for those values of ${\tau}$ and $T$, for which the sum

[TABLE]

where $\alpha$, $\beta$ and $\gamma$ are defined in Eq.(13), is negative. A condition $\delta L<0$ should, therefore, signal the impossibility of assigning meaningful path probabilities $p[i]$ in (12), in the same way as the appearance of ”negative probabilities”, discussed in the previous Section. This is, however, a less direct approach, and we ask whether it is as efficient as the tests of the previous two Sections.

We already know that the LGI would be satisfied on the network of lines in Fig.4, since a suitable classical ensemble does exist on its the horizontal and diagonal lines, while on the vertical lines it can be found at laest in principle. Indeed, these lines divide the ($\tau$, $T$)-plane into the segments inside which the LGI is either violated (light colour), or satisfied (black), as shown in Fig.5. It is readily seen that the LGI leaves much of the (${\tau},T$)-plane black, being a much less sensitive indicator of ”signalling in time”, than the negative probability test of Sect.V. Such is the price of relying on the sums rules, satisfied by the averages, rather than on the averages themselves.

VI Conclusions and discussion

In summary, a sequence of quantum measurements, made on an elementary quantum system, can be described in terms of real observable paths, constructed from the virtual ones. The outcomes of previous observations can influence the results of the measurements yet to be made, provided the earlier measurements create new real scenarious, by destroying interference, otherwise existent between the virtual paths.

A simple illustration of such a description may be provided by a qubit, completing its Rabi cycle by $t=T=\pi$. In this case, two interfering paths, $\{3\}$ and $\{4\}$ in Fig.1, have amplitudes of the same magnitude, but of opposite sign, $\pm i\sin{\tau}\cos{\tau}$. Destructive interference prevents the system from reaching the state $|-{\rangle}$. An additional measurement at $t={\tau}$ makes both virtual paths real as shown if Fig. 2c, and, at $t=T$, the qubit is found in $|-{\rangle}$ with a probability $\sin^{2}({\tau})/2$. An earlier measurement at $t=\tau$ clearly affect the outcomes at $T$, or, if one prefers the language of Koff , ”signalling in time” occurs.

We also considered two other approaches, based on evaluation of two-times averages of the qubit’s variable. One approach assumes that the real scenarios (paths) exist at all times, and are not created by the measuring device(s). It fails, since meaningful path probabilities cannot, in general, be found where destruction of interference between virtual paths is known to take place. One exception are the vertical lines in Fig.4, where this ”negative probability test” errs by finding a spurious ensemble, consistent with the quantum mechanical averages (13), but misrepresenting the actual situation.

The second method, based on the Leggett-Garg inequalities, tests a sum rule, which the averages should satisfy in the absence of ”signalling”. As suggested in Koff , the LGI provide a sufficient, but not necessary condition, and detects the quantum behaviour in far fewer cases than the negative probability test, as shown in Fig. 5. Perhaps, one reason for the popularity of the approach is the LGI’s formal similarity to the celebrated Bell’s inequality Bell . We find, however, little advantage in using the analogy, and advocate much simpler elementary methods, serving the same purpose.

VII Acknowledgements

Financial support of MINECO and the European Regional Development Fund FEDER, through the grant FIS2015-67161-P (MINECO/FEDER,UE) and the Basque Government Grant No IT986-16 is acknowledged by DS.