Measure of macrocoherence

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Examensarbete 30 hp Februari 2021

Measure of macrocoherence

Patrik Bernhardsson Handledare: Erik Sjöqvist

Ämnesgranskare: Johan Larsson

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1 Sammanfattning

Inom många områden kan det vara av nytta att kunna kvantiera en resurs eller fysisk företeelse för att kunna jämföra eller mäta tillgången av något fenomen. Det kan handla om det mest vardagliga som vikt till det mer ab- strakta som till exempel makrokoherens; hur kvantfysiskt ett system kan fortsätta vara när systemet skalas upp till det man i vardaglig kontext kan kalla för makronivå. Det är vanligt att se kvantfysik som någonting atomer styrs av, men det bör i teorin inte vara så. På grund av detta kan det vara an- vändbart att kunna visa att ett system följer kvantfysikens lagar i jämförelse med det som kallas för klassisk eller realistisk fysik. Legget-Garg-olikheter är just olikheter som har blivit härledda från en realistisk, icke-kvantfysisk, bild och kan därför inte bryta mot en Leggett-Garg-olikhet om systemet verkar klassiskt. I denna artikel beskrivs hur en sådan olikhet kan användas för att kvantisera hur makrokoherent ett system är, samtidigt dras paralleler till den kvantfysiska egenskapen sammanätning som redan har etablerade mått och kan användas som en resurs i kommunikation. Slutledningsvis påpekas vikten av att ha en klar resurs anslutet till makrokoherens som riktmärke för att ta fram ett egentligt och rätt mått för makrokoherens.

2 Abstract

Macrocoherence is the concept of quantum mechanics being scaled up to the macroscopic level where everyday physical systems should inhibit quantum mechanical properties, however this is not what is observed. Through the use of Leggett-Garg inequalities, one can infer if there is a fundamental quantum mechanical behavior of the system being observed. Then, using violations of these inequalities, this paper discusses the possibility of extracting useful measures of how macroscopic a system can be. Utilizing an analogy with the measures of locality through Bell's inequalities, the scope of what a measure should consist of is discussed. A measure should be proper in the sense that a baseline of 0 should be obtained from system that never violates an LGI.

Further, it is proposed that a measure should extend naturally to all orders of LGIs without ranking quantum systems dierently. With these in mind two measures are proposed, one utilizing the integral over the violated area of a LGI over time whereas the other uses inner products over a matrix dened elementwise as a specied LGI. The measures scopes are discussed

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and their applications are showcased for some analytical quantum systems.

Though functional, the measures are found to lack a resource tied to its value complicating the conceptualization of what is being measured. It is concluded that a new eort to nd a true measure of macrocoherence should start from the concept of a resource.

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3 Introduction

When discussing quantum mechanics usually microscopic systems are considered, but the laws of quantum mechanics should be able to be extrapolated to macroscopic systems. However, such objects are not observed in everyday settings and lead to the idea of Schrödinger's paradox where a cat is in a superposition of dead and alive [1]. That a system on a macroscale should exhibit counter-intuitive quantum behaviour but seemingly does not is the idea of macrocoherence and has been the discussion of many of the greatest physicists [2]. The question then is, what is quantum mechanics at a macroscopic scale and how is it detected? Leggett and Garg [3] suggested a way to dierentiate between a system that follows classical mechanics rules from that of the quantum mechanical by considering correlation functions. For that they dened the picture of macroscopic realism as a system that does not take on multiple realizations at once, such as Schrödinger's cat, and that also can be freely measured without interfering with the state or its evolution. Both of these assumptions are immediately disregarded in the quantum picture where superposition [4] violates the rst assumption and the second by the collapse of the wave function [5].

The idea of an inequality to describe otherwise impossible dynamics is not a novelty, to disprove the EPR paradox [6], Bell came up with his famous theorem [7]which later led to the inclusion of non-locality in the local-hidden variable theory by de Broglie-Bohm [8]. In analogy to Leggett-Garg and non- classicality, the Clauser-Horne-Shimony-Holt inequality [9] derived from this theorem can be used to detect quantum entanglement for a given density state [10]. In addition, the CHSH inequality is useful for measuring the amount of entanglement present in a system using the Horodecki measure [11], and together with other measures such as concurrency [12] and negativity [13]

can be used to properly quantify how entangled a system is. These measures are usually derived on properties of entanglement such as that of positivity under a positive partial transpose [11] or that of entanglement of formation [14] which is based on the information carry capacity of the system, or its entropy.

This suggests that there should be an idea of a measure of Leggett-Garg inequalities as a measure of quantum entanglement in time as said inequality have even been considered a temporal Bell's inequality[15]. Instead of measuring non-locality, decoherence or non-classicality is measured from the quantum picture to the macrorealistic picture originally described by

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Leggett-Garg.

4 Background

The assumptions of macrorealism are dened as follows [3]:

(A1) Macroscopic Realism (MR): a macroscopic system with two or more macroscopically distinct states available will at all times be in one or the other of these states.

(A2) Non-Invasive Measurability (NIM) at the macroscopic level:

it is possible, in principle, to determine the state of the system with arbitrarily small perturbation on its subsequent dynamics.

To complete the picture of realism, Leggett in a later paper [16] added an assumption of induction:

(A3) Induction: the outcome of a measurement on the system cannot be aected by what will or will not be measured on it later.

In addition, one can dene a fourth assumption to relate macrorealism to the idea of locality as described by Einstein's theory of relativity[17, 16]:

(B1) Local Realism: an event (in the sense of special relativity) cannot be causally aected by any past events which lie outside its past light cone.

A1 and A2 are what combined would describe macrorealism in the broader sense, or simply classicity. A3 not being noted down in the original work might be explained by how intrinsic it is to the understanding of any event in the physical world. However, even though it is natural to consider causality (outside of special relativity) and the direction of time as being obvious, it is clear that the LGI requires this assumption (as do much of physics) to be derived. Leggett himself argues that one should take caution when assuming A3 and that the next breakthrough in physics might come from a reconsideration of this seemingly very natural fact [16]. B1, on the other hand, is not required for an LGI, but its inclusion is to have a direct analogy to the derivation of the CHSH inequality and describes the idea of locality which also has been used to describe realism [18].

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4.1 The Leggett-Garg inequality

From A1-A3 one can now derive the following inequality. First dene a correlation function Cij by considering a dichotomous variable Qi = ±1 where the subscript denotes time of measurement, ti thus Qi = Q(t_i). Also consider the joint probability Pij(Q_i, Q_j) which from previous denitions simply denotes the probability of obtaining Qi, Q_j from measurements at two dierent times. The Cij then reads as:

C_ij = X

Qi,Qj=±1

Q_iQ_jP_ij(Q_i, Q_j), (1) Intuitively, equation 1 describes the correlation of measurements at dif- ferent times and essentially adds up the averages. In a repeated experiment setting, one can dene Cij instead as the average of the measurements:

C_ij = 1 N

N

X

r=1

Q^(m)_i Q^(m)_j , (2)

where the superscript now denotes the realization of experiment m. What can be said about equation 1-2? Clearly, −1 ≤ Cij ≤ 1 and the value describes the correlation between measurements at ti, tj. For example, if the same value of Q is observed at both times in a realization of the experiment, whether it be +1 or -1, the value of Cij is 1; implying full correlation. On the other hand, -1 implies complete anticorrelation. Cij rst becomes interesting, in this setting, when one considers a third measurement so that one can investigate the switch of correlation. Consider three dierent times of measurement in order from rst to last; t1, t2, t3, then what conclusions can be drawn if C₁₃ = ±1? It is possible that Q(ti) for all i correlate so that Cij = 1 for all combinations (12, 23, 13). If C12 or C23 anti-correlate, on the other hand, one can gather information about the switch of the measurement variable Q. It should be clear now that by considering multiple correlation functions, more information can be gathered about the evolution of the system and this is the assumption that forms the backbone of an LGI.

Before the formal proof, the shortest of such a combination is:

K₃ = C₂₁+ C₃₂− C₃₁, (3)

where the subscript (on K) is a reminder for the number of correlation functions and the minus sign serves to cause an anti-symmetry between correla-

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possible correlations and anti-correlations (in the classical/intuitive picture of realism) that the value of K3 has to lie in the interval [-3, 1] so that the inequality becomes:

−3 ≤ K₃ ≤ 1. (4)

Having discussed the idea behind LGI, it is appropriate to show the formal proof. Again, using the correlation function dened in equation 1 and expand the probability density in a way that ts with the macrorealistic picture:

Pij(Qi, Qj) = X

Qk;k6=i,j

Pij(Q1, Q2, Q3). (5) Here the probability density P now includes a sum over a third measurement.

According to A1 measuring or not measuring Qk; k 6= i, j does not change the outcome of Qi for any i. In addition, A2 makes it possible to drop the subscript as it is possible to perform the measurements in such a way that future measurements will not be aected (A3 works in the other direction;

past experiments cannot have been aected). What would otherwise be independent probability densities over the permutations of Cij; instead becomes dependent, Pij(Q₃, Q₂, Q₁) = P (Q₃, Q₂, Q₁)and some important correlation functions can be written in full:

C₂₁= P (+, +, +) − P (+, +, −) − P (−, −, +) + P (−, −, −) (6)

−P (+, −, +) + P (+, −, −) + P (−, +, +) − P (−, +, −),

C32= P (+, +, +) + P (+, +, −) + P (−, −, +) + P (−, −, −) (7)

−P (+, −, +) − P (+, −, −) − P (−, +, +) − P (−, +, −),

C₃₁= P (+, +, +) − P (+, +, −) − P (−, −, +) + P (−, −, −) (8) +P (+, −, +) − P (+, −, −) − P (−, +, +) + P (−, +, −),

where + and - are standing in for +1 and -1 respectively. The combination of correlation functions is the same as in equation 3:

K₃ = C₂₁+ C₃₂− C₃₁. (3)

Using the fact that all permutations of the measurement outcomes sum to 1;

3 simplies to:

K₃ = 1 − 4[P (+, −.+) + P (−, +, −)]. (9)

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The range of K3 is obtained by letting P (+, −, +) + P (−, +, −) = 1 and P (+, −, +) = P (−, +, −) = 0 respectively. Resulting in the bounds:

−3 ≤ K3 ≤ 1, (4)

which was to be proven. It is also possible to prove 4 using an ontic model[19, 20, 21] which is comparable to local-hidden-variable theory. Ontic state describes a state, a realistic model which denes a state as a system from which all physical properties can be derived from; unlike the quantum mechanical model. Thus an LGI cannot be used as a violation of local realism, as is the case of Bell's Theorem [7] (although some may still argue that point [22]). The proof still considers a correlation function Cij now using an ontic state ξ. The assumption is that the system is prepared over a probability distribution µ(ξ) and the measurement outcome is represented by φ(Qi|ξ), which is the probabilty that given the ontic state ξ; Qi is measured, where again the subscript implies measurement at time ti. Further, a probability of disturbance is introduced, γi(ξ⁰|Q_i, ξ), which reads as the probability of ξ → ξ⁰ given outcome of measurement Qi. Then P in equation 1 becomes [23, 24, 25]:

P (Qi, Qj) = Z

dξ⁰dξφj(Qj | ξ⁰) γi(ξ⁰ | Qi, ξ) ξi(Qi | ξ) µ(ξ). (11) Since under the assumption of non-invasive measurability the state should be left unperturbed, the probability of disturbance becomes the delta function γi(ξ⁰|Qi, ξ) = δ (ξ⁰− ξ) and the correlation function now looks like:

C_ij = Z

dξ X

Qi,Qj=±1

Q_iQ_jφ_j(Q_j | ξ) φ_i(Q_i | ξ) µ(ξ)

= Z

dξµ(ξ) hQ_ii_ξhQ_ji_ξ,

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where the second inequality is achieved simply by the sum dening the expectation value of the system and according to A1, the probability functions φkare independent. Again, it does not matter if the measurement is made or what the outcome is; there is still a denitive value inherent to the system.

Now, the sum of correlation functions forming K3 is simply:

K =

Z

dξµ(ξ)h

hQ i hQ i + hQ i hQ i − hQ i hQ i i

, (13)

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which is bounded as −3 ≤ K3 ≤ 1since for a dichotomous observable −1 ≤ hQ_ii_ξ ≤ 1.

4.2 Higher order LGIs

As could be deduced from the previous summing of correlation functions in 3, it is of course possible to increase the number of correlations to achieve higher order inequalities. The derivation does not dier much from the equations of 4-9 [26], but dier slightly for odd and even n; Kn:

K_n= C₂₁+ C₃₂+ ... + C_n(n−1)− C_n1, (14)

−n ≤ K_n≤ n − 2, n odd,

−(n − 2) ≤ K_n ≤ n − 2, n even.

For odd n, only the upper bound is of relevance. Since the expectation value is bounded by ±1, it is impossible for the terms to equal less than

−n. However, for even n both bounds are of relevance. A question arises then, for the family of increasing Kn, if a higher order inequality is violated, are then the above inequalities automatically violated for suitable choices of time parameters? It turns out that for n ≥ 4 the inequalities can be derived considering triangle inequalities and symmetry transformations (such as for some ti, Qi → −Qi) [27] and in that sense are reducible. However, there are higher order LGIs which are irreducible that are composed dierently, such as the pentagon inequality:

X

i≤j≤5

C_ji+ 2 ≥ 0. (15)

Even the reducibility of higher order LGIs do not make them completely useless. For example on the topic of measures, it is desirable that any measure on LGI violations should extend naturally to any order of LGI violations.

Further, if such a measure keeps its domain order invariant (such that for a measure µ on Knand Km and input a, b; if for Kn, µ(a) < µ(b)then the same inequality holds for Km for any n, m). Also, repeated measurements (and thus higher order LGIs) can be used to remove the clumsiness loophole

(2.8 see Violation of Leggett Garg inequalities) by removing the error an invasive measurement would have and thus strengthen the argument that macrorealism is broken rather than the clumsiness of the observer [28].

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4.3 CHSH inequality

Whereas the Leggett-Garg measurements concern the same system with ob- servations separated in time, the CHSH inequality [9] concerns measurements on two systems (usually denoted Alice's (A) and Bob's (B)) separated in space. Here, separation in space is simply according to locality dened in B1; Alice and Bob should be outside of each other's light cones. First, consider the arithmetic inequality:

|ab − ab⁰+ a⁰b + a⁰b⁰|

= |a (b − b⁰) + a⁰(b + b⁰)| ≤ 2 (16) for a, a⁰, b, b⁰ = ±1, which can be easily veried through substitution. This inequality is of course also valid for the interval [−1, 1] for each variable. The next step is to dene expectation values for four observables; A, A⁰ chosen by Alice and B, B⁰ chosen by Bob; all which admit a value ±1. Similar to how equation 3 relies on A1 to dene a joint probability as the margin of a distribution over all variables, the CHSH inequality relies on a joint probability distribution over the results of A, A⁰, B, B⁰. Such a probability distribution can be shown to be equivalent to there being a deterministic local-hidden variable theory [29] and, as will be shown, that the CHSH inequality holds.

Since if there exists a well-dened P (a, a⁰, b, b⁰)then from the inequality in 16 one can derive expectation values for the observables as the marginal probability of P, i.e. P (a, b) = P_a⁰_,b⁰P (a, a⁰, b, b⁰). According to B1, the value of A cannot be aected by the observed value of B and such a marginal probability should exist:

B = hABi_ab+ hAB⁰i_ab0 + hA⁰Bi_ab− hA⁰B⁰i_ab0 ≤ 2. (17) Even though orginally 17 was derived without the use of assumption A1 and A3, they are implicitly made and have been included in later works [16]. For example, A1 allows for the values of Alice's and Bob's system to simultaneously exist throughout the course of the experiment regardless of order of measurement and, A3 as natural as it is, is implicitly assumed in B1, however, if an experimental violation of 17 occurs over an ensemble of experiments, A3 says that the choice of future experiments do not alter the previous ones. The paradox of violating B1 was rst proposed by Einstein- Podolsky-Rosen [6] (commonly referred to as the EPR-paradox) to claim that the quantum mechanical description was incomplete. According to the

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Heisenberg uncertainty [30], momentum and position of a quantum particle cannot be known to arbitrary precision. However, if two particles A and B interact with each other then position of A and momentum of B can be measured and through their interaction the momentum and position of the other particle should be known. This caveat is what Einstein referred to as spooky-action-at-a-distance [31], as quantum mechanics predict the entanglement of these particles and an instantaneous collapse of the wave function of the distant particle in the event of measurement of an individual particle. Similar to the no-cloning theorem [32, 33], where if one were allowed to clone a system then measurements about the original particle could be deduced from the cloned particle without wave collapse. This cannot be allowed, so there is no such thing as perfect cloning.

4.4 No-go theorems

The concept of action at a distance seemingly leads to the violation of special relativity, that no information travels quicker than the speed of light [34], a violation of the CHSH inequality should in theory mean having to accept instantaneous signaling. However, in the setting as described above between Alice and Bob, even with entangled particles there is no way for Alice to send information towards Bob with measuring her particle alone [35, 36, 37].

Bob cannot distinguish between the collapse of his particle's wave function and the randomness inherent in his measurement. Information can only be gained through Alice communicating via a classical channel to give Bob information about her results. It would then appear as entanglement do not have applications in communication, but it does not end there. Since a measurement uses up the entanglement procedure so that Bell's inequality is no longer violated, a quantum channel can be used to assure Alice and Bob that there were no eavesdroppers in their communication. Such an application is possible through the E91 protocol [38]. In addition, information can be compressed more optimal through quantum teleportation [39, 40, 41] using the fact that Alice can alter Bob's system from a distance. The teleportation scheme is limited by the randomness of Alice's measurement and thus still require Alice to communicate with Bob classically what the measurement outcome was.

Analogously on the topic of LGI, a no-signalling in time theorem has been proposed [42, 24, 43, 44], where again the probability for a measurement at time tk(Q_k) is determined as the margin of a joint probability for two

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measurements (Qi, Q_j):

Pk(Qk) =X

Qi

Pik(Qi, Qk), (18)

which can be interpreted as a statistical version of A2. If no-signaling in time (NSIT) is violated then macrorealism would break down just between two points (LGIs use at least three measurements, smallest one being K3). It is thus considered a stronger proof of macrorealism (and non-invasive measurability) and reduces the discussion of violations down to the existence of such a joint probability, similar to Bell's inequality. It is straightforward to show that quantum mechanics as formulated does not allow for such a probability to exist [45]. The probability of achieving Qk is in the Heisenberg picture:

P (Q_k) = T r(Q(t_k)ρ), (19)

where Q(tk) is the appropriate projection observable and ρ is the density matrix. The joint probability takes the form:

P_ik(Q_k, Q_i) = T r(Q(t_i)Q(t_k)ρQ(t_k)). (20) Summing over Qi results in:

X

Qi

P_ik(Q_i, Q_k) = T r(Q(t_k)(X

Qi

Q(t_i)ρQ(t_i))), (21) which clearly does not simplify to 19 in the general case. Even though violations of this equality seem enough to question macrorealism, it runs into the problem of being ambiguous in what a violation of NSIT would imply;

a failure of macrorealism or non-invasive measurement which can always be argued by a persistent macrorealist. Further, NSIT can result in more measurements having to take place since all possible states in the Hilbert space belonging to the system will have to be measured. NSIT is still of interest since the possible violation can be larger than that of LGI, signifying maybe a stronger macrocoherence than that of an LGI [44].

4.5 Measurement tactic and the clumsiness loophole

As is well known, there is no such thing as a denitive quantum measurement which does not alter the system [46, 47]. However, since the LGI was rst

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proposed as a way of disproving the realistic picture for an insistent macrorealist, this is in general not a problem. A quantum physicist in dispute with the macrorealist does not have to agree that NIM is an impossible assumption to disprove realism. The problem therein lies not with the apparent impossi- bility of performing a non-invasive measurement, but in how a macrorealist might always refer to the observer as a clumsy one, where any violation of the inequality is due to faulty apparatus or incompetence of the experimenter [28]. There are two main methods of avoiding this, on the surface, irrefutable argument, the rst being ideal negative measurements proposed rst in the original paper [3] and the second being weak measurements [48] which aim to disturb the system as little as possible in the quantum mechanical sense with the concession being less information gained.

The biggest obstacle in proving that a measurement is non-invasive is that even if one can show that the measurement at time t and time t + δt always correspond (so that one can argue the system was never disturbed), it would be impossible to show that a hidden-variable was not disturbed [49].

This problem is not seen as readily in Bell's inequality since the seperation is spatial and not temporal. Even though a realist can claim a similar loophole, the communication loophole [50], stating that the measurement of Alice's particle inuences Bob's. However, this could indeed be seen as the point of the experiment, the loophole instead insists that the type of communication would not be instantaneous but align with that of special relativity. By performing the measurements outside of each other's light cones such a loophole can be readily closed. Several such Bell experiments have been proposed [51, 52, 53, 54]. Yet, no such experiment have been shown to exist for the LGI-case. The temporal disturbance is harder to close seeing as it does not take on limitations from any physical law, but it can be narrowed to seem implausible if not impossible. The idea behind ideal negative measurements in the classical sense is that if Q is to be measured and the results ±1 are again of interest; instead of probing the system the detector only sends back the result if it is +1. The absence of a result then infers the result −1, however, no actual measurement took place. Quantum theory rejects such a measurement since the measurement still gives information [55] and thus causes wave function collapse. Even though weak measurements would not convince a macrorealist of non-intrusion, suering from the same problem as a projective measurement, they are still less intrusive in the quantum mechanical sense. Weak measurements are nonetheless able to violate LGIs although they now take on a dierent form. For a weak measurement of a di-

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chotomous variable, the result is no longer limited to the two values and can now be continuous. In the statistical limit, the weak measurement scheme will correspond to the projective measurements, or an unambiguous classical measurement. As an example, weak measurements (q) can be modeled using Kraus operators (E(q)) [56] as a representation of a linear map taking the system from ρ → EρE^† where E(q) can look like [57]:

E(q) = (2λ/π)^1/4exp−λ(q − Q)² , (22) which takes the form of a Gaussian. The value of λ ≥ 0 determines the strength of the measurement where λ = 0 would denote no measurement at all, whereas λ → ∞ would be the projective measurement. The problem of violation now becomes more of a statistical matter where the amount of experiments has to increase for the result to be signicant.

4.6 Analytical solution in the quantum picture

To investigate the violations that occur for LGIs in the quantum realism,

rst it is needed to nd a quantum analog for the correlation functions which can take dierent forms depending on type and order of measurements. This paper will use projective measurements as was rst assumed by Leggett-Garg and then the correlation functions take on a simpler form [58]:

C_ij = 1

2h{Q_i, Q_j}i, (23)

as will be shown. Consider a single quantum system |ψi (assumed pure for notational convenience) with governing mechanics described by a time- independent Hamiltonian H with a dichotomous von Neumann [59, 60] observable Q. The general result for projective operators is that they satisfy the condition:

Q² = 1. (24)

Upon measurement with Q, the system collapses to an eigenvector corresponding to that outcome. This is especially important since this will make the initial state of the system irrelevant; at the initial measurement the system will simply collapse into either | + 1i or | − 1i regardless of its initial state. The corresponding projectors into the eigenspace are for +1 and -1 respectively:

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1 + Q

2 ,1 − Q

2 , (25)

which can be generalized to:

1 + qQ

2 , (26)

where q ∈ −1, 1 denotes the eigenvalue.

Working within the Heisenberg picture, the observable Q will be evolving instead of the state |ψi and thus the operator at tj will look like:

Q(tj) ≡ e^−iHt^jQe^iHt^j, (27) where ~ = 1 and i denotes the complex constant. Next is to nd P (qi, q_j) for the correlation function in equation 1. First the measurement at ti is performed, collapsing the wave function which is then followed by measurement at tj. The probability of this occurring given by the axioms of quantum mechanics is:

P (q_i, q_j) =

ψ

1 + qiQ(t_i)

2 · 1 + qjQ(j)

2 ·1 + qiQ(i) 2

ψ

, (28) which, when multiplied, out looks like:

P (q_i, q_j) = 1 4 +1

4q_ihψ|Q(t_i)|ψi + 1

8q_jhψ|Q(t_j)|ψi+ (29) 1

8q_iq_jhψ|{Q(t_i), Q(t_j)}|ψi + 1

8q_jhψ|Q(t_i)Q(t_j)Q(t_i)|ψi.

Here the curly brackets {·, ·} denotes the anti-commutator. Even though 29 looks messy, the correlation function sums over the eigenvalues and from the above equation only the term with the anticommutator has any correlation.

Therefore the end result is simply:

C_ij =P

Qi,Qj=±1Q_iQ_jP_ij(Q_i, Q_j) =

P (1, 1) + P (−1, −1) − P (1, −1) − P (−1.1) =

1

2h{Q_i, Q_j}i,

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This truly gives a name to the idea of quantum entanglement in time. This also has a direct analogy to the Bell's inequality where the correlation functions are the expectation value of the tensor product of the observables while in the temporal LGI it is the antisymmetry over the same operator in time.

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4.7 Maximum violation

Answering the question of how large a maximum violation of an inequality can be can determine how non-local, for the Bell's inequality, or how non- macrorealistic, for the Leggett-Garg inequality, a system can be for given parameters. The maximum violation seems to have some physical signi- cance, for example, for the Bell's inequality, the Cirelson's bound [61] sets the limit that B = 2√

2 . This can be linked to limits in quantum computation [62] and information sharing [63], whereas a higher bound could break the laws of quantum mechanics such as the no-signaling theorem laid out in a previous section. To investigate the maximum for the Leggett-Garg inequality, using equation 23, rst consider a specic case with a qubit and H = ¹₂Ωσ_x and Q = σz. From equation 28 Q as a function of time can be found. First, consider the following identity:

eⁱ¹²^Ωσ^x^t = 1 cos1

2Ωt + iσxsin1

2Ωt, (31)

which follows from the denition of the exponential function for operators and the fact that σ²x = 1 together with the series expansion of sin and cos.

Then equation 27 for this specic case becomes:

Q(t) = e⁻ⁱ¹²^Ωσ^x^tσ_zeⁱ¹²^Ωσ^x^t =

cos(Ωt) i sin(Ωt)

−i sin(Ωt) − cos(Ωt)

. (32)

The product Q(ti)Q(t_j) then becomes:

Q(t_i)Q(t_j) = cos(Ω(t_i− t_j)) −i sin(Ω(t_i− t_j))

−i sin(Ω(t_i− t_j) cos(Ω(t_i− t_j))

. (33)

The two o-diagonal elements are antisymmetric in the transformation ti ↔ t_j while the diagonal elements are symmetric immediately signaling that equation 23 becomes:

1

2{Q_i, Q_j} = 1 cos(Ω(ti − t_j). (34) It is clear that this operator, as expected, does not depend on initial conditions. The expectation value is always going to be:

C_ij = cos(Ω(t_i− t_j). (35)

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Simplifying further by considering time between two measurements to be constant τ gives the value for Kn in equation 14 the analytical value for the system:

K_n= (n − 1) cos(Ωτ ) − cos((n − 1)Ωτ ). (36) The maximum violation is achieved when τ = _Ωn^π , where Kn becomes:

Kn= (n − 1) cos ^π_n − cos (n − 1)^π_n

= (n − 1) cos ^π_n + cos (n − 1)^π_n+ π

= (n − 1) cos ^π_n + cos (n − 1)^π_n +^nπ_n

= (n − 1) cos ^π_n + cos (2n − 1)^π_n = n cos ^π_n ,

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which incidentally for K4 = 2√

2, while for K3 = 3/2. That the bound for LGI for the case n = 4 and Bell's inequality coincide is not a coincidence [64]. Both inequalities are derived using the same macrorealistic principles with the dierence being separation in time and space respectively as seen above and a direct mapping can be achieved by a Leggett-Garg-inequality letting Q1 = B⁰, Q₂ = A, Q₃ = B, Q₄ = A⁰. If somehow the bounds would deviate from each other, it would imply that one of the inequalities impose a stronger limitation for the condition of realism or quantum correlation.

The system above can be generalized to show that all time-independent Hamiltonians will cause the correlation functions to behave like a cosine function. Using the fact that any normalized unitary operator can be pa- rameterized using the pauli matrices such that Q = n · σ where σ is the pauli vector and n a unit vector. Then any time-independent Hamiltonian will simply rotate the operator in time and the correlation function will be the angle between them as shown by:

1 2

Dn

Q_i, ˆQ_joE

= n_i· n_jh1i = n_i· n_j, (38) using the identity (ni· σ) (n_j · σ) = n_i · n_j1 + iσ · (ni× n_j) together with the symmtery and anti-symmetry of the dot- and cross product respectively.

Again, in the general case the initial conditions do not play a part as expected.

The dot product between two unit vectors is simply given by cos(θ) where θ is the angle between the vectors. If the angle between all vectors are constantly θ, Kn takes on the familiar form of:

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Kn =

n−1

X

m=1

cos (θm) − cos

n−1

X

m=1

θm

!

. (40)

As within quantum mechanics, most deviations of observables and measurements stem from the non-commuting nature of the observables, thus for LGIs it is the non-commutativity with itself at dierent times (or non- commutativity with the Hamiltonian) that causes the violations. Using the same identity as in 38, the commutators can be derived as:

[Q_i, Q_j] = 2iσ · (n_i× n_j). (41) The points where the absolute values of the commutators are maximized correspond to the point of maximum violation for the system [65].

Even though the above derivation was for qubits, it still holds in the general case [58] and thus the derivation for the maximum size in this setting also holds for any arbitrary size of the system (up to the macrolevel) [66].

However, this is not the actual maximum violation for general measurements as the less invasive measurements allow for higher violations [67, 68, 69].

4.8 Violation of Leggett Garg inequalities

It is appropriate to analyze the circumstances which allow a violation of an LGI to occur. It is clear from the above background that A1-3 are all necessary for the inequality to hold, thus a violation would indicate that one of the assumptions would be false. It turns out that any non-trivial time- independent and non-classical Hamiltonian under projective measurement can violate an LGI [42] and as such there is not any pure quantum system adhering to the macrorealism assumptions of A1-3. Thus, LGIs can be used as a macrocoherent test, either for A1 or A3 if one is feeling adventurous, as long as the clumsiness loophole is avoided as discussed in the section on measurement tactics. To date, several experiments have been able to witness violations of LGIs. The rst violation [70] being a superconducting qubit [71]

using the continous weak measurement scheme, several additional violations have been achieved using weak measurements on photons [72, 25, 73]. The projective measurement scheme has also shown violations on photons [74] as well as nuclear magnetic resonance [75, 76]. Furthermore, violations have been observed with ideal negative measurements [77], which was also used in

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are anywhere near the size of Shrödinger's cat, they do provide a framework for how to grow the violations to the macroscopic scale. On a further note, only the experiments using ideal negative measurements actually avoid the clumsiness loophole and leaves the violation to break either A1 and A3 and should thus be considered real evidence that would make the macrorealist not be able to claim clumsiness. In this sense LGIs can be used to show the predictions of quantum mechanics in several systems, or quantumness [78]

to test whether quantum mechanics or classical mechanics would be a better

t for a system. In this way, LGIs are a direct measure of the quantum coherence a system inhibits and if the emergence of non-classical behavior or decoherence can serve a role in physical reactions [79, 80].

5 Measures

Before proposing measures for the Leggett-Garg case, some existing measures for entanglement are presented.

5.1 Measures of entanglement

The idea of quantifying entanglement can be physically motivated by its use as a resource in LOCC (Local Operations Classical Communication), i.e.

quantum teleportation that uses local operations on an entangled state as the resource together with classical communication to transmit perfect quantum states [81]. From Bell's inequality and the Cirelson bound it is understood that there are constraints on LOCC and it is what motivates a measure of the bit carrying-capacity of a quantum state. However, there is still not a single unique measure that quanties this resource perfectly due to the diculty in ordering LOCC operations on mixed states [82, 83]. In addition, a measure can have intrinsic value as a mathematical tool or for quantifying entanglement in dierent situations [84, 85]. It is therefore not completely useless to base a measure through a purely axiomatic consideration; writing down a wish list of requirements and nding a suitable positive function afterwards; without considerations for a physical resource. Such axioms can be as follows [86]:

1. A bipartite entanglement measure E(ρ) is a mapping from bipartite density matrices into positive real numbers.

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2. E(ρ) = 0 if the state ρ is separable.

3. E does not increase on average under LOCC.

4. For pure state, the measure reduces to the entropy of entanglement E(|ψihψ|) = (S ◦ tr_B) (|ψihψ|). (42) The rst two axioms describe the very basis of a measure, that it is positive and that a state having no trace of entanglement has a 0 measure. Item 3 links the measure to the physical resource being used up; the requirement is that on use the measure does not increase the amount of entanglement in the system. The fourth item is not as obvious as it relates to a measure which is considered appropriate for pure states, thus any more general measure should have the property of ordering the pure states similarly to the entropy of entanglement. This list is not complete, there are several more items that can be included based on the purposes of the measure.

Now for a few examples of measures that satisfy the axioms described above. Entanglement of formation is a measure representing the minimal possible average entanglement over all pure state decompositions of a mixed state:

EF(ρ) := inf (

X

i

piE (|ψii hψi|) : ρ =X

i

pi|ψii hψi| )

. (43)

This measure is greatly discussed as it relates to many open problems in quantum information [87]. Another measure which relates to the entanglement of formation is that of concurrence:

C(ρ) = max {0, λ₁− λ₂− λ₃− λ₄} , (44) where λi are the singular values of ρσy⊗ σ_yρ^∗σ_y⊗ σ_y (where ρ^∗ is the complex conjugate of ρ in the computational basis) in decreasing order. The two measures are related as for two-qubit states [88, 89]:

E_F(ρ) = s

1+√

1−C²(ρ) 2

s(x) = −x log₂x − (1 − x) log₂(1 − x).

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Another example is Negativity dened as [13]:

N (ρ) :=

ρ^T^B − 1

2 , (46)

with the double brackets denoting the trace norm kXk := tr√

X^†X. Neg- ativity uses the fact that a state (limited to systems in 2 ⊗ 2 and 2 ⊗ 3) is entangled if it has negative eigenvalues under partial transpose [90].

An example which is closely related to the background is based on the maximum violation of Bell's inequality [72, 91, 92, 93] and is simply dened to be:

B(ρ) ≡p

max[0, M (ρ) − 1], (47)

where the numberM is dened as the violation of CHSH according to:

BmaxCHSH

Tr (ρBCHSH) = 2p

M (ρ). (48)

M can be computed as the sum of the two largest singular values squared of the matrix Tij = T r(ρσ_iσ_j).

5.2 Proposed measures for macrocoherence

In accordance with the above background, a similar list can be constructed for the case of LGIs. Item 1 and 2 have a simple analog in the LGI-picture;

the rst one being identical. The second item, in the context of using LGIs as a measure of macrocoherence, should instead state that a system dened by a classical Hamiltonian, and thus never violates equation 2, should never have a measure greater than zero. The third and fourth item, however, does not have a natural equivalent statement in the LGI setting, partly because there is not a resource found in the literature attributed to the cause of LGI other than its quantumness. Perhaps one could argue that macrocoherence, or a system's resilience to decoherence have usefulness in quantum computation [94], however it is not clear that LGIs would be the best t to quantify such a measure. For example, it is more natural to quantify decoherence in classical computation with the non-classical depth [95] which has been tied to generating entanglement [96] and the ecacy of quantum teleportation [97].

An analogy to item number 4 suers from the lack of already established measures of entanglement in time that are outside the non-commuting nature of the observable with the Hamiltonian. In regards to this, a suggested list of axioms for an LGI measure could be as follows:

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1. A general measure of a quantum system's (an observable, Hamilto- nian and quantum state) macrocoherence in the setting of an LGI µ(Q, H, ρ, t) is a mapping from an observable, Hamiltonian, density matrix and a time array containing time of measurements ti into positive real numbers.

2. µ(Q, H, ρ, t) = 0 if the system never violates an LGI.

3. µ extends naturally to all orders of LGIs and ranks the systems in the same order for any of those LGIs.

Here item 1 and 2 are naturally extended to the case of LGIs while 3 takes on a dierent meaning. It should be preferential for a measure to not dierentiate between dierent LGIs as to not introduce an arbitrariness in which measure to choose. It might be natural to pick K3, as it consists of the smallest amount of correlation functions and therefore measurements. However, K4 bares the closest resemblance to the well-studied Bell's inequality and could be a more natural t for comparison. Additionally, item 3 states that the measure should have a natural extension and not look too dissimilar to the choice of K_n. This is mathematically justied as deviations from this rule could cause results or contradictions where a state can be seen as macrocoherent and non-macrocoherent by the same measure. Below are two proposed measures satisfying the obvious 1 and 2, but not necessarily 3. These measures are dened for the case of K3, see discussion for the extensibility to higher order.

The rst prospect for a measure is inspired by a measure of non-markovianity [98] and is dened as:

Z

K3≥1

K₃(H, Q, τ ) dτ, (49)

where the time between tα, t_β, t_γ is taken to be uniformly τ.

A second proposed measure is as follows. Dene a matrix L as:

L_ij = hQ(t_i)Q(0)i + hQ(t_i+ t_j)Q(t_i)i − hQ(t_i+ t_j)Q(0)i , (50) where tk is now an array of n time intervals so that L eectively holds n² dierent K3 for dierent times tα and tβ sharing only the time for the initial measurement (here dened at t = 0). To satisfy axiom 1, the measure is dened using the eigenvalues of L^†L which are guaranteed to be real and positive. Such a metric is inspired by earlier works on the Bell's inequality

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[11]. However, to make this measure proper, the matrix is adjusted so that for systems with no violations the measure gives 0, in accordance with axiom 2. Thus, the matrix is instead dened as:

L_ij =

(L_ij − 1, if Lij ≥ 1,

0, otherwise. (51)

This ensures that if no violations of LGI occurs, L becomes the zero matrix and its singular values are obviously 0. The subtraction of 1 from the value of Lij serves to make the measure continuous as a function of τ, whereas otherwise the measure would have signicant cuts due to sudden violations or compliance to the inequality.

6 Discussion and application

The application of the two measures on two dierent systems is considered.

First, the qubit system which was solved in equations 31-37 and admits the violations indenitely for its evolution, therefore not exhibiting any decoherence. Secondly, a system modeled by the Lindblad equation [99]:

˙ ρ = −i

~[H, ρ] +X

n≥1

LnρL^†_n−1

2L^†_nLn, ρ

, (52)

which models non-unitary evolution while preserving the laws of quantum mechanics by being trace-preserving and completely positive. By using non- unitary transformation, the system can be modeled as losing energy to its environment and therefore being submitted to decoherence. The second system uses the same Hamiltonian, but with Lindblad operator L = 0.1σx to model spin damping. It evolves similar to the rst but the spin gets damped over time to the point where K3 is no longer violated. As such, it can be considered to go from a pure quantum realization to one where the spin is no longer present and exhibiting decoherence. The rst measure has the caveat that for some systems, like the rst qubit system, the measure is in-

nite since the measure never stops violating K3 for all time, as shown in Figure 1. However, this can be an advantage since such a system is governed by unitary transformations and therefore does not exhibit any decoherence.

As such it can be extended in the measure to include the case of being innite as the benchmark for a pure evolution that is forever entangled in time. This

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Figure 1: Showing the K3 violations as a function of τ with the shaded area indicating the measure of equation 49 for the system without collapse operators.

Measure of macrocoherence