Quantum Objective Realism

The question whether quantum measurements reflect some underlying objective reality has no generally accepted answer. We show that description of such reality is possible under natural conditions such as linearity and causality, although in terms of moments and cumulants of finite order and without relativistic invariance. The proposed construction of observations' probability distribution originates from weak, noninvasive measurements, with detection error replaced by some external finite noise. The noise allows to construct microscopic objective reality, but remains dynamically decoupled and hence unobservable at the macroscopic level.

Invasiveness cannot be avoided completely, but one can introduce weak measurements [7], with the system coupled so weakly to the detector that it remains almost undisturbed. The price is a large detection noise/error, although completely independent of the system. Subtracting of the detector noise, the statistics of the measurements has a well-defined limit for vanishing coupling, which is described, for non-compatible observables, by a quasiprobability (real but often not positive) distribution [8,9]. Although the most common weak-measurement theories assume instantaneous system-detector interaction [10][11][12][13][14][15], the theory has been generalized to time-extended interaction (with memory, non-Markovian) [16], parametrized by detector temperature, with the measured correlations vanishing in total detector-system equilibrium.
In this work, we further generalize models of weak, noninvasive measurement in the framework of relativity and discuss the problem of frame invariance. Importantly, noninvasive measurements cannot be timesymmetric [17], but we will check continuous transformations of reference frame. We will show that the underlying quasiprobability can be constructed in an invariant form. However, the detection noise is never invariant, which follows from Cauchy-Schwarz type inequalities and universal properties of correlations. It is not in conflict with relativistic collapse theories [18], which are invariant at the level of stochastic equations, but neither consider weak limit nor the problem of stationarity and invariance of the final state -the detector eventually drives the measured state either to constantly heated, nonstationary state, or an equilibrium but not invariant. In contrast, we will show that expectation values of the measured quantities cannot be invariant in the limit of noninvasive measurement if the error of detector, inevitable to turn the quasiprobability into normal probability -realism, is taken into account, and stationarity is demanded.
Construction. We start by developing a general framework of weak quantum measurement based upon the POVM formalism including non-Markovian features, as in the recent work [16]. We use standard relativistic notation, with c = = k B = 1, real coordinates x = (x µ , µ = 0, 1, 2, 3) with single time x 0 , (flat) metric tensor g µν = g µν = diag(1, −1, −1, −1) and Minkowski product We consider a set of n independent detectors recording n spacetime-assigned signals a j (x) for j = 1, . . . , n. Each detector is related to an observableÂ j (the index will be dropped when unimportant). Classical correlators of measured quantities like a 1 (x 1 ) · · · a n (x n ) should be related to their equivalents for weak quantum measurements, replacing a(x) in the correlator by a superoperator dx Ǎ x−x (x ) and perform time order [16]: Here, T denotes time order with respect to the arguments x 0 in brackets, · · · means quantum average Tr · · ·ρ with the density matrixρ. Superoperators should fulfill linearity and causality -outcomes can depend only linearly on operators within their lightcones, which imposes the form The superoperatorsǍ c/q [19] act on any operatorX as an anticommutator/commutator:Ǎ cX = {Â,X}/2 anď A qX = [Â,X]/i. Alternatively 2Ǎ c =Ǎ + +Ǎ − and iǍ q =Ǎ + −Ǎ − withǍ +X =ÂX andǍ −X =XÂ. In the above expressions we assumed for simplicity that the detectors are in a stationary state so that only relative differences x − x matter. Please note that the time order is irrelevant for spacelike intervals (y = x − x with y · y < arXiv:1411.0337v1 [quant-ph] 2 Nov 2014 0) because then the operators commute. The commuting property can be shown using closed time path theory [20,21] for each Feynman diagram [22]. Hence, time order is here in fact causal order, consistent with relativity. We will also assume that the average of single measurements coincides with the usual average for projective measurements, i.e. a(x) w = Â (x) . This implies g(x−x ) = δ(x−x ). Other choices of g simply mimic the effect of classical filters. Thus the only freedom left is the choice of the real memory function f that multipliesǍ q . Note that f (x) can be non-zero for x 0 > 0 without violating causality, since it is accompanied byǍ q and only future measurements are affected. For the last measurement, future effects disappear because the leftmostǍ q vanishes under the trace in Eq. (1). In the Markovian case f = 0. Now we want to show that correlations obeying these requirements can be obtained from the general quantum measurement formalism. Based on Kraus operatorsK [23], the probability distribution of the measurement results is ρ = Ǩ forǨX =KXK † , where the only condition onK is that the outcome probability is normalized regardless of the input stateρ. Here we needK to be spacetime-dependent. In general, we assume thatK[Â, a] is a functional of the whole spacetime history of observ-ablesÂ(x) and outcomes a(x). We shall assume that the functionalK is stationary so it depends only on relative spacetime arguments.
The desired correlation function (1) can be derived by the following limiting procedure for an almost general POVM, where the average on the right-hand side is with respect to ρ η . We assume the absence of internal cor- Here, |k[a]| 2 is a functional probability of time-resolved outcomes independent of the properties of the system which represents the detection error. As we want the measurement to be noninvasive to lowest order, we impose the condition that F [a, x ]|k[a]| 2 Da vanishes; Da is the functional measure. Our conditions imply that Thus, the most general weak Kraus operator takes the form given in Eq. (4). We postpone discussion of the actual form Kraus operators, focusing first on properties of f and g (= δ if not stated otherwise).
Properties. In [16] there was constructed f of the detector temperature-dependent form. At finite temperatures we cannot expect invariance at all. At zero temperature, it was f (x) = µ=1,2,3 δ(x µ )/πx 0 . Certainly this form is still not invariant, depends on the reference frame. It is only that correlations at zero temperature vanish in vacuum, in all frames.
Invariance implies f in the different form Although with any new f we cannot get exactly the same expectations as in the previously derived scheme [16], we can demand partial equality at the level of two point correlations of the measured quantities. Let us define Fourier transform f (p) = dxf (x)e ix·p . Then previously f (p) = isgn p 0 . The zero-temperature vacuum state is invariant under frame transformations, which is commonly believed but highly nontrivial to prove [21]. The new function, invariant and giving the same second order vacuum correlations, obtained by natural extension of the previous one, is f (p) = isgnp 0 θ(p·p). In space-time domain, it means the singular function sgnx 0 δ (x · x)/π 2 . The consistency follows from the fact correlations vanish at zero temperature for spacelike p (p · p < 0) and analogous considerations as in [16] (eqs. 6-8 therein).
Fourth order correlations in equilibrium vacuum still vanish. It follows from unitarity which implies vanishing Ǎ qǍqǍqǍq . Then Ǎ cǍcǍcǍc = Ǎ +Ǎ−Ǎ−Ǎ− + (+ ↔ −) (hight hand side symmetrized). By Feynman rules nonvanishing crossing +− must be timelike (p · p > 0). Hence, our expression vanishes if all the Fourier arguments of A are spacelike which is the case when all functions f are zero.
Higher order correlation will not vanish even in equilibrium vacuum state.
As example, let us take |j 0 (p A )j 0 (p B )j 0 (p C )| 2 where p X = (2q 0 /3, q ⊥ cos φ X , q ⊥ sin φ X , 0) with q 0 m, φ A,B,C = 0, 2π/3, 4π/3 so that spatial vectors point symmetrically in the plane while q 0 is just above decay threshold and q ⊥ > q 0 so that p · p < 0. The lowest order contribution has the form (depicted graphically in Fig. 1b) where q 0 = (q 0 , 0, 0, 0) and (X) 2 = X · X. In the expression one must perform full symmetrization of two separate sets ABC for the left and right branch. The δ ± correspond to transitions +− and here just make k almost irrelevant, k ∼ 0. In this case all denominators have almost the same strictly negative value. Evaluation of the numerator trace and symmetrization gives for q ⊥ q 0 the leading term q 4 ⊥ with some nonvanishing dimensionless factor so (7) is nonzero.
Broken invariance. Turning back to the question of invariant measurements we can think of the detection error that is added to the quasiprobability so that their convolution is positive definite (normal probability). For the normal probability, we have Cauchy-Schwarz type inequality with some regularization δ → δ , i.e. a(p) → dp a(p )δ (p − p ). Let us take a vector field a = j µ . Then, for an arbitrary stationary and relativistically invariant probability, we have j µ (p)j ν (q) = δ(p + q)(p µ p ν ξ(p · p) + g µν η(p · p)) which must be positive definite. For spacelike p it is positive only if η = 0 (we could even use charge conservation to set ξ = 0). Taking p A from our previous example (7) we have then |j 3 (p A )| 2 = 0 -for the regularization we should replace j 3 by (p A ) 2 j · z − (j · p A )(p A · z) with z = (0, 0, 0, 1). However, the expression Fig. 1c) does not vanish in the Markovian scheme (f = 0) analogously as (7) so the inequality (8) is clearly violated. In the case f (p) = isgn p 0 θ(p · p) we must take 6th order correlations, e.g. the Cauchy-Schwarz inequality is violated by nonvanishing (7). The important consequence is that we cannot construct any invariant noninvasive measurements for vector fields. For scalar fields we can formally make the noise |a(p)| 2 positive for all p. However, from the correspondence principle follows that zero-frequency (long-time) quantum equilibrium correlations satisfy classical fluctuation-dissipation theorem [24], which makes them vanish in vacuum (zerotemperature). This fact follows directly from closed time path formalism [20,21] and does not rely at all on relativity. However, only detector operating at zero temperature -e.g. quantum tape [25] -can lead to relativistic invariance (broken at finite temperatures). Therefore all vacuum zero-frequency correlations must vanish and, from invariance, |a(p)| 2 must vanish for all spacelike p so the inequality (8) or (9) will be violated by a scalar analogue of the previous example. One can certainly add an ex-nihilo noise for spacelike p but only in the noninvasive limit. For any finite measurement strength this will not help because the inequalities (8) and (9) must hold for actually measured correlations, without any artificially added noise. A solution would be a classical filter permitting only timelike momenta, i.e. g(p) = 0 for spacelike p. Unfortunately it would violate relativistic causality, as for invariant g, for spacelike x we get [26] which is always somewhere nonzero [27].
We cannot also take f (p) = ±iθ(−p · p) as it gives complex f (x). The invariance and correspondence principle at zero temperature will be hence broken in standard theory at least spontaneously, which is our main result, and. Once we accept this, it is easy to construct a particular Gaussian measurement scheme (Appendix), apparently invariant [18], but must be supplemented by a heat sink in a preferred frame to become stationary. It also shows that any quantum realism based on adding error to quasiprobability (to get real positive probability) will also break relativistic invariance. Only formalisms beyond standard correspondence principle or quantum field theory allow relativistically invariant measurements [28]. An inconvenience of the above provided examples is that they all work is the frequency scale comparable to electron mass (upon unit conversion) so they are far beyond experimental capabilities. However, they have lowenergy analogues although in higher order of interaction. Instead of writing lengthy and boring expressions it is better to depict them using standard closed time path formalism in Fig. 2. The subfigures correspond to previous examples in Fig. 1 with the respective letter.
Conclusion. We have shown that relativistic noninvasive measurement can be defined but they lose some properties of nonrelativistic measurements, such as vanishing of equilibrium correlations at the second order and vacuum correlations at sixth order when we try to build invariant and stationary outcome probability. It is also impossible to define quantum realism nor any stationary model of measurement in a relativistically invariant way consistent with standard quantum field theory and, for scalar fields, correspondence principle. The invariance at the level of measurements must be broken at least spontaneously.
I thank W. Belzig, P. Chankowski and P. Pearle for dis- cussions, hints and inspirations that helped me to complete this work.
Appendix. An example of a POVM leading to relativistic weak measurement is based on the Gaussian detector prepared in the initial state (wavefunction) φ(X) ∝ exp(−X 2 ), where X is not a fourvector x but simple one-dimensional measurable position-like parameter, interacting with the system by the time-dependent Hamiltonian density(in the interaction picture) H I (x ) = (δ(x − x )P + 2f (x − x )X)Â(x ). The momentum-like quantity P makes the shift X → X −Â(x). For the measurement of a(x) = X we get the Gaussian Kraus operator Here, the first term in the exponent is the Markovian part, while the second term describes the non-Markovian measurement process including a fixed but arbitrary real function f (x), characterizing the memory effect, as it makes the outcome depend on distant observables. The Heaviside function θ follows from the fact that P shifts the phase for x 0 < x 0 and ensures the normalization of the Kraus operator. By comparing with (4), we get |k[a]| 2 = 2/πe −2a 2 and F [a, x ] = 2a(x)(δ(x − x ) − if (x − x )), which in this special case are just usual functions. Following the standard procedure we find the Kraus superoperator in the form To prove the normalization, da Ǩ = 1, we perform the Gaussian integral over a (time order is no problem if kept up throughout the calculation) and get where we have ordered properlyǍ q (x ) andǍ c (x). In the power expansion, omitting the identity term, the leftmost superoperator is alwaysǍ q . Since TrǍ q · · · = 0 we obtain da Ǩ [Â, a] = 1 or daK †K =1. In general, we defineK[Â, a] for n measurements asK[Â, a] = T jK [Â j , a j ], takingĤ I = jĤ j,I . To get a weak measurement, we substituteK byK η which is obtained by replacingĤ I → ηĤ I and measuring a(x) = ηX. Note that puttingÂ = 0 gives Gaussian white noise ρ ∝ e −2a 2 , which leads to large detection noise in the weak limit, ρ η ∝ e −2η 2 a 2 , that has to be subtracted/deconvoluted from the experimental data. The scheme is apparently relativistically invariant just as shown in [18] but for any finite η the disturbance heats the system constantly making it impossible to reach a stationary state. The heat must be transferred to a sink in a preferred frame.