The Measurement Paradox in Quantum Mechanics
The indeterministic nature of the measurement process in quantum mechanics has been a major source of philosophical controversies since almost a century (Born 1926) (c.f., for instance, (Landsman 2017, Chapter 11) for a modern discussion). Its essence is illustrated on the diagram in Fig. 1.
There are many competing theoretical proposals on how to solve the measurement problem. One major branch of research relies on the concept of decoherence (Zurek 2003; Schlosshauer 2005), either through the scheme of quantum Darwinism (Zurek 2009) or spectrum broadcast structure (Horodecki et al. 2015; Mironowicz et al. 2017). Another approach seeks to relax the assumption about unitarity of the quantum states’ evolution, which could facilitate some “objective collapse” mechanism (Bassi et al. 2013). However, all these efforts are aimed at the “small” rather than the “big” measurement problem (Pitowsky 2006; Bub and Pitowsky 2010) (see also Bub 2015; Landsman 2017). The former asks how does a quantum superposition evolve into a classical statistical mixture, while the latter consists in the puzzle of how is a single outcome eventually chosen from a statistical mixture.
Both small and big measurement problems involve some kind of information loss. In the first case it is connected with the relative phases between the quantum states, in the second case these are the discarded classical elements of the statistical mixture. It seems therefore natural to argue that quantum mechanism is simply incomplete as a theory of natural phenomena. This point, in the context of composite—entangled—systems, has been the topic of the famous Einstein–Bohr debate (Einstein et al. 1935; Bohr 1935).
The question whether there could be a hidden local realistic theory behind quantum mechanics has been turned into a testable prediction by means of Bell’s theorem (Bell 1964). In the simplest scenario (Clauser et al. 1969) two parties perform simultaneous measurements on two particles from an entangled pair. If the measurement outcomes are determined by a “hidden variable” \(\lambda\), then a certain measure of correlations S between the outcomes cannot exceed the value of 2. On the other hand, the quantum formalism implies that S can be as large as \(2\sqrt{2}\) (Cirel’son 1980).
Since the 1970s multiple “Bell tests” on different physical systems have shown, with a high statistical significance, that the experimental value of S can easily exceed the classical bound and approach the quantum limit (Aspect 2015). However, as recognised already by John Bell himself, any Bell test relies on three external assumptions called “loopholes”. Two of them, “locality” imposing spacelike separation of the two measurements and “fair sampling” requiring a sufficiently sensitive detecting devices, were successfully closed very recently (Hensen et al. 2015; Giustina et al. 2015; Shalm et al. 2015; Rosenfeld et al. 2017).
Yet, there exists a third loophole, the “freedom of choice”, which assumes that the settings of the parties’ devices are uncorrelated with the hidden variable governing the outcomes. It is well known (Brans 1988; Hall 2010) that by relaxing this assumption one can easily attain, and even surpass, the quantum bound on S with a local realistic model. This loophole can be mitigated, for instance through explicit human choices (The BIG Bell Test Collaboration 2018) or signals from remote quasars (Rauch 2018), but it cannot be ultimately overcome (Hall 2016). Its status is therefore metaphysical rather than physical, as realised already by Bell (2001).
In conclusion, for any experiment demonstrating the validity of quantum mechanics one can find a local (or nonlocal, as in the case of Bohmian mechanics) realistic model, which explains its outcomes. Consequently, there exists no unequivocal way to prove that random events exist in nature. We note, however, that such models are operationally equivalent, because they offer exactly the same predictions. On the other hand, the quantum predictions can be conveniently regarded as conditional statements of the form: “If the experimental input was random, then the output will also be random”. This viewpoint is adopted, for instance, by Conway and Kochen in their “Free will theorem” (Conway and Kochen 2006, 2009). It can also be quantified in the “randomness amplification” schemes (Colbeck and Renner 2012; Brandão et al. 2016).Footnote 3
The (“big”) measurement problem in quantum mechanics is in fact an instance of the experiment paradox. It arises primarily because of the random input \(d_{\text {in}}\), which disturbs the system F in an uncontrollable way. The consequence of the latter is an unavoidable indeterminacy of some of the measurement outcomes. This harmonises with the “randomness amplification” viewpoint on quantum experiments. It can always be avoided by questioning the freedom of, at least a part of, the experimental input \(d_{\text {in}}\). However, the general experiment paradox induces also a second riddle—the “preparation paradox”, to which we now turn.
The Preparation Paradox
In contradistinction with the measurement paradox, the preparation paradox is present not only in quantum theory. A convenient universal framework for both classical and quantum mechanics uses the algebraic language (Strocchi 2008; Keyl 2002) of states—encoding the properties of a given physical system F and observables—measurable physical quantities. Any observable A has a spectrum \({\mathrm {sp}}(A) \subset {\mathbb {R}}\), that is a set of possible measurement outcomes and any state \(\omega\) defines a probability distribution \(\mu _{\omega ,A}\) over the set \({\mathrm {sp}}(A)\). Models of physical phenomena are formulated in terms of dynamical equations
$$\begin{aligned} f(\omega (t),t) = 0, \quad \text {for} \quad t \in [0,T], \end{aligned}$$
(1)
where t is a time parameter and f is functional (typically, a linear differential operator) acting on the space of states. Such a model specifies the time-evolution of the system’s state \(\omega (t)\) from an initial condition
$$\begin{aligned} g(\omega (t),t) \vert _{t=0} = 0, \end{aligned}$$
(2)
determined by a (collection of) functionals g. A typical example is a first order partial differential equation with an initial condition \(\omega (0) = \omega _0\).
The predictions of the model (1) are then formulated as follows: If the system was initially described by (2) and an observable A was measured at a time \(t > 0\), then an outcome \(a\in {\mathrm {sp}}(A)\) will be obtained with probability \(\mu _{\rho (t),A}(a)\)Footnote 4. Hence, to test a model determined by equation (1) one has to prepare the system in an initial condition (2) and then measure an observable A at some time \(t \in (0,T]\). Multiple experiments with different inputs \(g_1, g_2 \ldots , g_x\) would tell us whether the predicted probabilities match the observed ones.
Note that the experimental input listed above is indeed free within model (1), because the latter specifies neither the initial conditions (The model does specify the admissible forms of the initial conditions, but not the numerical values.) nor the observable A and the measurement time t. However, we have to admit that the studied system had been in some state \(g_0\) (for instance, the vacuum state) before it was prepared by the experimentalist.
This means that in every experiment the state of the studied system F changes from the \(g_0\), in which it would have been had the experiment not been performed, to a prepared state \(g_i\). Clearly, such a preparation procedure is physical as it effectuates a physical change in the system F. Yet, this change is not modelled within the model (1), for if it could have been modelled within (1), then the entire “experiment” would have been a natural evolution of the system, rather than a valid test.
To understand the ‘resetting problem’ we could construct an extended model
$$\begin{aligned} f(\rho (t),t) = 0, \quad \text {for} \quad t \in [t_0,T], \end{aligned}$$
(3)
describing the evolution of the system from some earlier time \(t_0 < 0\) until T. Then, we assume that its dynamics has been disturbed
$$\begin{aligned} f(\rho (t),t) = j(t), \quad \text {for} \quad t \in [t_0,0), \end{aligned}$$
(4)
with a suitable source j, so that the desired condition (2) at \(t=0\) is met, regardless of the primordial system’s initial conditions at \(t=t_0\). But, clearly, models (3) and (4) are different and the introduction of a source term forces us to change the model. Had the experiment not been performed, the object would evolve according to equation (3) rather than (4). If, on the other hand, we attempt to model the source itself we lose (or rather, shift to another level of complexity) its tunability—hence the ‘preparation paradox’ (see Fig. 2).
The preparation paradox applies equally well to classical and quantum mechanics. In the latter case one can take, for instance, a model (1) with a Schödinger or von Neumann equation describing the dynamics of a quantum state. We are free to prepare the initial quantum state of the system—for instance through projective measurement—, but we have to assume that the tested dynamical equation does not model the entire preparation process. The same line of reasoning could be followed in the Heisenberg picture, in which the system’s state remains steady, but the observables evolve in time. Let us also note that the source term j need not be a function—it could be, for instance, a time-dependent Hamiltonian appended to the Schrödinger equation of a given system.
In conclusion, regardless of whether the theory entails that the measurement—i.e. information acquisition—disturbs the system or not, the preparation procedure is always invasive.
Preparing a system “from outside” is not problematic if we consider a restricted model, say, of a bouncing basket ball. Obviously, one does not expect such a model to say anything about the motion of our hand, which prepares the initial state of the ball. The problems start if the tested model is universal, as we expect a law of physics to be. If we want to test, for instance, the Newton’s law of universal gravitation we have to assume that it does not model the entire preparation procedure. Consequently, there are phenomena to which it does not apply and hence it is not universal.
Typically, the experimental outcomes depend very weakly on how the system has been prepared. The ‘triggering effects’, that is the details of the source j, can usually be alleviated below the noise level shaped by the uncontrolled interaction of the system with its environment. The paradox is, however, more salient in the cosmological context, to which we now turn.
Cosmological Paradox
Modern cosmological models are formulated in the framework of field theory. Let us emphasise that the fields do not evolve per se—a solution to field equations specifies the field content in the entire spacetime. Therefore, any disturbance coming ‘from outside’ would effectuate a global change. In other words, a local terrestrial experiment affects both future and past states of the Universe (see Fig. 3). Note also that cosmological observations are indeed genuine experiments for, firstly, they might but need not be effectuated and, secondly, they involve a number of free input data, such as the telescope’s location and direction or electromagnetic spectrum sensitivity range.
As an illustration, let us consider a cosmological model based on Einstein’s equations
$$\begin{aligned} G_{\mu \nu } = \tfrac{8 \pi G}{c^4} T_{\mu \nu }, \end{aligned}$$
(5)
with a matter energy–momentum tensor \(T_{\mu \nu }\) (possibly including the “dark energy”, i.e. the cosmological constant term \(\varLambda g_{\mu \nu }\)). The geometrical Bianchi identity \(\nabla ^{\mu } G_{\mu \nu } = 0\) implies the local covariant conservation of energy and momentum \(\nabla ^{\mu } T_{\mu \nu } = 0\) (Wald 1984). But, if an ‘external’ source term \(j_{\nu }\) is introduced into (5), the conservation law is violated, \(\nabla ^{\mu } T_{\mu \nu } = -j_{\nu }\), explicitly breaking general covariance. In other words, if one introduces into the universe some information which was not there, one creates ex nihilo a local source of energy–momentum.
In quantum field theory, whereas the energy and momentum need not be conserved locally, the suitable expectation values ought to be conserved. Concretely, if \({\hat{T}}_{\mu \nu }\) is the energy–momentum operator constructed from quantum matter fields, then
$$\begin{aligned} \nabla _{\mu } \langle \psi \vert {\hat{T}}^{\mu \nu } \vert \psi \rangle = 0 \end{aligned}$$
(6)
should hold (Bertlmann 2000) for any state vector \(\vert \psi \rangle\). The introduction of a, possibly quantum, source \({\hat{j}}\) violates the constraint (6) leading to the Einstein anomaly and, eventually, to the breakdown of general covariance (Bertlmann 2000) (see also Bednorz 2016).
In order to perceive the experiment paradox from the perspective of ‘cosmic evolution’ we firstly need to choose a time function—that is an observer—, which fixes an effective splitting of the global spacetime into space and time (Wald 1984). Secondly, one has to assure that equations (5) allow for a well-defined Cauchy problem (Ringström 2009). The latter consists in imposing initial data on a time-slice, say at observer’s time \(t=0\), and studying its (maximal) hyperbolic development (see Fig. 3). This guarantees that both past and future field configurations are uniquely derived from the imposed initial data. The objectivity of the evolution is guaranteed by general covariance, which enables unequivocal transcription of the time-slice field configurations for different observers.
Now, a free perturbation, which means an abrupt change of initial data on a time-slice, in a region K of space inflicts a change in both causal future \(J^+(K)\) and causal past \(J^-(K)\) of K. The problem persists in the context of quantum field theory, because of the “time-slice axiom” (Haag 1996). This is independent from the fact that projective measurements are as harmful to quantum field theory as they are to the non-relativistic quantum theory.
As an illustration suppose that we put a satellite into orbit to test the validity of a cosmic model. Clearly, the matter distribution on a time-slice is slightly different with the satellite and without it. Then, if F is taken literally to be the whole universe, we would have to adjust the past states of the universe to match its present state with the satellite, which effectively means changing the model. Alternatively, one could maintain that the satellite was predetermined to be put into orbit, which means that \(G(d_{\text {in}}) \subset F\) and we could not have not performed the ‘experiment’.
Such absurd conclusions can avoided by recognising that it is more appropriate to say that modelling always involves some aspect of the system and not the system itself. In the cosmic context, one seeks to model the global properties of the universe, while neglecting its microscopic details. The latter could then be treated as the “environment” E, in which the consequences of the experiment paradox could be hidden. Indeed, if one adopts a coarse-grained model of matter, say with a pixel of the size of a galaxy, then obviously the presence or lack of a satellite in Earth’s orbit does not make any difference.
The cosmic example illustrates again that the experiment paradox is not about the unavoidable presence of systematic errors in any experiment, but about the existence of consistent foundations of natural sciences.