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A measure of physical reality

From the premise that an observable is real after it is measured, we envisage a tomography-based protocol that allows us to propose a quantifier for the degree of indefiniteness of an observable given a quantum state. Then, we find that the reality of an observable can be inferred locally only if this observable is not quantumly correlated with an informer. In other words, quantum discord precludes Einstein's notion of separable realities. Also, by monitoring changes in the local reality upon measurements on a remote system, we are led to define a measure of nonlocality. Proved upper bounded by discordlike correlations and requiring indefiniteness as a basic ingredient, our measure signals nonlocality even for separable states, thus revealing nonlocal aspects that are not captured by Bell inequality violations.

From Descartes' dictum cogito ergo sum one might be tempted to conclude that a brain-endowed system living in an empty universe could ensure its own existence. This position, however, cannot be maintained within a scientific framework. The reason is that physics is relative by essence, so it is not possible for an object to ascribe any physical state to itself; a reference frame is needed. It follows, therefore, that any attempt to build an empirically accessible notion of physical reality demands in the first place the definition of two entities, namely, an "observer" and an "observed", whose roles are interchangeable. These objects can interact and get to know about each other, in which case it then makes sense for one to speak of the physical reality of the other.
This is precisely what we have in ordinary situations. Our notion of reality is nurtured by the process of repeated measurements that takes place, e.g., every time we watch an object at rest. The huge amount of photons that reach our retina after being scattered by the object-without appreciably disturbing it-informs us that the object "exists" in a "definite" position, so it is real. Although the sensation of reality is granted to the person who receives the scattered photons, the information about the object was already encoded in the photons. Such information, which manifests as correlations in the system "object + photons", was generated via physical interactions which by no means depend on the very existence of a retina or a brain. The capability of a system of informing the presence of another is a primordial condition for physical reality. Without such a mechanism we cannot empirically probe reality.
In 1935, an attempt was put forward by Einstein, Podolsky, and Rosen (EPR) [1] aiming at defining the notion of element of reality: "If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." Along with the conception that "every element of the physical reality must have a counterpart" in a complete theory, this definition immediately implies that ei-ther "(1) the quantum-mechanical description of reality given by the wavefunction is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality." By tacitly assuming locality and then arguing that there are quantum states for which noncommuting observables can be simultaneously real, EPR proved (2) wrong and claimed the incompleteness of quantum theory. Later on Bell showed that any theory aiming at completing quantum mechanics would be unavoidably nonlocal [2], Bohmian mechanics [3] being a prominent illustration of that. Other approaches appeared defending purely statistical interpretations for the quantum formalism [4], in consonance with Einstein's view.
Discussions about foundational aspects of quantum theory, particularly on the wave function interpretation, entered the 21st century with physicists polarized in two main lines of thought, both supported by substantial amount of theoretical work. While on one hand ψ-ontic models ascribe to the wave function a realistic nature, on the other ψ-epistemic models suggest that it actually is knowledge about an underlying reality. Recent years have witnessed significant contributions in favor of ψ-ontic models [5][6][7][8][9][10][11][12][13] against a more modest number of works exploring ψ-epistemic ones [14][15][16][17]. Although general and powerful for their purposes, these works seem to offer no clear interpretation for mixed states.
Here we aim at discussing the notion of physical reality by focusing not only on the quantum state but also on observables and their measurements. In particular, we want to formulate a criterion grounded on the following protocol. Consider an experimental procedure that prepares a physical state for a multipartite system. A task is defined which consists of determining, via state tomography, the most complete description for this preparation. We are allowed to repeat the procedure as many times as necessary to get an ideal tomography. Thus, at the end of the day we get to know that every time the procedure runs the quantum mechanical description for the system will be, say, ρ (Fig. 1a). Then we are exposed to a differ- An observable O1 is secretly measured after the preparation, so it is surely real before the tomography, which then will give ΦO 1 (ρ). If ΦO 1 (ρ) = ρ, then the measurement just revealed a pre-existing element of reality. ent scheme (Fig. 1b). Again we are asked to propose a complete description for the system state, given the same preparation and tomography process, but now a measurement of an observable O 1 = k o 1k O 1k , with projectors O 1k = |o 1k o 1k | acting on H 1 , is performed by an agent in between the preparation and the tomography in every run of the procedure. Quantum theory predicts that the system will be in the state O 1k ⊗ ρ 2|o 1k with probability p o 1k after the measurement is performed, where ρ 2|o 1k = Tr 1 (O 1k ρ O 1k )/p o 1k is the state of the rest of the system given the outcome o 1k and p o 1k = Tr(O 1k ρ O 1k ). Without any information about agent's measurements, after the state tomography our best description will be Now, the agent is certain, by EPR's criterion, that the observable is real after the measurement. It follows, therefore, that our description (1) is epistemic with respect to O 1 , i.e., p o 1k reflects only our subjective ignorance about the actual value of O 1 .
The protocol proceeds with the comparison of the descriptions obtained in (a) and (b). When Φ O1 (ρ) = ρ the situation is such that the agent can conclude that an element of reality for O 1 is already implied by the preparation. In this case, the agent measurements do not create reality, but reveal a pre-existing one. This suggests the following criteria of reality.
The map Φ O1 defined by Eq. (1) denotes a procedure of unread measurements, as delineated by our protocol. Clearly, the criterion (2) agrees with EPR's on the reality of O 1 when the preparation is an eigenstate of this observable, i.e., ρ = O 1k . But it also predicts an element of reality for a mixture of eigenstates, Another interesting point is that the criterion (2) automatically incorporates the fact that a measurement preserves a preexisting reality, i.e., Φ O1O1 (ρ) ≡ Φ O1 (Φ O1 (ρ)) = Φ O1 (ρ). Finally, since a state in the form Φ O1 (ρ) can be viewed as an epistemic state, its collapse upon the readout of a measurement can be interpreted as mere information updating rather than a physical process.
The above criteria motivates us to quantify by how much a given state ρ is far from a state with O 1 real. We then define the indefiniteness (or irreality) of the observable O 1 given the preparation ρ ∈ H as the entropic distance where S(ρ) is the von Neumann entropy. Because projective measurements can never reduce the entropy [18] one has that I(O 1 |ρ) ≥ 0. Since the von Neumann entropy is concave, it follows that I will be zero (i.e., O 1 will be real) if and only if the condition (2) holds [19]. Now the question arises whether this measure can furnish insights to further aspects of quantum theory. To start with we invoke the Stinespring theorem [20], Φ(ρ) = Tr A U ρ ⊗ |a 0 a 0 |U † , according to which any quantum operation Φ can be viewed as a reduced evolution of the system coupled to an ancillary system A, where U is a unitary operator acting on H ⊗ H A and |a 0 ∈ H A . This observation suggests that reality emerges upon the dynamical generation of correlations between the system and some informer, i.e., a degree of freedom that records the information about the physical state of the system. This point is illustrated by Bohr's floating-slit experiment [21]. After interacting with a light floating slit S, a particle P moves towards a double-slit system, which is rigidly attached to the laboratory. Momentum conservation implies that in order for P to move towards the upper (lower) slit, S has to move downwards (upwards). If m and M denote the masses of P and S, respectively, then the correlation generated in this experiment can be described by the where v and mv/M -the speeds of P and S, respectively-are treated as discrete variables, for simplicity. It is just an exercise to show that I(v|ρ P ), where ρ P = Tr S |Ψ Ψ|, is a monotonically increasing function of x = | mv M | − mv M | and that I(v|ρ P ) = 0 only if x = 0. This shows that the velocity v of P given ρ P will be real only if the motion of S can be unambiguously identified, i.e., if the slit can properly play the role of an informer, in which case no interference pattern will be seen. Clearly, the reality of the velocity can be adjusted by the ratio m M , whose value is previously chosen by the observer (in consonance with Bohr's view [21]). When m M (a nearly fixed slit), the momentum conservation will not be able to reveal the path of the particle, so the velocity will be maximally indefinite and interference fringes will appear.
Reality inseparability.-Consider the indefiniteness I(O 1 |ρ) of the observable O 1 given the preparation ρ. It is straightforward to check that where ρ 1 = Tr 2 ρ and D [O1] (ρ) = I 1:2 (ρ) − I 1:2 (Φ O1 (ρ)). Up to a minimization, D [O1] is a discord-like measure [22][23][24] written in terms of the mutual information I j:k (ρ) = S(ρ||ρ j ⊗ ρ k ) of the parts j and k, where S(ρ||σ) = Tr(ρ ln ρ − ρ ln σ) is the relative entropy. I(O 1 |ρ 1 ) can be viewed as a measure of local indefiniteness, as it quantifies the indefiniteness of O 1 given the local state ρ 1 = Tr 2 ρ. This quantity has recently been used to quantify waviness and coherence [25,26]. The relation (4) can be rewritten as where Interestingly, this shows that an amount D 1 of quantum correlation prevents the indefiniteness of O 1 given the tomography ρ to be equal to its indefiniteness given the local tomography ρ 1 . Meaning that the reality of O 1 cannot be devised separately from the other subsystems, even when they are far apart, this constitutes a violation of Einstein's separability principle [27].
Turning to the floating-slit experiment, when the slit is light enough we have that I(v|ρ P ) = 0, so v is locally definite and, accordingly, no interference fringe will be observed. On the other hand, I(v||Ψ ) = D P (|Ψ ) = ln 2 (the amount of entanglement in |Ψ ), so v is globally indefinite. After all, is there an element of reality associated with v? The answer depends on how the reality is probed. In an interference experiment, only the particle is monitored, so that it is the local reality that is accessed. If we look at both the particle and the slit, then we will be able to identify correlations, which will blur the reality of the particle.
Nonlocality.-Can a physical action on a system influence the reality of other spacelike-separated system? Let us consider a typical EPR scenario, in which two subsystems, 1 and 2, prepared in a state ρ, are sent to distinct laboratories separated by a distance d. An ancillary system A, prepared in the state ρ A = |a 0 a 0 |, is allowed to locally interact with the subsystem 2 through a unitary transformation U 2A (t). Let τ be the time interval necessary for the completion of the injective information storage U 2A (τ )|o 2k |a 0 = |o 2k |a k , where {|a k } is an orthonormal basis. After the interaction ceases, the global state reads (τ ) The assumption of spacelike-separated systems demands that d >> cτ . We then consider the following measure of nonlocality: (7) Clearly, this measure can signal alterations in the reality of O 1 after a physical interaction has taken place in a remote place. It is not difficult to show, from the above assumptions, that Tr whose nonnegativity is implied by the theorem given below. Invariant under permutation of indices, as can be noted from its symmetric form N (O 1 , O 2 |ρ) = S(Φ O1 (ρ)) + S(Φ O2 (ρ)) − S(Φ O1O2 (ρ)) − S(ρ), this measure quantifies nonlocal aspects associated with the couple {O 1 , O 2 } given ρ. We also define the minimal nonlocality of a preparation ρ as the maximally restrictive optimization over the observables, i.e., whose bounds are defined by the following result. Let Theorem. Given an arbitrary preparation ρ ∈ H 1 ⊗ H 2 , it holds that 0 ≤ N min (ρ) ≤ D 12 (ρ). In particular, if ρ is pure, then N min (ρ) = 0.
The proof is postponed to the Appendix. As far as the couple {O 1 , O 2 } is concerned, the following aspects are implied by the measure (8). First, N = 0 for a fully uncorrelated state ρ = ρ 1 ⊗ ρ 2 , as expected. Second, it can be readily checked that N (O 1 , O 2 |Φ O k (ρ)) = 0 (k = 1, 2), which implies that nonlocality can manifest only if the observables are both unreal for a given preparation. Interestingly, N also vanishes when the whole system is accessed. This point can be shown as follows. Consider a preparation for a multipartite system and an arbitrary partition H x ⊗ H y for the Hilbert space. Let U y be a unitary transformation in H y and O 1 ∈ H 1 ∈ H x . Given that Φ O1 and U y commute, it follows that We see that the reality of a given observable can never be changed by physical actions occurring in other (eventually remote) parts of the system. This result predicts that, provided we can access the whole system, no nonlocality can be found in nature. (Recently, a similar conclusion has been reached in the framework of the manyworlds interpretation [29].) By confronting the results (8) and (10) we learn that nonlocality is an "optical illusion" that emerges when parts of the system are ignored. However, we should remind that there is a scenario in which discarding a system is fundamentally unavoidable: the measurement process. To appreciate this point, consider a preparation ρ for two subsystem 1 and 2. Let |a 0 a 0 | and |b 0 b 0 | be the initial states of apparatuses A and B which are allowed to interact with 1 and 2, respectively, via local unitary transformations U 1A and U 2B . After the pertinent correlations have been established, the global system is given by ρ 12AB . However, the reduced state ρ 12 can never be directly accessed, after all that is why we need the apparatuses in the first place. Therefore, what we actually access is ρ AB , which can exhibit nonlocality as quantified by the measure (8).
In what follows we illustrate the behavior of N min for specific two-qubit preparations, namely, the Werner state ρ W = (1−f ) 4 1 ⊗ 1 + f |s s|, where f measures the fidelity of ρ W with the singlet |s = 1 √ 2 | + r | − r −| − r | + r (r = x, y, z), and the so-called α-state, ρ α = 1⊗1 4 + α 4 (σ 1 ⊗ σ 1 − σ 2 ⊗ σ 2 ) + 2α−1 4 σ 3 ⊗ σ 3 . Although analytical results have been computed, they are not insightful and will be omitted. They are shown in Fig. 2 along with the results for the global quantum discord D 12 and entanglement E (quantified via concurrence). Two aspects are remarkable. First, unlike Bell nonlocality, N may exist even in the absence of entanglement (see Refs. [30][31][32][33][34] for other examples of nonlocality without entanglement and a link with discord). This suggests an interesting analogy: N is able to capture nonlocal aspects to which Bell nonlocality is insensitive just like quantum discord can detect correlations that are invisible to entanglement measures. Second, N min vanishes for pure states (f = α = 1), as anticipated by the Theorem. This does not mean that pure states prevent nonlocality to manifest, but that there exists al least a couple of observables for which N vanishes (see Appendix). In fact, it is not difficult to show that, if we take observables O 1 and O 2 whose eigenstates define the Schmidt basis, then N (O 1 , O 2 ||ψ ) = E(|ψ ), where E(|ψ ) = −Tr 1 (ρ 1 ln ρ 1 ), with ρ 1 = Tr 2 |ψ ψ|, gives the entanglement entropy of |ψ . The take-away message here comes as follows. Take the singlet ρ s = |s s| as an example. By direct calculation we can check that N (σ 1r , σ 2r |ρ s ) = δ r,r E(|s ), where r, r = x, y, z and δ r,r is the Kronecker delta. This shows that nonlocal aspects appear when we look at the couple of observables in which the entanglement has been encoded, namely, the observables that define the Schmidt basis. No nonlocality is detected when we look at a couple of MNO, one of them composing the Schmidt basis. This explains why N min (|ψ ) = 0 and stress that N is conceptually different from quantum discord, which reduces to the entanglement entropy for pure states.
Nonlocal constraints.-When a correlation is created via physical interactions in a closed quantum system, a constraint is established which manifests as a conservation law in the operator algebra. For instance, for the singlet |s one has that σ z = σ 1z ⊗ 1 2 + 1 1 ⊗ σ 2z = 0.
Here the situation is such that even though the total spin σ z is real (i.e., fully definite, as ensured by the physical interaction) the individuals σ 1z and σ 2z are not. The crux is that the constraint 1 1 ⊗ σ 2z = −σ 1z ⊗ 1 2 reduces the a priori independent indefiniteness of σ 1z and σ 2z to a state of conditional reality, a situation in which the reality of an observable becomes conditioned to the reality of another. Once one of them gets real, so does the other. By separating the subsystems without degrading the constraint, one will be able to define the reality in one place via a spacelike-separated action. This does not occur classically, because even though the notion of a nonlocally spread constraint keeps valid, there is no fundamental indefiniteness underlying the observables, i.e., their realities are already established before the separation. Therefore, irreality is the basic mechanism underlying quantum nonlocality.