*1.4. Preview*

Our first goal is to shore up quantum foundations—to understand the emergence of stable classical states from the quantum substrate, and to deduce the origin of the rules governing randomness at the quantum-classical border. To this end, in the next two sections we shall derive collapse axiom (iva) and Born's rule (v) from the core postulates (o)–(iii). We shall then, in Section 4, account for the "objective existence" of pointer states. This succession of results provides a wholly quantum account of the emergence of classical reality.

We start with a derivation of the preferred set of *pointer states*—(iva), the business end of the collapse postulate. We will show that the nature of the information transfer—nature of the coupling to the measuring device—determines this preferred set, and that any set of orthogonal states will do. We will also see how these states are (ein)selected by the dynamics of the process of information acquisition, thus following the spirit of Bohr's approach which emphasized the ability to communicate the results of measurements. Orthogonality of outcomes implies that repeatedly measurable quantum observable must be Hermitian. We shall then compare this approach (obtained without resorting to reduced density matrices or any other appeals to Born's rule) with a decoherence-based approach to pointer states and the usual view of einselection.

Pointer states—terminal states for quantum jumps—are determined by the dynamics of information transfer. They define the outcomes independently of the instantaneous reduced density matrix of the system and of its initial state. Fixed outcomes define events, and call for the derivation of probabilities. In Section 3 we also take a fresh and very fundamental look at decoherence: It arises—along with Born's rule—from the symmetries of entangled quantum states.

Given Born's rule and preferred pointer states one is still faced with a problem. Quantum states are fragile. An initially ignorant observer cannot find out an unknown quantum state without endangering its existence: Collapse postulate means that selection of what to measure implies a set of outcomes. Therefore, only a lucky guess of an observable could let the observer find out an unknown state without repreparing it. The criterion for pointer states implied by postulates (o)–(iii) turns out to be equivalent to their stability under decoherence, and still leaves one with the same difficulty: How to find out an effectively classical but ultimately quantum pointer state and leave it intact?

The answer turns out to be surprisingly simple: Continuous monitoring of S by its environment results in redundant records of its pointer states in E. Thus, observers can find out the state of the system indirectly, from small fragments of the same E that caused decoherence. Recent and still ongoing studies discussed in Section 4 show how this replication selects the "fittest" states that can survive monitoring, and yields copious qmemes2, their information-theoretic offspring: Quantum Darwinism favors pointer observables at the expense of their complements. Objectivity of the preferred states is quantified by their redundancy—by the number of copies of the state of the system deposited in E. Stability in spite of the interaction with the environment is clearly a prerequisite for large redundancy. Pointer states do best in this information—theoretic "survival of the fittest".

The classical world we perceive consists predominantly of macroscopic objects. Bohr decreed their states were classical "by fiat", so that information about them could be acquired without perturbing them, thus restoring classical independence of existence from information. We recognize instead that quantum theory is universal. States of macroscopic objects become effectively classical (as Bohr wanted), but as a consequence of decoherence and einselection. Objects are immersed in the decohering environment consisting of subsystems (such as photons). Superpositions of pointer states are unstable, quickly turning into their mixtures. Thus, predictably evolving quantum states of macroscopic objects are restricted to stable, effectively classical pointer states einselected by decoherence. In the course of decoherence fragments of the environment that monitors them become inscribed with the data about their pointer states.

*Extanton* is a composite entity with the object of interest in its core embedded in the information-laden halo, part of the environment that monitors its pointer states. Information about them is heralded by the fragments of the environment, and disseminated throughout the Universe. Fragments of the halo intercepted by observers inform about the state of its core. Extanton combines the source of information (extanton core) with the means of its transmission (halo, often consisting of photons).

John Bell (1975 [40]; 1987 [41]) imagined "beables" (as in "to be or not to be"). In contrast to observables, beables were supposed to be robust, much like states of macroscopic objects in the classical domain posited by Bohr. They would exist, and (in contrast to quantum states), their states would be immune to observation.

Extantons are quantum, but fulfill these desiderata. Environment determines pointer states through einselection. Pointer states of extanton cores persist (hence, exist) and the environment broadcasts information about them. That information reaches observers, revealing the pointer state of the macroscopic system at the extanton core without the need for direct measurement (hence, without disrupting the state preselected by the decoherence).

We are immersed in such extaton halos, inundated with the information about pointer states in their cores. This is how the classical world we perceive emerges from within our quantum Universe.

As we shall see, several steps based on interdependent insights are needed to account for quantum jumps, for the appearance of the collapse, for preferred pointer states, for the probabilities and Born's rule, and, finally, for the consensus, the essence of objective reality—for the emergence of 'the classical' from within a quantum Universe. It is important to take these steps in the right order, so that each step is based only on what is already established. This is our aim, and this order has determined the structure of this paper: The next three sections describe three crucial steps. Nevertheless, each section can be read separately: Preceding sections are important to provide the right setting, but are generally not essential as a background. An overview of the resulting quantum theory of the classical is presented and the interpretational implications are discussed in Section 5.2.

#### **2. Quantum Jumps and Einselection from Information Flows and Predictability**

This section shows how the core quantum postulates (o)–(iii) lead to the discreteness we regard as characteristic of the quantum world. In textbooks this discreteness is introduced via the collapse axiom (iva) designating the eigenstates of the measured observable as the only possible outcomes. Here, we show that discontinuous quantum jumps between

<sup>2</sup> This term is a quantum version of the familiar *meme*, that according to Wikipedia, stands for "...an image ... that is copied (often with slight variations) and spread rapidly".

a restricted set of orthogonal states turn out to be a consequence of symmetry breaking that resolves the tension between the unitarity of quantum evolutions and repeatability. We shall also see how preferred Hermitian observables defined by the resulting orthogonal basis are related to the familiar pointer states.

Unitary evolution of a general initial state of a system S interacting with an apparatus A leads—as illustrated by Equation (1.1)—to an entangled state of SA. Thus, there is no single outcome—no literal collapse—and an apparent contradiction with our immediate experience. It may seem that the measurement problem cannot be addressed unless unitarity is somehow circumvented (e.g., along the *ad hoc* lines of the Copenhagen Interpretation).

We start with the same assumptions and follow similar steps, but arrive at a different conclusion. This is because instead of demanding a single outcome we shall only require that the result of the measurement can be confirmed (by a re-measurement), or communicated (by making a copy of the record). In either case, copying some state (of the system or of the apparatus) is essential. As "perception" and "consciousness" presumably depend on copying and other such information processing tasks (as they undoubtedly do) then the necessity to deal with the Universe "one branch at a time" can produce symptoms of collapse while bypassing the need for it to be "literal".

Amplification—the ability to make copies, qmemes of the original—is the essence of the repeatability postulate (iii). It calls for nonlinearity (one needs to replicate the original state, or at least its salient features) that would appear to be in conflict with the unitarity (hence, linearity) demanded by postulate (ii).

As we shall see, copying is possible for orthogonal subsets of states of the original. Each such subset is determined by the measurement device—by the unitary evolution that implements copying. When, beforehand, the system is not in one of such copying eigenstates, its state is not preserved. This shows (Zurek, 2007 [42]; 2013 [43]) why one cannot find out an unknown quantum state. Most importantly, we reach this conclusion (where the role of the copying device parallels function of the classical apparatus in Bohr's Copenhagen Interpretation) without calling on the collapse axiom (iv) or on Born's rule, axiom (v).

#### *2.1. Repeatability and the Quantum Origin of Quantum Jumps*

Consider a quantum system S interacting with another quantum system E (which can be an apparatus, or—as the present notation suggests—an environment). Let us suppose (in accord with the repeatability postulate (iii)) that there are states of S that remain unperturbed by this operation, e.g., that this interaction implements a measurement—like information transfer from S to E:

$$|s\_k\rangle|\varepsilon\_0\rangle \implies |s\_k\rangle|\varepsilon\_k\rangle. \tag{2.1}$$

We now establish:

**Theorem 1.** *The set of the unperturbed states* {|*sk*-} *of the "control"—of the system* S *that is being measured or decohered—must be orthogonal.*

**Proof.** From the linearity implied by the unitarity of (ii) and Equation (2.1) we get, for an arbitrary initial state |*ψ*S - in HS (allowed by the superposition principle, postulate (i));

$$\left|\psi\_{\mathcal{S}}\right\rangle \left|\varepsilon\_{0}\right\rangle = \left(\sum\_{k} a\_{k} \left|s\_{k}\right\rangle\right) \left|\varepsilon\_{0}\right\rangle \Rightarrow \sum\_{k} a\_{k} \left|s\_{k}\right\rangle \left|\varepsilon\_{k}\right\rangle = \left|\Psi\_{\mathcal{S}\mathcal{E}}\right\rangle. \tag{2.2}$$

But, again by (ii), the norm must be preserved,

$$|\sum\_{k} \alpha\_{k} |s\_{k}\rangle|^{2} = |\sum\_{k} \alpha\_{k} |s\_{k}\rangle |s\_{k}\rangle|^{2},$$

so that elementary algebra leads to:

$$\operatorname{Re}\sum\_{j,k}a\_j^\*a\_k\langle s\_j|s\_k\rangle = \operatorname{Re}\sum\_{j,k}a\_j^\*a\_k\langle s\_j|s\_k\rangle\langle\varepsilon\_j|\varepsilon\_k\rangle.\tag{2.3}$$

This must hold for every |*ψ*S - in HS—for any set of complex {*<sup>α</sup>k*}. Thus, for any two states in the set {|*sk*-}:

$$
\langle s\_j | s\_k \rangle (1 - \langle \varepsilon\_j | \varepsilon\_k \rangle) \tag{2.4}
$$

This equality immediately implies that {|*sk*-} must be orthogonal if they are to leave any imprint—deposit any information—in E while remaining intact: It can be satisfied only when *sj*|*sk*- = *<sup>δ</sup>jk*, unless *<sup>ε</sup>j*|*<sup>ε</sup>k*- = 1—that is, unless the information transfer has failed, as - *εj* = |*<sup>ε</sup>k*-—the states of E bear no imprint of the states of S.

Equation (2.4) establishes postulate (iva)—the orthogonality of the outcome states (i.e., of the "originals" of the copying eigenstates). As we have noted, (iva) is the essence, the "business end" of the collapse axiom (iv). When the outcome states are orthogonal, any value of *<sup>ε</sup>j*|*<sup>ε</sup>k*- is admitted, including *<sup>ε</sup>j*|*<sup>ε</sup>k*- = 0, which corresponds to a perfect record.

Note that—as long as the state, Equation (2.1) is a direct product before and after the measurement—this conclusion holds for an arbitrary initial state of E, since Equation (2.4) demands orthogonality whenever there is any transfer of information from S to E—that is, whenever *<sup>ε</sup>j*|*<sup>ε</sup>k*- = 1. It is of course possible that there are subsets of orthogonal states that cannot be distinguished by the environment. We shall consider such degeneracy shortly.

The limitation of copying to distinguishable (orthogonal) outcome states is then a direct consequence of the uncontroversial core postulates (o)–(iii). It can be seen as a resolution of the tension between linearity of quantum theory (superpositions and unitarity of (i) and (ii)) and nonlinearity of the process of proliferation of information—of amplification. This nonlinearity is especially obvious in cloning, as cloning in effect demands "two of the same". The main difference is that in cloning copies must be perfect. Therefore, scalar products must be the same, *ςj*,*<sup>k</sup>* = *<sup>ε</sup>j*|*<sup>ε</sup>k*- = *sj*|*sk*-. Consequently, in cloning we have a special case of Equation (2.4): *<sup>ς</sup>j*,*<sup>k</sup>*(<sup>1</sup> − *<sup>ς</sup>j*,*<sup>k</sup>*) = 0. Clearly, there are only two possible solutions; *ςj*,*<sup>k</sup>* = 0 (which implies orthogonality), or the trivial *ςj*,*<sup>k</sup>* = 1.

Indeed, we can deduce orthogonality of states that remain unperturbed while leaving small but distinct imprints in E directly from the no-cloning theorem [44–46]) that limits copying allowing it for orthogonal sets of states (thus precluding use of entangled quantum states for superluminal communication): As the states of S remain unperturbed by assumption, arbitrarily many imperfect copies can be made. However, each extra imperfect copy brings the collective state of all copies correlated with, say, - *sj* , closer to orthogonality with the collective state of all of the copies correlated with any other state |*sk*-. Therefore, one could distinguish - *sj* from |*sk*- by a measurement on a collection of sufficiently many copies, and use that information to produce their "clones". As a consequence, also imperfect copying (any value of *<sup>ε</sup>j*|*<sup>ε</sup>k*- except 1) that preserves the "original" is prohibited.

We now have a useful definition of an *event*. Wheeler [47]—following Bohr—insisted that "No phenomenon is a phenomenon until it is a measured (recorded) phenomenon". Our contribution is to supply—using information transfer and the dynamics of copying— an operational definition of a "recorded phenomenon". We have just demonstrated that the ability to record events repeatedly associates them with a set of orthogonal states. This in turn implies discreteness, and the inevitability of jolts, quantum jumps that force the system to choose one of the items on the discrete menu of final (outcome) states.

Events that ge<sup>t</sup> recorded repeatedly precipitate quantum jumps. They emerge—as a consequence of the discreteness we have just deduced—from within the quantum measurement setting (as discussed, e.g., by von Neumann, 1932 [4]) where both the state of the measured system and of the apparatus are initially pure, and the final state (while entangled) is also pure. The defining characteristic of an event is a transition from before the measurement (from the old state of the system that was known, but it was not known

what will happen when the new measurement is made) to when the outcome of the new measurement can be confirmed by repeated re-measurements.

Appearance of events in a pure state case prompts the question about their probabilities. If we were to proceed logically we would suspend discussion of how the core postulates (o)–(iii) imply the essence of axiom (iv), derive Born's rule, and only then come back and consider how quantum jumps—the essence of the collapse—emerge in the mixed state case using the relation between pure states and reduced density matrices, the usual tools of decoherence. This course of argumen<sup>t</sup> would require a detour before we can come back and complete the discussion that we have already started.

We shall avoid this, but we shall also avoid using probabilities and Born's rule, as in [35,48]. Some readers may nevertheless prefer to take that detour on their own, "jump" to Section 3, and return to the discussion below after they are convinced that Born's rule emerges from the symmetries of entanglement in the pure state case. While the reasoning below does not depend on probabilities computed using *pk* = |*ψk*| 2, it employs ideas (such as purification) and mathematical tools (such as trace) that are suggested by decoherence and useful in the "Church of Larger Hilbert Space" approach to mixtures.

We also note that our tasks differ depending on whether mixed states of the control or mixed states of the target (see Figure 1) are the focus of attention. We start below with the simpler case—a target (e.g., an environment) that is in the mixed state. In that case generalization from pure states to mixtures is relatively straightforward, as the challenge is primarily technical [42,49].

Generalization of our discussion to the case when the control—the source of information— is allowed to be in a mixed state must take into account an additional complication: The state of control can change, and ye<sup>t</sup> result in the same copy—a quantum meme or a *qmeme*—of the essential information. This degeneracy is important in considering readout of information from a macroscopic apparatus pointer or any other macroscopic device that is supposed to keep reliable records [43]. Obviously, the detailed microscopic state of such a device is of little consequence—the information of interest is what gets copied. It resides in the corresponding (likely macroscopic) degrees of freedom (e.g., of an apparatus pointer). Many microscopic states may (and usually will) represent the same information. Therefore, degeneracy—the fact that many microstates represent the same record and will result in the same copy of that record—must be considered along with the possibility of mixed states of the control. We shall return to this case of mixed and degenerate control later in this section.

#### *2.2. Mixed States of the "Target"*

Equations (2.1)–(2.4) are based on idealizations that include purity of the initial state of E. Regardless of whether E designates an environment or an apparatus, this is unlikely to be a good assumption. However, this assumption is also easily bypassed: An unknown state of E can be represented as a pure state of an enlarged system. This is the purification (aka "Church of Larger Hilbert Space") strategy: Instead of a density matrix *ρ*E = ∑*i pi*|*<sup>ε</sup>i<sup>ε</sup>i*| of a mixed state one can deal with a pure entangled state of E and E defined in HE ⊗ HE :

$$\left| \varepsilon \varepsilon' \right\rangle = \sum\_{i} \sqrt{p\_i} \left| \varepsilon\_i \right\rangle \left| \varepsilon'\_i \right\rangle,\tag{2.5a}$$

so that;

$$\rho\_{\mathcal{E}} = \sum\_{i} p\_{i} |\varepsilon\_{i}\rangle\langle\varepsilon\_{i}| = \text{Tr}\_{\mathcal{E}'} |\varepsilon\varepsilon'\rangle\langle\varepsilon\varepsilon'| \,. \tag{2.5b}$$

Therefore, when the initial state of E is mixed, there is always a pure state in an enlarged Hilbert space. Instead of (2.1) we can then write |*sk*-|*<sup>ε</sup>*0*ε* - ⇒ |*sk*-|*<sup>ε</sup>kε* - in obvious notation, and all of the steps that lead to Equations (2.3)–(2.4) can be repeated, so that:

$$
\langle s\_j | s\_k \rangle (1 - \langle \varepsilon\_j \varepsilon' | \varepsilon\_k \varepsilon' \rangle) \tag{2.6}
$$

and forcing one to the same conclusions as Equation (2.4).

Purification relates pure states and density matrices by treating *ρ*E = ∑*i pi*|*<sup>ε</sup>i<sup>ε</sup>i*| as a result of a trace. The connection of *ρ*E with |*εε* - = ∑*i* √*pi* |*<sup>ε</sup>i*- - *ε i* does involve tracing. However, there is no need to regard weights *pi* as probabilities. They are just coefficients that relate a state of the whole |*εε* - and of its part *ρ*E by a mathematical operation—a trace. Thus, *ρ*E is a mathematical object that represents a reduction of a pure state that exists in the larger Hilbert space, but does not yet—in absence of Born's rule—merit statistical interpretation.

Indeed, there is no need to even mention *ρ*E . All of the above discussion can be carried out right from the start with a pure state in a larger Hilbert space. It suffices to assume only that *some* such pure state in the enlarged Hilbert space exists and that lack of purity of E is a result of its entanglement with the rest of the Universe. This does not rely on Born's rule, but it does assert that ignorance that is reflected in a mixed local state (here, of E) can be regarded as a consequence of entanglement. This assertion is established in the next section, so—as we have already noted—readers can break the order of the presentation, consult the derivation of Born's rule in the next section, and return here afterwards.

There is also an alternative way to proceed that leads to the same conclusions but does not require purification. Instead, we assume at the outset that we can represent states as density matrices. Unitary evolution preserves scalar products, i.e., Hilbert-Schmidt norm of density operators defined by Tr*ρρ*. Therefore, one is led to:

$$\text{Tr}|s\_j\rangle\langle s\_j|\rho\_{\mathcal{E}}|s\_k\rangle\langle s\_k|\rho\_{\mathcal{E}} = \text{Tr}|s\_j\rangle\langle s\_j|\rho\_{\mathcal{E}|j}|s\_k\rangle\langle s\_k|\rho\_{\mathcal{E}|k'} $$

where *ρ*E|*j* and *ρ*E|*k* are mixed states of E affected by the two states of S that are unperturbed by copying. This in turn yields;

$$|\langle s\_{\rangle}|s\_k\rangle|^2(\text{Tr}\rho\_{\mathcal{E}}^2 - \text{Tr}\rho\_{\mathcal{E}\backslash\vert}\rho\_{\mathcal{E}\backslash\vert k}) = 0,\tag{2.7}$$

which can be satisfied only in the same two cases as before: Either *sj*|*sk*- = 0, or Tr*ρ*<sup>2</sup> E = Tr*ρ*E|*jρ*E|*k* which implies (by Schwarz inequality) that *ρ*E|*j* = *ρ*E|*k* (i.e., there can be no record of nonorthogonal states of S).

This conclusion can be reached even more directly: Obviously, *ρ*E|*j* and *ρ*E|*k* have the same eigenvalues *pm* as *ρ*E = ∑*m pm*|*<sup>ε</sup>m<sup>ε</sup>m*| from which they have unitarily evolved. Consequently, they could differ from each other only in their eigenstates that could contain record of the state of S, e.g.,: *ρ*E|*k* = ∑*m pm*|*<sup>ε</sup>m*|*k<sup>ε</sup>m*|*k*|. However, Tr*ρ*E|*jρ*E|*k* = ∑*m pm*<sup>2</sup>|*<sup>ε</sup>m*|*j*|*<sup>ε</sup>m*|*k*-|2, coincides with Tr*ρ*E 2 iff |*<sup>ε</sup>m*|*j*|*<sup>ε</sup>m*|*k*-|2 = 1 whenever *pm* = 0. It follows that *ρ*E|*j* = *ρ*E|*k*. Therefore, unless *sj*|*sk*- = 0, states - *sj* and |*sk*- cannot leave any imprint that distinguishes them—cannot deposit any record—in E.

In other words, in case of mixed target we can establish our key result using only pure states in an enlarged Hilbert space (purification), or only density matrices. The only reason one might want to invoke Born's rule is to provide a physically (rather than only mathematically) motivated bridge between these two representations of "impure" states of E. Such a bridge is obviously useful, but it is not essential in arriving at the desired conclusions we reach in this section.

The economy of our assumptions stands in stark contrast with the uncompromising nature of our conclusions: Predictability—the demand that information transfer preserves the state of the system (embodied in postulate (iii))—was, along with the superposition principle (i) and unitarity of quantum evolutions (ii)—key to our derivation of the discreteness of states that can be repeatedly accessed. Discrete terminal states are behind the inevitability of quantum jumps.

We shall see in Section 4 that existence of stable terminal points allows for amplification and for the resulting preponderance of records about the states in which the system persist—in spite of the coupling to the environment—for long time periods. These sojourns of predictable evolution can be occasionally interrupted by a jump into another stable terminal state caused by perturbations that do not commute with the pointer observables monitored by the environment.

#### *2.3. Predictability Killed the (Schrödinger's) Cat*

There are several ways to describe our conclusions so far. To restate the obvious, we have established that repeatedly accessible outcome states must be orthogonal. This is the interpretation—independent part of axiom (iv)—all of it except for the literal collapse. The core quantum postulates alone make it impossible to find out preexisting quantum states.

This is enough for the relative state account of quantum jumps—collapse axiom (iv) is not necessary for that. So, a cat suspended between life and death [20] cannot be seen in the records it leaves in the monitoring environment. Repeated records of only one of these two options will be available because only the two stable states (unperturbed by copying) allow for repeatability (postulate (iii))—for predictability (hence the above title).

Another way of stating our conclusion is to note that a set of orthogonal states defines a Hermitian operator when supplemented with real eigenvalues. The above discussion is then a derivation of the Hermitian nature of observables. It justifies the focus on Hermitian operators often invoked in textbook version of measurement axioms [3].

We note that "strict repeatability" (that is, assertion that states {|*sk*-} cannot change at all in the course of a measurement) is not needed: They can evolve providing that their scalar products remain unaffected. That is,

$$\sum\_{j,k} a\_j^\* a\_k \langle s\_j | s\_k \rangle = \sum\_{j,k} a\_j^\* a\_k \langle \vec{s}\_j | \vec{s}\_k \rangle \langle \varepsilon\_j | \varepsilon\_k \rangle \tag{2.8}$$

leads to the same conclusions as Equation (2.2) as long as *sj*|*sk*- = *s*˜*j*|*s*˜*k*-. Thus, when - *s*˜*j* and |*s*˜*k*- are related with their progenitors by a transformation that preserves scalar product (as would, e.g., any reversible evolution) the proof of orthogonality goes through unimpeded. Both unitary and antiunitary transformations are in this class. Other similar generalizations are also possible [50,51].

We can also consider situations when this is not the case—*sj*|*sk*- = *s*˜*j*|*s*˜*k*-. An extreme example of this arises when the state of the measured system retains no memory of what it was before (e.g., - *sj* ⇒ |0-, |*sk*- ⇒ |0-). For example, photons are usually absorbed by detectors, and coherent states (that are not orthogonal) play the role of the outcomes. Then the apparatus can (and, indeed, by unitarity, has to) "inherit" the distinguishability—the information—previously residing in the system. In that case the need for orthogonality of - *sj* and |*sk*- disappears. Of course such measurements do not fulfill postulate (iii)—they are not repeatable.

We emphasize that Born's rule was not used above. The values of the scalar product that played a role in the proofs are *sj*|*sk*- = 0 or *sj*|*sk*- = 1, and the key distinction was between the zero and non-zero value of *sj*|*sk*-. Both "0" and "1" correspond to certainty. For instance, when we have asserted immediately below Equation (2.4) that *<sup>ε</sup>j*|*<sup>ε</sup>k*- = 1, this implies that these two states of E are certainly identical. We have therefore derived probability for a very special case already. We shall relying on this special case—certainty— in the derivation of probabilities in Section 3.

#### *2.4. Records and Branches: Degenerate "Control"*

Our discussion so far is based on one key assumption—repeatability of measurement outcomes—which we have usually simplified to mean "nondemolition measurements", i.e., repeatable accessibility of the same "original" state of the measured system. However, as we have already noted, for microscopic systems this is at best an exception. On the other hand, repeatability is essential for an apparatus A, at the level of measurement *records*. Pointer of an apparatus can be read out many times, and everyone should agree on where does it point—on what is the record. Indeed, this repeatable accessibility is a property of not just apparatus pointers, but a defining property of states that comprise "objective classical reality". So, while the repeatability postulate (iii) at the level of quantum systems is an idealization of a theorist (e.g., Dirac, [3]), persistence of records stored in A as well as of effectively classical states of macroscopic quantum systems we encounter in our everyday experience is an essential fact of life and, therefore, a key desideratum of a successful theory

of objective classical reality. Here, we extend our discussion of repeatability to account for it using our core quantum postulates.

We start by noting that one is almost never interested in the state of the apparatus as a whole: Finding out pure states of an object with Avogadro's number of atoms (and, hence, with Hilbert space dimension of the order of 1010<sup>23</sup> ) is impractical and unnecessary. Obviously, there are many microstates of the apparatus that correspond to the same memory state and yield the same readout. We have to modify our above "nondemolition" approach to allow for perturbations of the microscopic states and to account for this degeneracy. Once we have done this, we shall also find it easier to deal with mixed states of A that, for any macroscopic system, are certainly typical.

Consider two pure states |*<sup>v</sup>*1- and |*<sup>v</sup>*2- that represent the same record "*v*". We take this to mean that observers or other memory devices M, M, ... will register the same state after interacting with A in either |*<sup>v</sup>*1- or |*<sup>v</sup>*2-:

$$\begin{aligned} \left| \left| \upsilon\_{1} \right\rangle \left| \mu \right\rangle \left| \mu' \right\rangle \dots & \Rightarrow \left| \left| \psi\_{1} \right\rangle \left| \mu\_{\upsilon} \right\rangle \left| \mu' \right\rangle \dots \Rightarrow \left| \left| \overleftarrow{\psi}\_{1} \right\rangle \left| \mu\_{\upsilon} \right\rangle \left| \mu'\_{\upsilon} \right\rangle, \\ \left| \left| \upsilon\_{2} \right\rangle \left| \mu \right\rangle \left| \mu' \right\rangle \dots & \Rightarrow \left| \left| \psi\_{2} \right\rangle \left| \mu\_{\upsilon} \right\rangle \left| \mu' \right\rangle \dots \Rightarrow \left| \overleftarrow{\psi}\_{2} \right\rangle \left| \mu\_{\upsilon} \right\rangle \left| \mu'\_{\upsilon} \right\rangle. \end{aligned} \tag{2.9v}$$

Note that evolution of the "original" is allowed (e.g., |*<sup>v</sup>*1- ⇒ |*v*˜1- ⇒ | ˜*v*˜1-). as long as it does not affect the repeatability of what is read out by M, M, ....

It is straightforward to see that any superposition or any mixed state of |*<sup>v</sup>*1- and |*<sup>v</sup>*2- will also register the same way—as |*μv*-—in the memory M. Registration of a different outcome—different readout *w*—by memory M can be represented as:

$$\begin{aligned} \left| \left| w\_1 \right> \left| \mu \right> \left| \mu' \right> \dots \Rightarrow \left| \left| \vec{w}\_1 \right> \left| \mu\_{\left< \nu \right>} \right> \left| \mu' \right> \dots \Rightarrow \left| \vec{w}\_1 \right> \left| \left| \mu\_{\left< \nu \right>} \right> \right| \left| \mu'\_{\left< \nu \right>} \right>, \\ \left| w\_2 \right> \left| \mu \right> \left| \mu' \right> \dots \Rightarrow \left| \left| \vec{w}\_2 \right> \left| \mu\_{\left< \nu \right>} \right> \left| \mu' \right> \dots \Rightarrow \left| \vec{w}\_2 \right> \left| \mu\_{\left< \nu \right>} \right> \left| \mu'\_{\left< \nu \right>} \right>. \end{aligned} \tag{2.9w}$$

Again, there are many microstates—|*<sup>w</sup>*1-, |*<sup>w</sup>*2-, etc.—that yield the same readout |*μw*-.

The above account offers a model of what happens when an apparatus A is consulted by many observers that can be represented by distinct M's. They can perturb the microstate but leave the record intact.

We can now repeat the pure state reasoning from above, assuming that the "control"— which was before the measured system, and may now be the apparatus A—is in a pure state. We are led to an analogue of Equation (2.4) that can be satisfied in two different ways: Either the memory devices register the same readout regardless of the underlying microscopic state of the system, as in:

$$
\forall\_{k,l} \ \langle \boldsymbol{\upsilon}\_k | \boldsymbol{\upsilon}\_l \rangle = \langle \widetilde{\boldsymbol{\upsilon}}\_k | \widetilde{\boldsymbol{\upsilon}}\_l \rangle \langle \boldsymbol{\mu}\_{\overline{\boldsymbol{v}}} | \boldsymbol{\mu}\_{\overline{\boldsymbol{v}}} \rangle,\tag{2.10}
$$

(so that *μv*|*μv*- = 1, in which case scalar products between the underlying states of A can take any value), or the readouts can differ,

$$
\forall\_{k,l} \ \langle \upsilon\_k | \upsilon\_l \rangle = \langle \psi\_k | \vec{\upsilon} \eta \rangle \langle \mu\_v | \mu\_w \rangle,\tag{2.11}
$$

and *vk*|*wl*- have to be orthogonal when they lead to distinct records in M, M, ....

The relation between the states of the control defined by the readout—by the imprints they leave on the state on the "target" M—is reflexive, symmetric and transitive. Hence, it defines equivalence classes: States that leave imprint "*v*" form a class V distinct from states in W that imprint "*w*". Record states in V (W, etc.) should retain class membership under the evolutions generated by readouts (otherwise they cannot be repeatedly consulted and keep the record). It is natural to represent such equivalence classes of states with orthogonal subspaces in the Hilbert space HA. It is also possible to define probabilities as measures on such equivalence classes and regard them as (macroscopic or coarse-grained) "events" (see e.g., Gnedenko, 1968, [52]).

Generalization to when the apparatus is in a mixed state can be carried out using purification strategy as before (this time purifying the "control" A) or by using preservation of the Hilbert-Schmidt product. Thus, unitary evolutions;

$$
\rho \overline{\wp}^4 |\mu\rangle\langle\mu| \implies \overline{\wp}^4\_{\overline{\wp}} |\mu\_{\overline{\wp}}\rangle\langle\mu\_{\overline{\wp}}|,\tag{2.12v}
$$

$$
\langle \rho\_W^A | \mu \rangle \langle \mu | \implies \overline{\rho}\_W^A | \mu\_W \rangle \langle \mu\_W | \, \_\prime \tag{2.12w}
$$

where *ρ*A V (*ρ*<sup>A</sup> W, etc.) is any density matrix with support restricted to only V (W, etc.) imply equality:

$$\left| \text{Tr} \rho\_{\mathcal{V}}^{\mathcal{A}} \rho\_{\mathcal{W}}^{\mathcal{A}} | \langle \mu | \mu \rangle |^{2} = \text{Tr} \tilde{\rho}\_{\mathcal{V}}^{\mathcal{A}} \tilde{\rho}\_{\mathcal{W}}^{\mathcal{A}} | \langle \mu\_{\mathcal{V}} | \mu\_{\mathcal{W}} \rangle |^{2}. \tag{2.13}$$

This is an analogue of the derivation of Equations (2.4)–(2.7) when the mixed state of the control is represented by a density matrix. As before, we conclude that Tr*ρ*<sup>A</sup> V *ρ*A W = 0 unless |*μv*|*μw*-|2 = 1.

In contrast to Equation (2.6) (where "control" S was pure, but the state of the "target" E was mixed) now the target is the memory M, and its state starts and remains pure. This shifting of "mixedness" from the target (as in Equation (2.7)) to the control may seem somewhat arbitrary, but—in the present setting—it is well justified. The motivation before was the process of decoherence or measurement, and the focus of attention was the system S. Now, the motivation is the readout of the state of the apparatus pointer by observers (but information flow in decoherence and in quantum Darwinism we discuss in Section 4 can be treated in the same manner).

#### 2.4.1. Repeatability and Actionable Information

Previously we have modeled the acquisition of information about a system by a (possibly macroscopic) apparatus or by the environment in the course of decoherence. In either case "target" could be expected to be in a mixed state but the "control" was pure. Now we are dealing with an apparatus acting as a macroscopic control. Its microscopic state is in general mixed, and can be influenced by the readout, but we still expect it to retain the record (e.g., of a measurement outcome). This is possible because of degeneracy—many microscopic states represent the same record.<sup>3</sup>

This record should be repeatedly accessible and unambiguous. Before, in the discussion following Equations (2.1)–(2.4), repeatability was assured by insisting that the state of the system—of the control—should remain unchanged during the readout. Now, we can no longer count on the preservation of the *state* of the original to establish repeatability. Instead, we demand—as a criterion for repeatable accessibility—that; (i) the copies should contain the same information, and; (ii) that information should suffice to distinguish record V from W.

Above, we have seen how this demand can be implemented when the states of the memories M,M, etc. are pure. Relaxing the assumption of pure memory states is possible. One can also allow for decoherence caused by the environment E. Thus, consider sequence of copying operations that, along with decoherence, lead to:

$$
\rho\_V^A \rho\_0^{\mathcal{M}} \rho\_0^{\mathcal{M}'} \dots \rho\_0^{\mathcal{E}} \implies \rho\_V^{A, \mathcal{M}, \mathcal{M}' \dots \mathcal{E}},\tag{2.14v}
$$

$$
\rho\_{\mathcal{W}}^{\mathcal{A}} \rho\_0^{\mathcal{M}} \rho\_0^{\mathcal{M}'} \dots \rho\_0^{\mathcal{E}} \implies \rho\_{\mathcal{W}}^{\mathcal{A}, \mathcal{M}, \mathcal{M}' \dots \mathcal{E}}.\tag{2.14w}
$$

.

Note that we allow the apparatus A that contains the original record, various memories, as well as E to remain correlated. Such a general final state suggests an obvious question: How can we test whether, say, memory M has indeed acquired a copy of the record in A that offers (at least partial) distinguishability of V from W?

To address this question we propose an operational criterion: The information contained in each of the memories should be *actionable*—it should allow one to alter the state

<sup>3</sup> Schrödinger cat comes to mind, with many microscopic states consistent with "alive" or "dead" 

of a test system T . Thus, copy in M will be certified as "actionable" when there is a conditional unitary transformation *U*(T |M) that alters the state of the test system so that:

$$
\rho\_{\mathcal{V}}^{\mathcal{A},\mathcal{M},\mathcal{M}'-\mathcal{E}} \otimes \rho\_0^{\mathcal{T}} \overset{\mathcal{U}(\mathcal{T}|\_{\mathcal{M}}^{\mathcal{M}})}{\underset{\cdot}{\longrightarrow}} \rho\_{\mathcal{V}}^{\mathcal{A},\mathcal{M},\mathcal{M}'-\mathcal{E}} \otimes \rho\_{\mathcal{V}'}^{\mathcal{T}} \tag{2.15v}
$$

$$
\rho\_{\mathcal{W}}^{\mathcal{A}\mathcal{M}\mathcal{M}^{\prime}\dots\mathcal{E}} \otimes \rho\_{0}^{\mathcal{T}} \stackrel{\mathcal{U}(\mathcal{T}|\_{\mathcal{M}}^{\mathcal{M}})}{\Longrightarrow} \tilde{\rho}\_{\mathcal{W}}^{\mathcal{A}\mathcal{M}\mathcal{M}^{\prime}\dots\mathcal{E}} \otimes \rho\_{\mathcal{W}}^{\mathcal{T}}.\tag{2.15w}$$

The test of actionability will be successful—the information in M will be declared actionable— when there exists an initial state *ρ*T0 of the test system such that:

$$\text{Tr}(\rho\_0^T)^2 > \text{Tr}\rho\_V^T \rho\_W^T. \tag{2.16a}$$

Preservation of Schmidt–Hilbert norm under unitary evolutions implies that—unless *<sup>ρ</sup>*AMM...<sup>E</sup> V and *<sup>ρ</sup>*AMM...<sup>E</sup> W are orthogonal to begin with—the overlap between the two "branches" should increase to compensate for the decrease of overlap in the test system states, Equation (2.16a), so that;

$$\text{Tr}\rho^{A\mathcal{M}\mathcal{M}'\dots\mathcal{E}}\_{\mathcal{V}}\rho^{A\mathcal{M}\mathcal{M}'\dots\mathcal{E}}\_{\} \prec \text{Tr}\bar{\rho}^{A\mathcal{M}\mathcal{M}'\dots\mathcal{E}}\_{\mathcal{V}}\bar{\rho}^{A\mathcal{M}\mathcal{M}'\dots\mathcal{E}}.\tag{2.16b}$$

Moreover, as we have assumed that in M, M...M(*k*)... there are many copies of the original record in A, this test of actionability can be repeated. However, the overlap of the density matrices of AMM...E on the RHS above cannot increase indefinitely as a result of such multiple iterations of actionability tests, as it is bounded from above by unity. Consequently, we conclude via this *reductio ad absurdum* reasoning that the ability to make multiple copies implies Tr*ρ*AMM...<sup>E</sup> V*<sup>ρ</sup>*AMM...<sup>E</sup> W = 0. Therefore,

$$\text{Tr}\rho \vec{\sqrt{\rho}} \rho \vec{\sqrt{\nu}} = 0 \tag{2.17}$$

is needed to allow for repeatable copying of the original record (or, more generally, the features distinguishing V from W of the original macroscopic state) in A.

We have assumed above that there is no preexisting correlation between the test system and the rest, AMM...M(*k*)...E. We could have actually assumed a pure state of T : When there is a *U*(T |M) that alters a mixed state of T , there will certainly be a pure state of T that can be altered.

We also note that (as before) the whole argumen<sup>t</sup> can be recast in the language of the "Church of Larger Hilbert Space" [43]. That is, one can carry it out without any appeals to Born's rule. There is an interesting subtlety in such treatments: The actionable information we have tested for above is local—it resides in a specific M(*k*). This need not be always the case: Actionable information may be nonlocal—it may reside in correlations between systems. Such an example is discussed in [43]. The locus of actionable information is assured by the selection of the conditional evolution operator *U*(T |M(*k*)) that—above— couples only specific M(*k*) to T .

We conclude that only orthogonal projectors (above, of A) can act as "originals" for unlimited numbers of copies. Of course, many of the outcome states of quantum system S inferred from the measurement records in A are not orthogonal. Measurements that result in such outcomes are not repeatable: State of S is perturbed, but its record in A is repeatedly accessible, so there is no contradiction. Repeatability of the records is therefore possible even when the recorded states of S at the roots of the corresponding branches are not orthogonal. Positive operator valued measures (POVM's, that is generalized measurements with outcomes that do not represent orthogonal states of the measured system, see [5]) arise naturally in this setting [43].

The reasoning behind the conclusions of this subsection parallels the pure states case, Equations (2.1)–(2.4), but the mathematics and, above all, the physical motivation, differ. Before we were dealing with the abstract postulate of repeatability that is found in Dirac [3] and other textbooks, but this idealized version is almost never implemented in the laboratory practice in measurements of microscopic quantum systems. In spite of its

idealizations, the abstract Dirac version of repeatability of measurements should not be dismissed too easily: Being able to confirm that a state is what it is known to be is essential to justify the very idea of a state in general, and of a quantum state in particular. The role of the state is, after all, to enable predictions, and the simplest prediction (captured by repeatability, no matter how difficult nondemolition measurements are to implement in practice) is that existence of a state can be confirmed. Indeed, this is how quantum states can give rise to "existence" we have become accustomed to in our quantum Universe.

In a classical Universe repeatability is taken for granted, as an unknown classical state can be found out without endangering its existence. Repeatability in a quantum setting allows one to use fragile quantum states as building blocks of classical reality, as we shall see in more detail in Section 4.

In practice predictability and even repeatability are encountered not in the measured microscopic quantum system S but, rather, in the memory of the measuring apparatus A, and, indeed, in the states of macroscopic systems. Apparatus pointer can be, after all, repeatedly consulted, as can be any effectively classical state. Moreover, our perception of the collapse arises not from the direct evidence of the behavior of some microscopic quantum S, but, rather, from the records of its state inscribed in the memory of a macroscopic (albeit still quantum) A.

We have seen that the same condition of repeatability that led to orthogonality (and, hence, discreteness) in the set of possible outcomes in the pure case of S enforces orthogonality of the subspaces of A (even when the microscopic state of A is allowed to change). Thus, while Equations (2.1)–(2.4) account for quantum jumps in the idealized case of quantum postulates (Dirac, 1958) [3], this subsection shows that discrete quantum jumps can occur as a result of orthogonality of the whole subspaces of the Hilbert space HA corresponding to repeatedly accessible records—to macroscopic pointer subspaces of the measuring device.

#### *2.5. Pointer Basis, Information Transfer, and Decoherence*

We are now equipped with a set of measurement outcomes or, to put it in a way that ties in with the study of probabilities we shall embark on in Section 3, with a set of possible *events*. Our derivation above did not appeal to decoherence, but decoherence yields einselection (which is, after all, due to the information transfer to the environment). We will now see that einselection based on repeatability and einselection based on decoherence are in effect two views of the same phenomenon.

Popular accounts of decoherence and its role in the emergence of 'the classical' often start from the observation that when a quantum system S interacts with some environment E "phase relations in S are lost". This is, at best, incomplete if not misleading, as it begs the more fundamental question: "Phases between *what*?". This in turn leads directly to the main issue addressed by einselection: "*What is the preferred basis*?". This key question is often muddled in the "folklore" accounts of decoherence.

The crux of the matter—the reason why interaction with the environment can impose classicality—is precisely the emergence of the preferred states. The basic criterion that selects preferred pointer states was discovered when the analogy between the role of the environment in decoherence and the role of the apparatus in a nondemolition measurement was recognized: What matters is that there are interactions that transfer information and ye<sup>t</sup> leave some states of the system unaffected [24].

The criterion for selecting such preferred states is persistence of correlation between two systems (e.g., system S and apparatus A). For the preferred pointer states this correlation should persist in spite of immersion of A in the environment. It is obvious that states (of, e.g., A) that are best at retaining correlations (with, e.g., S) also retain identity—i.e., correlation with me, the observer—and resist entanglement with the environment.

Our discussion above confirms that the simple idea of preserving a state while transferring the information about it—also the central idea of einselection—is powerful, and can be analyzed using minimal purely quantum ingredients—core postulates (o)–(iii). It leads to

breaking of the unitary symmetry and singles out preferred states of the apparatus pointer (supplied in textbooks by axiom (iv)) without any need to invoke physical (statistical) view of the reduced density matrices (which is central to the decoherence approach to collapse).

This is important, as partial trace (understood as an averaging procedure) and reduced density matrices (understood as probability distributions) employed in decoherence theory rely on Born's rule (which endows them with physical significance). Our goal in the next section will be to arrive at Born's rule, axiom (v)—to relate state vectors and probabilities. Obtaining preferred basis and deducing events without invoking density matrices and trace—without relying on Born's rule—is essential if we are to avoid circularity in its derivation.

To compare derivation of the preferred states in decoherence with their emergence from symmetry breaking imposed by axioms (o)–(iii) we return to Equation (2.2). We also temporarily suspend prohibition on the use of partial trace to compute reduced density matrix of the system:

$$\rho\_{\mathcal{S}} = \sum\_{j,k} a\_j a\_k^\* \langle \varepsilon\_k | \varepsilon\_j \rangle \langle s\_j \rangle \langle s\_k | = Tr\_{\mathcal{E}} | \Psi\_{\mathcal{E}\mathcal{S}} \rangle \langle \Psi\_{\mathcal{E}\mathcal{S}} |. \tag{2.18}$$

Above we have expressed *ρ*S in the pointer basis defined by its resilience in spite of the monitoring by E and not in the Schmidt basis. Therefore, until decoherence in that basis is complete, and the environment acquires perfect records of pointer states;

$$
\langle \varepsilon\_j | \varepsilon\_k \rangle = \delta\_{jk\prime} \tag{2.19}
$$

the eigenstates of *ρ*S do not coincide with the pointer states selected for their resilience in spite of the immersion in E.

Resilience—quantified by the ability to retain correlations in spite of the environment, and, hence, by persistence, as in Equations (2.4) and (2.7)—is the essence of the original definition of pointer states and einselection [24,25]. Such pointer states will be in general different from the instantaneous Schmidt states of S—the eigenstates of *ρ*S. They will coincide with the Schmidt states of

$$\left| \Psi\_{\mathcal{ES}} \right> = \sum\_{k} a\_{k} \left| s\_{k} \right> \left| \varepsilon\_{k} \right> \tag{2.20}$$

only when {|*<sup>ε</sup>k*-}—their records in E—become orthogonal. We did not need orthogonality of {|*<sup>ε</sup>k*-} to prove orthogonality of pointer states earlier in this section. It will be, however, useful in the next section, as it assures additivity of probabilities of the pointer states.

For pure states this discussion of additivity can be carried out in a setting that is explicitly free of any reference to density matrices or trace, and relies only on correlations (Zurek, 2005) [48]. Born's rule would be needed to establish the connection between them and to endow reduced density matrix with physical (statistical) interpretation, but—as we have seen—orthogonality of outcomes central for the definition of events can be established without Born's rule.

So, a piece of decoherence "folklore"—responsible for statements such as "decoherence causes reduced density matrix to be diagonal"—is at best imprecise, and often incorrect. The error is mathematical and obvious: *ρ*S is Hermitian, so it is always diagonalized by the Schmidt states of S. In addition, what we want in pointer states is preservation of their identity.

Still, "folklore" often assigns classicality to the eigenstates of *ρ*S, and that would make them candidate to the status of events. This was even occasionally endorsed by some of the practitioners of decoherence (Zeh, 1990; [53]; 2007 [54]; Albrecht, 1992 [55], but see Albrecht et al., 2021 [56,57]) and taken for granted by others (see, e.g., [58]). However, by and large it is no longer regarded as viable [31,32,36]: The eigenstates of *ρ*S are not stable. They depend on the time and on the initial state of S, which disqualifies them as events in the sense needed to develop probability theory, and do not fit the bill as "elements of

classical reality". There are also situations where the eigenstates of *ρ*S can be (in contrast to pointer states that are local whenever the coupling with the environment is local) very nonlocal [59–61].

Nonlocality of pointer states need not necessarily be a problem. The role of decoherence is to predict what happens—what states are "pointer" given the physical context (including Hamiltonians, the nature of the environment, etc.). Thus, testing it in situations when its predictions clash with our classical intuition is of interest (see, e.g., Poyatos, Cirac, and Zoller, 1996 [62]) . The problem with the eigenstates of *ρ*S is—primarily—their dependence on the initial state of the system. This is eliminated by the predictability sieve ([6,28,63]) and the repeatability-based approach (see also Refs. [42,43,49]).

As is often the case with folk wisdom, a grain of truth is nevertheless reflected in such oversimplified "proverbs": When the environment acquires perfect knowledge of the states it monitors without perturbing them, and *<sup>ε</sup>j*|*<sup>ε</sup>k*- = *<sup>δ</sup>jk*, pointer states "become Schmidt", and end up on the diagonal of *ρ*S. Effective decoherence favors such alignment of Schmidt states with the pointer states. Given that decoherence is—at least in the macroscopic domain—very fast, this can happen essentially instantaneously.

Still, this coincidence should not be used to attempt a redefinition of pointer states as instantaneous eigenstates of *ρ*S—instantaneous Schmidt states. As we have already seen, and as will become even clearer in the rest of this paper, it is essential to distinguish the process that fixes preferred pointer states (that is, dynamics of the information transfer that results in the measurement as well as decoherence, but does not depend on the initial state of the system) from the reasoning that assigns probabilities to these outcomes. These probabilities are determined by the initial state.

#### *2.6. Irreversibility of Perceived Events, or "Don't Blame the* 2*nd Law—Wavepacket Collapse Is Your Own Fault!"*

Irreversibility has been blamed for the collapse of the wavepacket since at least von Neumann (1932) [4]. The causes of irreversibility invoked in this context have typically classical analogues that go back to Boltzmann and the loss of information implicated in the Second Law (Zeh, 2007) [54].

Discrete quantum jumps occur as a consequence of the collapse. They are uniquely quantum, and a central conundrum of quantum physics. They reset the evolution relevant for the future of the observer putting it onto a course consistent with the measurement outcome (and prima facie at odds with the unitarity of quantum evolutions).

We have just seen how the discreteness of quantum jumps follows from the quantum core postulates. We now point out that—in addition to the "usual suspects" traditionally blamed for irreversibility—there is a uniquely quantum reason why events associated with quantum jumps are fundamentally irreversible. It is distinct from the information loss associated with the dynamics that is responsible for the Second Law.

This uniquely quantum source of irreversibility is a result of the information *gain* (rather than its loss). It is noteworthy that quantum physics provides a uniquely quantum key that solves the distinctly quantum conundrum of the wavepacket collapse.

We shall see below that information about the measurement outcome does not preclude reversal of the classical measurement, but makes it impossible to undo evolution that leads to a quantum measurement whenever a superposition of the potential outcomes—hence, the wavepacket collapse—is involved.

2.6.1. Classical Measurement Can Be Reversed Even when Record of the Outcome is Kept

Let us first examine a measurement carried out by a classical agent/apparatus **A** on a classical system **S**. The state s of **S** (e.g., location of **S** in phase space) is measured by a classical **A** that starts in the "ready to measure" state A0:

$$\mathfrak{a}\mathsf{A}\_{\mathsf{O}} \stackrel{\mathfrak{C}\_{\mathsf{S},\mathsf{A}}}{\rightleftharpoons} \mathfrak{a}\mathsf{A}\_{\mathsf{n}}.\tag{2.21a}$$

The question we address is whether the combined state of **SA** can be restored to the premeasurement sA0 even after the information about the outcome is retained somewhere— e.g., in the memory device **D**.

The dynamics E*SA* responsible for the measurement is assumed to be reversible and, in Equation (2.21*a*), it is classical. Therefore, classical measurement can be undone simply by implementing E−<sup>1</sup> *SA*. An example of E−<sup>1</sup> *SA* is (Loschmidt inspired) instantaneous reversal of velocities.

Our main point is that the reversal;

$$\mathfrak{a}\mathfrak{A}\_{\mathfrak{A}} \stackrel{\mathfrak{C}\_{\mathfrak{A}}^{-1}}{\rightleftharpoons} \mathfrak{a}\mathfrak{A}\_{\mathbb{O}}.\tag{2.21a'}$$

can be accomplished even after the measurement outcome is copied onto the memory device **D**;

$$\mathsf{B}\mathsf{A}\_{\mathsf{B}}\mathsf{D}\_{\mathsf{O}} \stackrel{\mathfrak{C}\_{\mathsf{A}\mathsf{D}}}{\rightleftharpoons} \mathsf{B}\mathsf{A}\_{\mathsf{B}}\mathsf{D}\_{\mathsf{B}}\tag{2.22a}$$

so that the pre-measurement state of **S** is recorded elsewhere (here, in **D**). Above, E*AD* plays the same role as E*SA* in Equation (2.21*a*). That is, the states of **S** and **A** separately, or the combined state **SA** will not reveal any evidence of irreversibility. After the reversal;

$$\mathfrak{a}\mathsf{A}\_{\mathfrak{a}}\mathsf{D}\_{\mathfrak{a}} \stackrel{\mathfrak{a}\_{\overline{\operatorname{SA}}}^{-1}}{\longrightarrow} \mathfrak{a}\mathsf{A}\_{\mathsf{O}}\mathsf{D}\_{\mathfrak{a}\mathsf{A}}\tag{2.23a}$$

the state of **SA** is identical to the pre-measurement state, even though the recording device still has the copy of the outcome. Starting with a partly known state of the system does not change this conclusion [23].

#### 2.6.2. Quantum Measurement Can't Be Reversed when the Record of the Outcome is Kept

Consider now a measurement of a quantum system S by a quantum A:

$$\left(\sum\_{\mathbf{s}} \alpha\_{\mathbf{s}} |s\rangle\right) |A\_0\rangle \stackrel{\natural \mathcal{L}\_{\mathbf{s}} \mathcal{A}}{\Longrightarrow} \sum\_{\mathbf{s}} \alpha\_{\mathbf{s}} |s\rangle |A\_{\mathbf{s}}\rangle. \tag{2.21b}$$

The evolution operator USA is unitary (for example, USA = ∑*<sup>s</sup>*,*<sup>k</sup>* |*ss*||*Ak*<sup>+</sup>*s*-*Ak*| with orthogonal {|*s*-}, {|*Ak*-} would do the job). Therefore, evolution that leads to a measurement is in principle reversible. Reversal implemented by <sup>U</sup>†SA will restore the pre-measurement state of SA:

$$
\sum\_{\mathbf{s}} \alpha\_{\mathbf{s}} |\mathbf{s}\rangle |A\_{\mathbf{s}}\rangle \xrightarrow{\Delta t\_{S,d}^{\dagger}} \left(\sum\_{\mathbf{s}} \alpha\_{\mathbf{s}} |\mathbf{s}\rangle\right) |A\_{0}\rangle. \tag{2.21b'}
$$

Let us, however, assume that the measurement outcome is copied before the reversal is attempted:

$$\mathbb{E}\left(\sum\_{\mathbf{s}} \alpha\_{\mathbf{s}} |\mathbf{s}\rangle \, |A\_{\mathbf{s}}\rangle\right) |D\_0\rangle \stackrel{\scriptstyle{\underline{\mathbf{M}}}{\Longrightarrow}}{\Longrightarrow} \mathbb{E}\_{\mathbf{s}} \left|\mathbf{s}\right\rangle |A\_{\mathbf{s}}\rangle \, |D\_{\mathbf{s}}\rangle. \tag{2.22b}$$

Here UAD plays the same role and can have the same structure as USA.

Note that unitary evolutions above implement *repeatable* measurement/copying on the states {|*s*-}, {|*As*-} of the system and of the apparatus, respectively. That is, these states of S and A remain untouched by the measurement and copying processes. As we have seen, such repeatability implies that the outcome states {|*s*-} as well as the record states {|*As*-} are orthogonal.

When the information about the outcome is copied, the pre-measurement state ∑*s <sup>α</sup>s*|*s*-|*<sup>A</sup>*0- of SA pair cannot be restored by <sup>U</sup>†SA. That is:

$$\left(\Delta\_{\mathcal{SA}}^{\dagger}\left(\sum\_{\mathbf{s}} \alpha\_{\mathbf{s}} |\mathbf{s}\rangle \, |A\_{\mathbf{s}}\rangle \, |D\_{\mathbf{s}}\rangle\right)\right) \\ = |A\_0\rangle\left(\sum\_{\mathbf{s}} \alpha\_{\mathbf{s}} |\mathbf{s}\rangle \, |D\_{\mathbf{s}}\rangle\right). \tag{2.23b}$$

The apparatus is restored to the pre-measurement |*<sup>A</sup>*0-, but the system remains entangled with the memory device. On its own, its state is represented by the mixture:

$$\boldsymbol{\varrho}^{\mathcal{S}} = \sum\_{\mathbf{s}} w\_{\mathbf{s}\mathbf{s}} |\mathbf{s}\rangle \langle \mathbf{s}|\_{\mathcal{A}}$$

where *wss* = |*<sup>α</sup>s*| 2. Reversing quantum measurement of a state that corresponds to a superposition of the potential outcomes is possible only providing the memory of the outcome is no longer preserved anywhere else in the Universe. Moreover, that means that the information transfer has to be "undone"—scrambling the record makes it inaccessible, but does not ge<sup>t</sup> rid of the evidence of the outcome.

We have now demonstrated the difference between the ability to reverse quantum and classical measurement. Information flows do not matter for classical, Newtonian dynamics. However, when information about a quantum measurement outcome is communicated— copied and retained by any other system—the evolution that led to that measurement cannot be reversed.

Quantum irreversibility can result from the information gain rather than just its loss— rather than just an increase of the (von Neumann) entropy. Recording of the outcome of the measurement resets, in effect, initial conditions within the observer's (branch of) the Universe, resulting in an irreversible, uniquely quantum "wavepacket collapse". Thus, from the point of view of the measurer, information retention about an outcome of a quantum measurement implies irreversibility. Quantum states are epiontic.

#### *2.7. Summary: Events, Irreversibility, and Perceptions*

What the observer knows is inseparable from what the observer is: the physical state of the agent's memory represents the information about the Universe. The reliability of this information depends on the stability of its correlation with external observables. In this very immediate sense, decoherence brings about the apparent collapse of the wavepacket: after a decoherence time scale, only the einselected memory states will exist and retain useful correlations [27,36,64,65]. The observer described by some specific einselected state (including a configuration of memory bits) will be able to access ("recall") only that state.

Collapse is a consequence of einselection and of the one-to-one correspondence between the physical state of the observer's memory and of the information encoded in it. Memory is simultaneously a description of the recorded information and part of an identity tag, defining the observer as a physical system. It is as inconsistent to imagine an observer perceiving something other than what is implied by the stable (einselected) records as it is impossible to imagine the same person with a different DNA. Both cases involve information encoded in a state of a system inextricably linked with the physical identity of an individual.

Distinct memory/identity states of the observer (which are also his states of knowledge) cannot be superposed. This censorship is strictly enforced by decoherence and the resulting einselection. Distinct memory states label and inhabit different branches of Everett's many-worlds universe. The persistence of correlations between the records (data in possession of the observers) and the recorded states of macroscopic systems is all that is needed to recover objective classical reality. In this manner, the distinction between ontology and epistemology—between what is and what is known to be—is dissolved. There can be *no information without representation* [66].Quantum states are *epiontic*.

The discreteness underlying "collapse of the wavepacket" has a well-defined origin— it resolves the conflict between the linearity of the unitary quantum evolutions and the nonlinearity associated with the amplification of information in measurements but also in the monitoring by the environment—in decoherence. Any process that involves (even modest) amplification—that leads to copies, qmemes of an "original state" (which, in view of the demand of repeatability, should survive the copying)—demands orthogonality.

Copying (as any other quantum evolution) is unitary, so it will not result in collapse. However, perception of the collapse will arise as a consequence of the irreversibility induced

by the information transfer we have just discussed. This purely quantum irreversibility provides a mechanism for collapse of the wavepacket that was not available (and not needed) in the classical setting—a mechanism that is fundamental, and uniquely quantum. The old question about the origins of irreversibility acquires a new quantum aspect especially apparent in the context of quantum measurements: Thus, while in the classical setting measuring of an evolving state of the system need not alter its evolution, in the quantum setting measurement derails evolution and redirects it onto the track consistent with the record made by the observer. One might say that a measurement re-sets the initial condition of the evolving quantum system [23].

Thus, while the irreversible wavepacket collapse was sometimes blamed on the consciousness of the observer (von Neumann, 1932 [4]; London and Bauer, 1939 [67]; Wigner, 1961 [68]), we have identified a purely physical cause of the collapse: Observer retaining information about the outcome precludes the reversal.

In the next section, we derive Born's rule. We build on einselection, but in a way ([7,35,48]) that does not rely on axioms (iv) or (v). In particular, the use of reduced density matrices we allowed temporarily shall be prohibited. We shall be able to use them again only after Born's rule has been derived.

From the point of view of axiom (v) and the rest of this paper the most important conclusion of the present section is that *repeatability requires distinguishability*. In a quantum setting of Hilbert spaces and unitarity of evolutions—postulates (i) and (ii)—this means that *repeatability begets orthogonality*. to assure repeatability—ability to reconfirm what is known—one must focus on mutually exclusive events represented by orthogonal states.

We end this section with a simple purely quantum definition of events in hand: Record made in the measurement resets initial conditions relevant for the subsequent evolution of the branch of the universal state vector tagged by that record. We now take up the question: What is the probability of a particular record—specific new initial condition—given the preexisting superposition of the possible outcome states.

#### **3. Born's Rule from the Symmetries of Entanglement**

The first widely accepted definition of probability was codified by Laplace (1820) [69]: When there are *N* possible distinct outcomes and the observer is ignorant of what will happen, all alternatives appear equally likely. Probability observer should then assign to any one outcome is 1/*N*. Laplace justified this *principle of equal likelihood* using *invariance* encapsulated in his 'principle of indifference': Player ignorant of the face value of cards (Figure 2a) will be *indifferent* when they are swapped before he gets the card on top, even when one and only one of the cards is favorable (e.g., a spade he needs to win).

Laplace's invariance under swaps captures *subjective* symmetry: Equal likelihood is a statement about observers 'state of mind' (or, at best, his records), and *not* a measurable property of the real physical state of the system (which is, after all, altered by swaps; see Figure 2b). In the classical setting probabilities defined in this manner are therefore ultimately unphysical. Moreover, indifference, likelihood, and probability are all ill-defined attempts to quantify ignorance. Expressing one undefined concept in terms of another is not a definition.

It is therefore no surprise that equal likelihood is no longer regarded as a sufficient foundation for classical probability, and several other attempts are vying for primacy [70]. Among them, relative frequency approach has perhaps the largest following, although it needs infinite (hence, fictitious) ensembles (von Mises, 1939) [71], and, thus, it is doubtful if it addresses the issue of "subjectivity".<sup>4</sup> Nevertheless, popularity of relative frequency approach made it an obvious starting point in the attempts to derive Born's rule, especially in the relative states context where "branches" of the universal state vector can be counted. However, attempts to date (Everett [10,11]; DeWitt [14,15]; Graham [72]; Geroch [73]) have been found lacking. This is because counting of many world branches does not suffice.

<sup>4</sup> After all, there are no infinite ensembles in our Universe, so the ones used to compute relative frequencies are abstract imaginary ensembles defined by *subjective* extrapolation from an assessment based on finite data sets.

"Maverick" branches that have frequencies of events very different from those predicted by Born's rule are also a part of the universal state vector. Relative frequencies alone do not imply that an observer is unlikely to be found on such a branch. To ge<sup>t</sup> rid of them one would have to assign to them—without physical justification—vanishing probabilities related to their small amplitudes. This amplitude-probability connection goes beyond the relative frequency approach, in effect requiring—in addition to frequencies—another measure of probability. Papers mentioned above introduce it *ad hoc*. This is consistent with Born's rule, but deriving it on this basis is circular (Stein [74]; Kent [38]; Joos [75]; Weinberg [76]). Indeed, formal attempts based on the "frequency operator" lead to mathematical inconsistencies (Squires [39]).

**Figure 2. Probabilities and symmetry**: (**a**) Laplace (1820) [69] appealed to subjective invariance (associated with 'indifference' based on observer's ignorance of the real physical state) to define probability via *principle of equal likelihood*: When ignorance means observer is indifferent to swapping (e.g., of cards), alternative events should be considered equiprobable. So, for the cards above, subjective probability *p*♠ = 12 would be inferred by an observer who does not know their face value, but knows that one (and only one) of the two cards is a spade. (**b**) Nevertheless, the real physical state of the system is independent from what is known about it. Moreover, the order of the cards is altered by the swap—it is not 'indifferent'—illustrating subjective nature of Laplace's approach. Subjectivity of equal likelihood probabilities poses foundational problems for, e.g., statistical physics. This led to an alternative definition employing relative frequency—an objective property (albeit of a fictitious—and, hence, subjective—infinite ensemble). (**c**) Quantum theory allows for an objective definition of probabilities based on *a perfectly known state* of a composite system and on the symmetries of entanglement [7,35,48,77,78]. When two systems (S and E) are maximally entangled (i.e., Schmidt coefficients differ only by phases, as in the Bell state above), a swap |♠-♥| + |♥-♠| in S can be undone by 'counterswap' |♣-♦| + |♦-♣| in E. Certainty about entangled state of the whole in combination with the symmetry between |♠- and |♥-—certainty that *p*♠ = *p*♥—means that *p*♠ = *p*♥ = 12 , Equiprobability follows from the objective symmetry of entanglement. This *en*tanglement— assisted in*variance* (*envariance*) also establishes decoherence of Schmidt states, allowing for additivity of probabilities of the effectively classical pointer states. Probabilities derived through envariance quantify indeterminacy of the state of S alone given the global entangled state of SE.

The problem can be "made to disappear"—coefficients of maverick branches become vanishingly small (along with coefficients of *all* branches)—in the limit of *infinite* and fictitious (and, hence, subjectively assigned) ensembles (Hartle, 1968 [79]; Farhi, Goldstone, and Guttman, 1989 [80]). Such infinite ensembles—one might argue—are always required by the frequentist approach in the classical case, but this is a poor excuse (see Kent, 1990) [38]. As noted above, infinite ensembles are unphysical, subjective, and a weak spot of relative frequencies approach also in a classical setting. Moreover, in quantum mechanics infinite ensembles may pose problems that have to do with the structure of infinite Hilbert spaces (Poulin, 2005 [81]; Caves and Shack, 2005 [82]). It is debatable whether these mathematical problems are fatal, but it is also difficult to disagree with Kent (1990) [38] and Squires (1990) [39] that the need to go to a limit of infinite ensembles to define probability in a finite Universe disqualifies use of relative frequencies in the relative states setting.

The other way of dealing with this issue is to modify physics so that branches with small enough amplitude simply do not count (Geroch, 1984 [73]; Buniy, Hsu, and Zee, 2006 [83]). At least until experimental evidence for the required modifications of quantum theory is found one can regard such attempts primarily as illustration of the seriousness of the problem of the origin of Born's rule.

Kolmogorov's approach—probability as a measure (see, e.g., Gnedenko, 1968 [52])— bypasses the question we aim to address: How to relate probabilities to (quantum) states. It only shows that any sensible assignment (non-negative numbers that for a mutually exclusive and exhaustive set of events sum up to 1) will do. Moreover, Kolmogorov assumes additivity of probabilities while quantum theory insists—via superposition principle—on the additivity of state vectors. These two additivity requirements are at odds, as double slit experiment famously demonstrates.

Gleason's theorem (Gleason, 1957) [84] implements Kolmogorov's axiomatic approach to probability by looking for an additive measure on Hilbert spaces. It leads to Born's rule, but provides no physical insight into why the result should be regarded as probability. Clearly, it has not settled the issue: Rather, it is often cited [38,72,79,80] as a motivation to seek a physically transparent derivation of Born's rule.

We shall now demonstrate how quantum entanglement leads to probabilities based on a symmetry, but—in contrast to subjective equal likelihood based on ignorance—on an *objective* symmetry of *known* quantum states.
