Philosophical Issues in Quantum Theory

First published Mon Jul 25, 2016; substantive revision Wed Mar 23, 2022

This article is an overview of the philosophical issues raised byquantum theory, intended as a pointer to the more in-depth treatmentsof other entries in the Stanford Encyclopedia of Philosophy.

1. Introduction

Despite its status as a core part of contemporary physics, there is noconsensus among physicists or philosophers of physics on the questionof what, if anything, the empirical success of quantum theory istelling us about the physical world. This gives rise to the collectionof philosophical issues known as “the interpretation of quantummechanics”. One should not be misled by this terminology intothinking that what we have is an uninterpreted mathematical formalismwith no connection to the physical world. Rather, there is a commonoperational core that consists of recipes for calculatingprobabilities of outcomes of experiments performed on systemssubjected to certain state preparation procedures. What are oftenreferred to as different “interpretations” of quantummechanics differ on what, if anything, is added to the common core.Two of the major approaches, hidden-variables theories and collapsetheories, involve formulation of physical theories distinct fromstandard quantum mechanics; this renders the terminology of“interpretation” even more inappropriate.

Much of the philosophical literature connected with quantum theorycenters on the problem of whether we should construe the theory, or asuitable extension or revision of it, in realist terms, and, if so,how this should be done. Various approaches to what is called the“Measurement Problem” propose differing answers to thesequestions. There are, however, other questions of philosophicalinterest. These include the bearing of quantum nonlocality on ourunderstanding of spacetime structure and causality, the question ofthe ontological character of quantum states, the implications ofquantum mechanics for information theory, and the task of situatingquantum theory with respect to other theories, both actual andhypothetical. In what follows, we will touch on each of these topics,with the main goal being to provide an entry into the relevantliterature, including the Stanford Encyclopedia entries on thesetopics. Contemporary perspectives on many of the issues touched on inthis entry can be found inThe Routledge Companion to Philosophyof Physics (Knox and Wilson, eds., 2021);The Oxford Handbookof the History of Quantum Interpretations (Freire, et al. eds.,2022) contains essays on the history of discussions of theseissues.

2. Quantum Theory

In this section we present a brief introduction to quantum theory; seethe entry onquantum mechanics for a more detailed introduction.

2.1 Quantum states and classical states

In classical physics, with any physical system is associated a statespace, which represents the totality of possible ways of assigningvalues to the dynamical variables that characterize the state of thesystem. For systems of a great many degrees of freedom, a completespecification of the state of the system may be unavailable orunwieldy; classical statistical mechanics deals with such a situationby invoking a probability distribution over the state space of thesystem. A probability distribution that assigns any probability otherthan one or zero to some physical quantities is regarded as anincomplete specification of the state of the system. In quantummechanics, things are different. There are no quantum states thatassign definite values to all physical quantities, and probabilitiesare built into the standard formulation of the theory.

In formulating a quantum theory of some system, one usually beginswith the Hamiltonian or Lagrangian formulation of the classicalmechanical theory of that system. In the Hamiltonian formulation ofclassical mechanics, the configuration of a system is represented by aset of coordinates. These could be, for example, the positions of eachof a set of point particles, but one can also consider more generalcases, such as angular coordinates that specify the orientation of arigid body. For every coordinate there is an associatedconjugatemomentum. If the coordinate indicates the position of someobject, the momentum conjugate to that coordinate may be what weusually call “momentum,” that is, the velocity of the bodymultiplied by its mass. If the coordinate is an angle, the momentumconjugate to it is an angular momentum.

Construction of a quantum theory of a physical system proceeds byfirst associating the dynamical degrees of freedom withoperators. These are mathematical objects on which operationsof multiplication and addition are defined, as well as multiplicationby real and complex numbers. Another way of saying this is that theset of operators forms analgebra. Typically, it is said thatan operator represents anobservable, and the result of anexperiment on a system is said to yield a value for some observable.Two or more observables are said to becompatible if there issome possible experiment that simultaneously yields values for all ofthem. Others require mutually exclusive experiments; these are said tobeincompatible.

Of course, in a classical theory, the dynamical quantities that definea state also form an algebra also, as they can be multiplied andadded, and multiplied by real or complex numbers. Quantum mechanicsdiffers from classical mechanics in that the order of multiplicationof operators can make a difference. That is, for some operators\(A\),\(B\), the product \(AB\) is not equal to the product \(BA.\) If\(AB = BA,\) the operators are said tocommute.

The recipe for constructing a quantum theory of a given physicalsystems prescribes algebraic relations between the operatorsrepresenting the dynamical variables of the system. Compatibleobservables are associated with operators that commute with eachother. Operators representing conjugate variables are required tosatisfy what are called thecanonical commutation relations.If \(q\) is some coordinate, and \(p\) its conjugatemomentum, the operators \(Q\) and \(P\) representing themare required to not commute. Instead, the difference between\(PQ\) and \(QP\) is required to be a multiple of theidentity operator (that is, the operator \(I\) that satisfies,for all operators \(A\), \(IA = AI).\)

Aquantum state is a specification, for every experiment thatcan be performed on the system, of probabilities for thepossible outcomes of that experiment. These can be summed up as anassignment of an expectation value to each observable. These statesare required to belinear. This means that, if an operator\(C\), corresponding to some observable, is the sum of operators\(A\) and \(B\), corresponding to other observables, thenthe expectation value that a quantum state assigns to \(C\) mustbe the sum of the expectation values assigned to \(A\) and\(B\). This is a nontrivial constraint, as it is required to holdwhether or not the observables represented are compatible. A quantumstate, therefore, relates expectation values for quantities yielded byincompatible experiments.

Incompatible observables, represented by noncommuting operators, giverise to uncertainty relations; see the entry onthe uncertainty principle. These relations entail that there are no quantum states that assigndefinite values to the observables that satisfy them, and place boundson how close they can come to be simultaneously well-defined in anyquantum state.

For any two distinct quantum states, \(\rho\), \(\omega\), and anyreal number between 0 and 1, there is a correspondingmixedstate. The probability assigned to any experimental outcome bythis mixed state is \(p\) times the probability it is assigned by\(\rho\) plus \(1-p\) times the probability assigned to it by\(\omega\). One way to physically realize the preparation of a mixedstate is to employ a randomizing device, for example, a coin withprobability \(p\) of landing heads and probability \(1-p\) oflanding tails, and to use it to choose between preparing state\(\rho\) and preparing state \(\omega\). We will see another way toprepare a mixed state after we have discussed entanglement, in section3. A state that is not a mixture of any two distinct states is calledapure state.

It is both useful and customary, though not strictly necessary, toemploy aHilbert space representation of a quantum theory. Insuch a representation, the operators corresponding to observables arerepresented as acting on elements of an appropriately constructedHilbert space (see the entry onquantum mechanics for details). Usually, the Hilbert space representation isconstructed in such a way that vectors in the space represent purestates; such a representation is called anirreduciblerepresentation. Irreducible representations, in which mixedstates are also represented by vectors, are also possible.

A Hilbert space is a vector space. This means that, for any twovectors \(|\psi\rangle\), \(|\phi\rangle\) , in the space,representing pure states, and any complex numbers \(a\),\(b\), there is another vector, \(a |\psi\rangle + b|\phi\rangle\), that also represents a pure state. This is called asuperposition of the states represented by \(|\psi\rangle\)and \(|\phi\rangle\) . Any vector in a Hilbert space can be written asa superposition of other vectors in infinitely many ways. Sometimes,in discussing the foundations of quantum mechanics, authors fall intotalking as if some state are superpositions and others are not. Thisis simply an error. Usually what is meant is that some states yielddefinite values for macroscopic observables, and others cannot bewritten in any way that is not a superposition of macroscopicallydistinct states.

The noncontroversial operational core of quantum theory consists ofrules for identifying, for any given system, appropriate operatorsrepresenting its dynamical quantities. In addition, there areprescriptions for evolving the state of system when it is acted uponby specified external fields or subjected to various manipulations(seesection 1.3). Application of quantum theory typically involves a distinctionbetween the system under study, which is treated quantum mechanically,and experimental apparatus, which is not. This division is sometimesknown as theHeisenberg cut.

Whether or not we can expect to be able to go beyond thenoncontroversial operational core of quantum theory, and take it to bemore than a means for calculating probabilities of outcomes ofexperiments, remains a topic of contemporary philosophicaldiscussion.

2.2 Quantum mechanics and quantum field theory

Quantum mechanics is usually taken to refer to the quantizedversion of a theory of classical mechanics, involving systems with afixed, finite number of degrees of freedom. Classically, a field, suchas, for example, an electromagnetic field, is a system endowed withinfinitely many degrees of freedom. Quantization of a field theorygives rise to aquantum field theory. The chief philosophicalissues raised by quantum mechanics remain when the transition is madeto a quantum field theory; in addition, new interpretational issuesarise. There are interesting differences, both technical andinterpretational, between quantum mechanical theories and quantumfield theories; for an overview, see the entries onquantum field theory andquantum theory: von Neumann vs. Dirac.

The standard model of quantum field theory, successful as it is, doesnot yet incorporate gravitation. The attempt to develop a theory thatdoes justice both the quantum phenomena and to gravitational phenomenagives rise to serious conceptual issues (see the entry onquantum gravity).

2.3 Quantum state evolution

2.3.1 Schrödinger and Heisenberg pictures

When constructing a Hilbert space representation of a quantum theoryof a system that evolves over time, there are some choices to be made.One needs to have, for each timet, a Hilbert spacerepresentation of the system, which involves assigning operators toobservables pertaining to timet. An element of conventioncomes in when deciding how the operators representing observables atdifferent times are to be related.

For concreteness, suppose that have a system whose observables includea position, \(x\), and momentum, \(p\), with respect to someframe of reference. There is a sense in which, for two distinct times,\(t\) and \(t'\),position at time \(t\) andposition at time \(t'\) are distinct observables, and alsoa sense in which they are values, at different times, of the sameobservable. Once we have settled on operators \(\hat{X}\) and\(\hat{P}\) to represent position and momentum at time \(t\), westill have a choice of which operators represent the correspondingquantities at time \(t.\) On theSchrödingerpicture, the same operators \(\hat{X}\) and \(\hat{P}\) are usedto represent position and momentum, whatever time is considered. Asthe probabilities for results of experiments involving thesequantities may be changing with time, different vectors must be usedto represent the state at different times.

The equation of motion obeyed by a quantum state vector is theSchrödinger equation. It is constructed by first formingthe operator \(\hat{H}\)corresponding to the Hamiltonian of thesystem, which represents the total energy of the system. The rate ofchange of a state vector is proportional to the result of operating onthe vector with the Hamiltonian operator \(\hat{H}\).

\[ i \hbar {\,\D}/{\D t}\, \ket{\psi (t)} = \hat{H} \ket{\psi (t)}. \]

There is an operator that takes a state at time 0 into a state at time\(t\); it is given by

\[ U(t) = \exp\left(\frac{{-}i H t}{\hbar}\right). \]

This operator is a linear operator that implements a one-one mappingof the Hilbert space to itself that preserves the inner product of anytwo vectors; operators with these properties are calledunitaryoperators, and, for this reason, evolution according to theSchrödinger equation is calledunitary evolution.

For our purposes, the most important features of this equation is thatit isdeterministic andlinear. The state vector atany time, together with the equation, uniquely determines the statevector at any other time. Linearity means that, if two vectors\(\ket{\psi_1(0)}\) and \(\ket{\psi_2(0)}\) evolve into vectors\(\ket{\psi_1(t) }\) and \(\ket{\psi_2(t)}\), respectively, then, ifthe state at time 0 is a linear combination of these two, the state atany time \(t\) will be the corresponding linearcombination of \(\ket{\psi_1(t)}\) and \(\ket{\psi_2(t)}\).

\[ a\ket{\psi_{1}(0)} + b\ket{\psi_{2}(0)} \rightarrow a\ket{\psi_{1}(t)} + b\ket{\psi_{2}(t)} . \]

The Heisenberg picture, on the other hand, employs different operators\(\hat{X}(t)\), \(\hat{X}(t')\) for position, depending on the timeconsidered (and similarly for momentum and other observables). If\(\hat{A}(t)\)is a family of Heisenberg picture operators representingsome observable at different times, the members of the family satisfy the Heisenberg equation of motion,

\[i \hbar d/dt \; \hat{A}(t) = \hat{A}(t) \hat{H} - \hat{H} \hat{A}(t).\]

One sometimes hears it said that, on the Heisenberg picture, the stateof the system is unchanging. This is incorrect. It is true that thereare not different state vectors corresponding to different times, butthat is because a single state vector serves for computingprobabilities for all observables pertaining to all times. Theseprobabilitiesdo change with time.

2.3.2. The collapse postulate

As mentioned, standard applications of quantum theory involve adivision of the world into a system that is treated within quantumtheory, and the remainder, typically including the experimentalapparatus, that is not treated within the theory. Associated with thisdivision is a postulate about how to assign a state vector after anexperiment that yields a value for an observable, according to which,after an experiment, one replaces the quantum state with an eigenstatecorresponding to the value obtained. Unlike the unitaryevolution applied otherwise, this is a discontinuous change of thequantum state, sometimes referred to ascollapse of the statevector, orstate vector reduction. There are twointerpretations of the postulate about collapse, corresponding to twodifferent conceptions of quantum states. If a quantum state representsnothing more than knowledge about the system, then the collapse of thestate to one corresponding to an observed result can be thought of asmere updating of knowledge. If, however, quantum states representphysical reality, in such a way that distinct pure states alwaysrepresent distinct physical states of affairs, then the collapsepostulate entails an abrupt, perhaps discontinuous, change of thephysical state of the system. Considerable confusion can arise if thetwo interpretations are conflated.

The collapse postulate occurs already in the general discussion at thefifth Solvay Conference in 1927 (see Bacciagaluppi and Valentini, 2009,437–450). It is also found in Heisenberg’sThePhysical Principles of the Quantum Theory, based on lecturespresented in 1929 (Heisenberg, 1930a, 27; 1930b, 36). Von Neumann, inhis reformulation of quantum theory a few years later, distinguishedbetween two types of processes: Process 1:, which occurs uponperformance of an experiment, and Process 2:, the unitary evolutionthat takes place as long as no measurement is made (von Neumann, 1932;1955, §V.I). He does not take this distinction to be a differencebetween two physically distinct processes. Rather, the invocation ofone process or the other depends on a somewhat arbitrary division ofthe world into an observing part and an observed part (see vonNeumann,1932, 224; 1955, 420).

The collapse postulate does not appear in the first edition (1930) ofDirac’sPrinciples of Quantum Mechanics; it isintroduced in the second edition (1935). Dirac formulates it asfollows.

When we measure a real dynamical variable \(\xi\), the disturbanceinvolved in the act of measurement causes a jump in the state of thedynamical system. From physical continuity, if we make a secondmeasurement of the same dynamical variable \(\xi\) immediately afterthe first, the result of the second measurement must be the same asthat of the first. Thus after the first measurement has been made,there is no indeterminacy in the result of the second. Hence, afterthe first measurement has been made, the system is in an eigenstate ofthe dynamical variable \(\xi\), the eigenvalue it belongs to beingequal to the result of the first measurement. This conclusion muststill hold if the second measurement is not actually made. In this waywe see that a measurement always causes the system to jump into aneigenstate of the dynamical variable that is being measured, theeigenvalue this eigenstate belongs to being equal to the result of themeasurement (Dirac 1935: 36).

Unlike von Neumann and Heisenberg, Dirac is treating the“jump” as a physical process.

Neither von Neumann nor Dirac take awareness of the result by aconscious observer to be a necessary condition for collapse. For vonNeumann, the location of the cut between the “observed”system and the “observer”is somewhat arbitrary. It may beplaced between the system under study and the experimental apparatus.On the other hand, we could include the experimental apparatus in thequantum description, and place the cut at the moment when lightindicating the result hits the observer’s retina. We could alsogo even further, and include the retina and relevant parts of theobserver’s nervous system in the quantum system. That the cutmay be pushed arbitrarily far into the perceptual apparatus of theobserver is required, according to von Neumann, by theprincipleof psycho-physical parallelism.

A formulation of a version of the collapse postulate according towhich a measurement is not completed until the result is observed isfound in London and Bauer (1939). For them, as for Heisenberg, this isa matter of an increase of knowledge on the part of the observer.

Wigner (1961) combined elements of the two interpretations. Like thosewho take the collapse to be a matter of updating of belief in light ofinformation newly acquired by an observer, he takes collapse to takeplace when a conscious observer becomes aware of an experimentalresult. However, like Dirac, he takes it to be a real physicalprocess. His conclusion is that consciousness has an influence on thephysical world not captured by the laws of quantum mechanics. Thisinvolves a rejection of von Neumann’s principle ofpsycho-physical parallelism, according to which it must be possible totreat the process of subjective perception as if it were a physicalprocess like any other.

There is a persistent misconception that, for von Neumann, collapse isto be invoked only when a conscious observer becomes aware of theresult. As noted, this is the opposite of his view, as the cut may beplaced between the observed system and the experimental apparatus, andit is for him an important point that the location of the cut besomewhat arbitrary. In spite of this, von Neumann’s position issometimes conflated with Wigner’s speculative proposal, andWigner’s proposal is sometimes erroneously referred to as thevon Neumann-Wigner interpretation.

None of the standard formulations are precise about when the collapsepostulate is to be applied; there is some lee-way as to what is tocount as an experiment, or (for versions that require reference to anobserver) what is to count as an observer. Some, including von Neumannand Heisenberg, have taken it to be a matter of principle that therebe some arbitrariness in where to apply the postulate. It is commonwisdom that, in practice, this arbitrariness is innocuous. The rule ofthumb that seems to be applied, in practice, in setting the splitbetween the parts of the world treated quantum-mechanically and thingstreated as classical objects has been formulated by J. S. Bell as,“[w]hen in doubt enlarge the quantum system,” to the pointat which including more in the quantum system makes negligibledifference to practical predictions (Bell 1986, 362; Bell 2004, 189).If anything is to be counted as “standard” quantummechanics, it is the operational core we have discussed, supplementedby a heuristic rule of application of this sort. Standard quantummechanics works very well. If, however, one seeks a theory that iscapable of describing all systems, including macroscopic ones, and canyield an account of the process by which macroscopic events, includingexperimental outcomes, come about, this gives rise to the so-called“measurement problem”, which we will discuss after we haveintroduced the notion of entanglement (seesection 3).

2.3.3. Wave functions

Among the Hilbert-space representations of a quantum theory arewave-function representations.

Associated with any observable is itsspectrum, the range ofpossible values that the observable can take on. Given anyphysical system and any observable for that system, one can alwaysform a Hilbert-space representation for the quantum theory of thatsystem by considering complex-valued functions on the spectrum of thatobservable. The set of such functions form a vector space. Givena measure on the spectrum of the observable, we can form a Hilbert spaceout of the set of complex-valued square-integrable functions on thespectrum by treating functions that differ only on a set of zeromeasure as equivalent (that is, the elements of our Hilbert space arereally equivalence classes of functions), and by using the measure todefine an inner product (seeentry on Quantum Mechanics if this terminology is unfamiliar).

If the spectrum of the chosen observable is a continuum (as it is, forexample, for position or momentum), a Hilbert-space representation ofthis sort is called awave function representation, and thefunctions that represent quantum states,wave functions (also“wave-functions,” or “wavefunctions”). Themost familiar representations of this form are position-space wavefunctions, which are functions on the set of possible configurationsof the system, and momentum-space wave functions, which are functionsof the momenta of the systems involved.

3. Entanglement, nonlocality, and nonseparability

Given two disjoint physical systems, \(A\) and \(B\),with which we associate Hilbert spaces \(H_{A}\) and\(H_{B}\), the Hilbert space associated with the composite system isthe tensor product space, denoted \(H_{A} \otimes H_{B}\).

When the two systems are independently prepared in pure states\(\ket{\psi}\) and \(\ket{\phi}\), the state of the composite systemis theproduct state \(\ket{\psi} \otimes \ket{\phi}\)(sometimes written with the cross, \(\otimes\), omitted).

In addition to the product states, the tensor product space containslinear combinations of product states, that is, state vectors of theform

\[ a\ket{\psi_{1}} \otimes \ket{\phi_{1}} + b\ket{\psi_{2}} \otimes \ket{\phi_{2}} \]

The tensor product space can be defined as the smallest Hilbert spacecontaining all of the product states. Any pure state represented by astate vector that is not a product vector is anentangledstate.

The state of the composite system assigns probabilities to outcomes ofall experiments that can be performed on the composite system. We canalso consider a restriction to experiments performed on system \(A\),or a restriction to experiments performed to \(B\). Such restrictionsyields states of \(A\) and \(B\), respectively, called thereducedstates of the systems. When the state of the composite system\(AB\) is an entangled state, then the reduced states of \(A\) and\(B\) are mixed states. To see this, suppose that in the above statethe vectors \(\ket{\phi_{1}}\) and \(\ket{\phi_{2}}\) representdistinguishable states. If one confines one’s attention toexperiments performed on \(A\), it makes no difference whether anexperiment is also performed on \(B\). An experiment performed on\(B\) that distinguishes \(\ket{\phi_{1}}\) and \(\ket{\phi_{2}}\)projects the state of \(A\) into either \(\ket{\psi_{1}}\) or\(\ket{\psi_{2}}\), with probabilities \(\abs{a}^{2}\) and\(\abs{b}^{2}\), respectively, and probabilities for outcomes ofexperiments performed on \(A\) are the corresponding averages ofprobabilities for states \(\ket{\psi_{1}}\) and\(\ket{\psi_{2}}\). These probabilities, as mentioned, are the same asthose for the situation in which no experiment is performed on\(B\). Thus, even if no experiment is performed on \(B\), theprobabilities of outcomes of experiments on \(A\) are exactly as ifsystem \(A\) is either in the state represented by \(\ket{\psi_{1}}\)or the state represented by \(\ket{\psi_{2}}\), with probabilities\(\abs{a}^{2}\) and \(\abs{b}^{2}\), respectively.

In general, any state, pure or mixed, that is neither a product statenor a mixture of product states, is called anentangledstate.

The existence of pure entangled states means that, if we consider acomposite system consisting of spatially separated parts, then, evenwhen the state of the system is a pure state, the state is notdetermined by the reduced states of its component parts. Thus, quantumstates exhibit a form ofnonseparability. See the entry onholism and nonseparability in physics for more information.

Quantum entanglement results in a form of nonlocality that is alien toclassical physics. Even if we assume that the reduced states of \(A\)and \(B\) do not completelycharacterize their physical states, but must be supplemented by somefurther variables, there are quantum correlations that cannot bereduced to correlations between states of \(A\) and \(B\);see the entries onBell’s Theorem andaction at a distance in quantum mechanics.

4. The measurement problem

4.1 The measurement problem formulated

If quantum theory is meant to be (in principle) a universal theory, itshould be applicable, in principle, to all physical systems, includingsystems as large and complicated as our experimental apparatus. It iseasy to show that linear evolution of quantum states, when applied tomacroscopic objects, will routinely lead to superpositions ofmacroscopically distinct states. Among the circumstances in which thiswill happen are experimental set-ups, and much of the earlydiscussions focussed on how to construe the process of measurement inquantum-mechanical terms. For this reason, the interpretational issueshave come to be referred to as themeasurement problem. Inthe first decades of discussion of the foundations of quantummechanics, it was commonly referred to as theproblem ofobservation.

Consider a schematized experiment. Suppose we have a quantum systemthat can be prepared in at least two distinguishable states, \(\ket{0}_{S}\) and \(\ket{1} _{S}\). Let \(\ket{R} _{A}\) be a ready state ofthe apparatus, that is, a state in which the apparatus is ready tomake a measurement.

If the apparatus is working properly, and if the measurement is aminimally disturbing one, the coupling of the system \(S\)with the apparatus \(A\) should resultin an evolution that predictably yields results of the form

\[ \ket{0} _{S} \ket{R} _{A} \Rightarrow \ket{0}_{S}\ket{“0” } _{A} \]\[ \ket{1} _{S} \ket{R} _{A} \Rightarrow \ket{1}_{S}\ket{“1”} _{A} \]

where \(\ket{“0” } _{A}\) and \(\ket{“1”}_{A}\) are apparatus states indicating results 0 and 1,respectively.

Now suppose that the system \(S\) is prepared in asuperposition of the states \(\ket{0} _{S}\) and \(\ket{1}_{S}\).

\[ \ket{\psi(0)} _{S} = a\ket{0} _{S} + b\ket{1} _{S}, \]

where \(a\) and \(b\) are both nonzero.If the evolution that leads from the pre-experimental state to thepost-experimental state is linear Schrödinger evolution, then wewill have

\[ \ket{\psi(0)} _{S} \ket{R} _{A} \rightarrow a\ket{0} _{S}\ket{“0” } _{A} + b\ket{1} _{S}\ket{“1” }_{A}. \]

This is not an eigenstate of the instrument reading variable, but is,rather, a state in which the reading variable and the system variableare entangled with each other. The eigenstate-eigenvalue link, appliedto a state like this, does not yield a definite result for theinstrument reading. The problem of what to make of this is called the“measurement problem” which is discussed in more detailbelow.

4.2 Approaches to the measurement problem

If quantum state evolution proceeds via the Schrödinger equationor some other linear equation, then, as we have seen in the previoussection, typical experiments will lead to quantum states that aresuperpositions of terms corresponding to distinct experimentaloutcomes. It is sometimes said that this conflicts with ourexperience, according to which experimental outcome variables, such aspointer readings, always have definite values. This is a misleadingway of putting the issue, as it is not immediately clear how tointerpret states of this sort as physical states of a system thatincludes experimental apparatus, and, if we can’t say what itwould be like to observe the apparatus to be in such a state, it makesno sense to say that we never observe it to be in a state likethat.

Nonetheless, we are faced with an interpretational problem. If we takethe quantum state to be a complete description of the system, then thestate is, contrary to what would antecedently expect, not a statecorresponding to a unique, definite outcome. This is what led J.S.Bell to remark, “Either the wavefunction, as given by theSchrödinger equation, is not everything, or it is notright” (Bell 1987: 41, 2004: 201). This gives us a (primafacie) tidy way of classifying approaches to the measurementproblem:

There are approaches that involve a denial that a quantum wavefunction (or any other way of representing a quantum state) yields acomplete description of a physical system.
There are approaches that involve modification of the dynamics toproduce a collapse of the quantum state in appropriatecircumstances.
There are approaches that reject both horns of Bell’sdilemma, and hold that quantum states undergo unitary evolution at alltimes and that a quantum state-description is, in principle,complete.

We include in the first category approaches that deny that a quantumstate should be thought of as representing anything in reality at all.These include variants of the Copenhagen interpretation, as well aspragmatic and other anti-realist approaches. Also in the firstcategory are approaches that seek a completion of the quantum statedescription. These include hidden-variables approaches and modalinterpretations. The second category of interpretation motivates aresearch programme of finding suitable indeterministic modificationsof the quantum dynamics. Approaches that reject both horns ofBell’s dilemma are typified by Everettian, or“many-worlds” interpretations.

4.2.1 The “Copenhagen interpretation”

Since the mid-1950’s, the term “Copenhageninterpretation” has been commonly used for whatever it is thatthe person employing the term takes to be the ‘orthodox’viewpoint regarding the philosophical issues raised by quantummechanics. According to Howard (2004), the phrase was first used byHeisenberg (1955, 1958), and is intended to suggest a commonality ofviews among Bohr and his associates, included Born and Heisenberghimself. Recent historiography has emphasized diversity of viewpointsamong the figures associated with the Copenhagen interpretation; seethe entry onCopenhagen interpretation of quantum mechanics, and references therein. Readers should be aware that the term is notunivocal, and that different authors might mean different things whenspeaking of the“Copenhagen interpretation.”

4.2.2 Non-realist and pragmatist approaches to quantum mechanics

From the early days of quantum mechanics, there has been a strain ofthought that holds that the proper attitude to take towards quantummechanics is an instrumentalist or pragmatic one. On such a view,quantum mechanics is a tool for coordinating our experience and forforming expectations about the outcomes of experiments. Variants ofthis view include some versions of the Copenhagen interpretation. Morerecently, views of this sort have been advocated by physicists,including QBists, who hold that quantum states represent subjective orepistemic probabilities (see Fuchset al., 2014). Thephilosopher Richard Healey defends a related view on which quantumstates, though objective, are not to be taken asrepresentational (see Healey 2012, 2017a, 2020). For more onthese approaches, see entry onQuantum-Bayesian and pragmatist views of quantum theory.

4.2.2 Hidden-variables and modal interpretations

Theories whose structure include the quantum state but includeadditional structure, with an aim of circumventing the measurementproblem, have traditionally been called “hidden-variablestheories”. That a quantum state description cannot be regardedas a complete description of physical reality was argued for in afamous paper by Einstein, Podolsky and Rosen (EPR) and by Einstein insubsequent publications (Einstein 1936, 1948, 1949). See the entry ontheEinstein-Podolsky-Rosen argument in quantum theory.

There are a number of theorems that circumscribe the scope of possiblehidden-variables theories. The most natural thought would be to seek atheory that assigns to all quantum observables definite values thatare merely revealed upon measurement, in such a way that anyexperimental procedure that, in conventional quantum mechanics, wouldcount as a “measurement” of an observable yields thedefinite value assigned to the observable. Theories of this sort arecallednoncontextual hidden-variables theory. It was shown byBell (1966) and Kochen and Specker (1967) that there are no suchtheories for any system whose Hilbert space dimension is greater thanthree (see the entry onthe Kochen-Specker theorem).

The Bell-Kochen-Specker Theorem does not rule out hidden-variablestheoriestout court. The simplest way to circumvent it is topick as always-definite some observable or compatible set ofobservables that suffices to guarantee determinate outcomes ofexperiments; other observables are not assigned definite values andexperiments thought of as “measurements” of theseobservables do not reveal pre-existing values.

The most thoroughly worked-out theory of this type is the pilot wavetheory developed by de Broglie and presented by him at the FifthSolvay Conference held in Brussels in 1927, revived by David Bohm in1952, and currently an active area of research by a small group ofphysicists and philosophers. According to this theory, there areparticles with definite trajectories, that are guided by the quantumwave function. For the history of the de Broglie theory, see theintroductory chapters of Bacciagaluppi and Valentini (2009). For anoverview of the de Broglie-Bohm theory and philosophical issuesassociated with it see the entry onBohmian mechanics.

There have been other proposals for supplementing the quantum statewith additional structure; these have come to be calledmodalinterpretations; see the entry onmodal interpretations of quantum mechanics.

4.2.3 Dynamical Collapse Theories

As already mentioned, Dirac wrote as if the collapse of thequantum state vector precipitated by an experimental intervention onthe system is a genuine physical change, distinct from the usualunitary evolution. If collapse is to be taken as a genuine physicalprocess, then something more needs to be said about the circumstancesunder which it occurs than merely that it happens when an experimentis performed. This gives rise to a research programme of formulating aprecisely defined dynamics for the quantum state that approximates thelinear, unitary Schrödinger evolution in situations for whichthis is well-confirmed, and produces collapse to an eigenstate of theoutcome variable in typical experimental set-ups, or, failing that, aclose approximation to an eigenstate. The only promising collapsetheories are stochastic in nature; indeed, it can be shown that adeterministic collapse theory would permit superluminal signalling.See the entry oncollapse theories for an overview, and Gao, ed. (2018) for a snapshot of contemporarydiscussions.

Prima facie, a dynamical collapse theory of this type can bea quantum state monist theory, one on which, in Bell’s words,“the wave function is everything”. In recent years, thishas been disputed; it has been argued that collapse theories require“primitive ontology” in addition to the quantum state. SeeAlloriet al. (2008), Allori (2013), and also the entry oncollapse theories, and references therein. Reservations about this approach have beenexpressed by Egg (2017, 2021), Myrvold (2018), and Wallace (2020).

4.2.4 Everettian, or “many worlds” theories

In his doctoral dissertation of 1957 (reprinted in Everett 2012), HughEverett III proposed that quantum mechanics be taken as it is, withouta collapse postulate and without any “hidden variables”.The resulting interpretation he called therelative stateinterpretation.

The basic idea is this. After an experiment, the quantum state of thesystem plus apparatus is typically a superposition of termscorresponding to distinct outcomes. As the apparatus interacts withits environment, which may include observers, these systems becomeentangled with the apparatus and quantum system, the net result ofwhich is a quantum state involving, for each of the possibleexperimental outcomes, a term in which the apparatus readingcorresponds to that outcome, there are records of that outcome in theenvironment, observers observe that outcome,etc.. Everettproposed that each of these terms be taken to be equally real. From aGod’s-eye-view, there is no unique experimental outcome, but onecan also focus on a particular determinate state of one subsystem,say, the experimental apparatus, and attribute to the other systemsparticipating in the entangled state arelative state,relative to that state of the apparatus. That is, relative to theapparatus reading ‘+’ is a state of the environmentrecording that result and states of observers observing that result(see the entry onEverett’s relative-state formulation of quantum mechanics, for more detail on Everett’s views).

Everett’s work has inspired a family of views that go by thename of “Many Worlds” interpretations; the idea is thateach of the terms of the superposition corresponds to a coherentworld, and all of these worlds are equally real. As time goes on,there is a proliferation of these worlds, as situations arise thatgive rise to a further multiplicity of outcomes (see the entrymany-worlds interpretation of quantum mechanics, and Saunders 2007, for overviews of recent discussions; Wallace 2012is an extended defense of an Everettian interpretation of quantummechanics).

There is a family of distinct, but related views, that go by the nameof “Relational Quantum Mechanics”. These views agree withEverett in attributing to a system definite values of dynamicalvariables only relative to the states of other systems; they differ inthat, unlike Everett, they do not take the quantum state as theirbasic ontology (see the entry onrelational quantum mechanics for more detail).

4.3 Extended Wigner’s friend scenarios as a source of no-go theorems

As mentioned, quantum theory, as standardly formulated, employs adivision of the world into a part that is treated with the theory, anda part that is not. Both von Neumann and Heisenberg emphasized anelement of arbitrariness in the location of the division. In someformulations, the division was thought of as a distinction betweenobserver and observed, and it became common to say that quantummechanics requires reference to an observer for its formulation.

The founders of quantum mechanics tended to assume implicitly that,though the “cut” is somewhat moveable, in any givenanalysis a division would be settled on, and one would not attempt tocombine distinct choices of the cut in one analysis of an experiment.If, however, one thinks of the cut as marking the distinction betweenobserver and observed, one is led to ask about situations involvingmultiple observers. Is each observer permitted to treat the other as aquantum system?

The consideration of such scenarios was initiated by Wigner(1961). Wigner considered a hypothetical scenario in which a friendconducts an observation, and he himself treats the joint system,consisting of the friend and the system experimented upon, as aquantum system. For this reason, scenarios of this sort have come tobe known as “Wigner’s friend” scenarios. Wigner wasled by consideration of such scenarious to hypothesize that consciousobservers cannot be in a superposition of states corresponding todistinct perceptions; the introduction of conscious observersinitiates a physical collapse of the quantum state; this involves,according to Wigner, “a violation of physical laws whereconsciousness plays a role” (Wigner 1961, 294 ;167, 181).

Frauchiger and Renner (2018) initiated the discussion of scenariosof this sort involving more than two observers, which have come to becalled “extended Wigner’s friend” scenarios. Furtherresults along these lines include Brukner (2018), Bong et al. (2020),and Guérin et al. (2021). The strategy of these investigationsis to present some set of plausible-seeming assumptions (a differentset, for each of the works cited), and to show, via consideration of ahypothetical situation involving multiple observers, the inconsistencyof that set of assumptions. The theorems are, therefore, no-gotheorems for approaches to the measurement problem that would seek tosatisfy all of the members of the set of assumptions that has beenshown to be inconsistent.

An assumption common to all of these investigations is that it isalways permissible for one observer to treat systems containing otherobservers within quantum mechanics and to employ unitary evolution forthose systems. This means that collapse is not regarded as a physicalprocess. It is also assumed that each observer always perceives aunique outcome for any experiment performed by that observer; thisexcludes Everettian interpretations. Where the works cited vary is inthe other assumptions made.

It should be noted that each of the major avenues of approach to themeasurement problem is capable of giving an account of goings-on inany physical scenario, including the ones considered in these works.Each of them, therefore, must violate some member of the set ofassumptions shown to be inconsistent. These results do not poseproblems for existing approaches to the measurement problem; rather,they are no-go theorems for approaches that might seek to satisfy allof the set of assumptions shown to be inconsistent. As the assumptionsconsidered include both unitary evolution and unique outcomes ofexperiments, and the scenarios considered involved situationsinvolving superpositions of distinct experimental outcomes, theseresults concern theories on which the quantum state, as given by theSchrödinger equation, is not a complete description of reality,as it fails to determine the unique outcomes perceived by theobservers. These preceptions could be thought of as supervening onbrain states, in which case there is physical structure not includedin the quantum state, or as attributes of immaterial minds. On eitherinterpretation, the sorts of theories ruled out fall under the firsthorn of Bell’s dilemma, mentioned in section 4.2, and theseno-go results in part reproduce, and in part extend, no-go results forcertain sorts of modal interpretations (see entry onmodal interpretations of quantum mechanics).

These results involving extended Wigner’s friend scenarios have engenderedconsiderable philosophical discussion; see Sudbery (2017, 2019),Healey (2018, 2020), Dieks (2019), Losada et al. (2019), Dascal(2020), Evans (2020), Fortin and Lombardi (2020), Kastner (2020),Muciño & Okon (2020), Bub (2020, 2021), Cavalcanti (2021),Cavalcanti and Wiseman (2021), and Żukowski and Markiewicz(2021).

4.4 The role of decoherence

A quantum state that is a superposition of two distinct terms, suchas

\[ \ket{\psi} = a \ket{\psi_{1}} + b \ket{\psi_{2}} , \]

where \(\ket{\psi_{1}}\) and \(\ket{\psi_{2}}\) are distinguishablestates, is not the same state as a mixture of \(\ket{\psi_{1}}\) and\(\ket{\psi_{2}}\), which would be appropriate for a situation inwhich the state prepared was either \(\ket{\psi_{1}}\) or\(\ket{\psi_{2}}\), but we don’t know which. The differencebetween a coherent superposition of two terms and a mixture hasempirical consequences. To see this, consider the double-slitexperiment, in which a beam of particles (such as electrons, neutrons,or photons) passes through two narrow slits and then impinges on ascreen, where the particles are detected. Take \(\ket{\psi_{1}}\) tobe a state in which a particle passes through the top slit, and\(\ket{\psi_{2}}\), a state in which it passes through the bottomslit. The fact that the state is a superposition of these twoalternatives is exhibited in interference fringes at the screen,alternating bands of high and low rates of absorption.

This is often expressed in terms of a difference between classical andquantum probabilities. If the particles were classical particles, theprobability of detection at some point \(p\) of thescreen would simply be a weighted average of two conditionalprobabilities: the probability of detection at \(p\),given that the particle passed through the top slit, and theprobability of detection at \(p\), given that theparticle passed through the bottom slit. The appearance ofinterference is an index of nonclassicality.

Suppose, now, that the electrons interact with something else (call ittheenvironment) on the way to the screen, that could serveas a “which-way” detector; that is, the state of thisauxiliary system becomes entangled with the state of the electron insuch a way that its state is correlated with \(\ket{\psi_{1}}\) and\(\ket{\psi_{2}}\). Then the state of the quantum system, \(s\),and its environment, \(e\), is

\[ \ket{\psi} _{se} = a \ket{ \psi_{1}} _{s} \ket{ \phi_{1}} _{e} + b \ket{ \psi_{2}}_{s} \ket{ \phi_{2}} _{e} \]

If the environment states \(\ket{\phi_{1}} _{e}\) are\(\ket{\phi_{2}}_{e}\) are distinguishable states, then thiscompletely destroys the interference fringes: the particles interactwith the screen as if they determinately went through one slit or theother, and the pattern that emerges is the result of overlaying thetwo single-slit patterns. That is, we can treat the particles as ifthey followed (approximately) definite trajectories, and applyprobabilities in a classical manner.

Now, macroscopic objects are typically in interaction with a large andcomplex environment—they are constantly being bombarded with airmolecules, photons, and the like. As a result, the reduced state ofsuch a system quickly becomes a mixture of quasi-classical states, aphenomenon known asdecoherence.

A generalization of decoherence lies at the heart of an approach tothe interpretation of quantum mechanics that goes by the name ofdecoherent histories approach (see the entry onthe consistent histories approach to quantum mechanics for an overview).

Decoherence plays important roles in the other approaches to quantummechanics, though the role it plays varies with approach; see theentry onthe role of decoherence in quantum mechanics for information on this.

4.5 Comparison of approaches to the measurement problem

Most of the above approaches take it that the goal is to provide anaccount of events in the world that recovers, at least in someapproximation, something like our familiar world of ordinary objectsbehaving classically. None of the mainstream approaches accord anyspecialphysical role to conscious observers. There have,however, been proposals in that direction (see the entry onquantum approaches to consciousness for discussion).

All of the above-mentioned approaches are consistent with observation.Mere consistency, however, is not enough; the rules for connectingquantum theory with experimental results typically involve nontrivial(that is, not equal to zero or one) probabilities assigned toexperimental outcomes. These calculated probabilities are confrontedwith empirical evidence in the form of statistical data from repeatedexperiments. Extant hidden-variables theories reproduce the quantumprobabilities, and collapse theories have the intriguing feature ofreproducing very close approximations to quantum probabilities for allexperiments that have been performed so far but departing from thequantum probabilities for other conceivable experiments. This permits,in principle, an empirical discrimination between such theories andno-collapse theories.

A criticism that has been raised against Everettian theories is thatit is not clear whether they can even make sense of statisticaltesting of this kind, as it does not, in any straightforward way, makesense to talk of the probability of obtaining, say, a ‘+”outcome of a given experiment when it is certain that all possibleoutcomes will occur on some branch of the wavefunction. This has beencalled the “Everettian evidential problem”. It has beenthe subject of much recent work on Everettian theories; see Saunders(2007) for an introduction and overview.

If one accepts that Everettians have a solution to the evidentialproblem, then, among the major lines of approach, none is favored in astraightforward way by the empirical evidence. There will not be spacehere to give an in-depth overview of these ongoing discussions, but afew considerations can be mentioned, to give the reader a flavor ofthe discussions; see entries on particular approaches for moredetail.

Everettians take, as a virtue of the approach, the fact that it doesnot involve an extension or modification of the quantum formalism.Bohmians claim, in favor of the Bohmian approach, that a theory onthese lines provides the most straightforward picture of events;ontological issues are less clear-cut when it comes to Everettiantheories or collapse theories.

Another consideration is compatibility with relativistic causalstructure. See Myrvold (2021) for an overview of relavisticconstraints on approaches to the measurement problem.The deBroglie-Bohm theory requires a distinguished relation of distantsimultaneity for its formulation, and, it can be argued, this is anineliminable feature of any hidden-variables theory of this sort, thatselects some observable to always have definite values (see Berndlet al. 1996; Myrvold 2002, 2021). On the other hand, thereare collapse models that are fully relativistic. On such models,collapses are localized events. Though probabilities of collapses atspacelike separation from each other are not independent, thisprobabilistic dependence does not require us to single one out asearlier and the other later. Thus, such theories do not require adistinguished relation of distant simultaneity. There remains,however, some discussion of how to equip such theories with beables(or “elements of reality”). See the entry oncollapse theories and references therein; see also, for some recent contributions tothe discussion, Fleming (2016), Maudlin (2016), and Myrvold(2016). In the case of Everettian theories, one must first thinkabout how to formulate the question of relativistic locality. Severalauthors have approached this issue in somewhat different ways, with acommon conclusion that Everettian quantum mechanics is, indeed, local.(See Vaidman 1994; Bacciagaluppi 2002; Chapter 8 of Wallace 2012; Tipler2014; Vaidman 2016; and Brown and Timpson 2016.)

5. Ontological Issues

As mentioned, a central question of interpretation of quantummechanics concerns whether quantum states should be regarded asrepresenting anything in physical reality. If this is answered in theaffirmative, this gives rise to new questions, namely, what sort ofphysical reality is represented by the quantum state, and whether aquantum state could in principle give an exhaustive account ofphysical reality.

5.1 The question of quantum state realism.

Harrigan and Spekkens (2010) have introduced a framework fordiscussing these issues. In their terminology, a completespecification of the physical properties is given by theonticstate of a system. An ontological model posits a space of onticstates and associates, with any preparation procedure, a probabilitydistribution over ontic states. A model is said to be\(\psi\)-ontic if the ontic state uniquely determines thequantum state; that is, if there is a function from ontic states toquantum states (this includes both cases in which the quantum statealso completely determines the physical state, and cases, such ashidden-variables theories, in which the quantum state does notcompletely determine the physical state). In their terminology, modelsthat are not \(\psi\)-ontic are called \(\psi\)-epistemic. Ifa model is not \(\psi\)-ontic, this means that it is possible for someontic states to be the result of two or more preparations that lead todifferent assignments of pure quantum states; that is, the same onticstate may be compatible with distinct quantum states.

This gives a nice way of posing the question of quantum state realism:are there preparations corresponding to distinct pure quantum statesthat can give rise to the same ontic state, or, conversely, are thereontic states compatible with distinct quantum states? Pusey, Barrett,and Rudolph (2012) showed that, if one adopts a seemingly naturalindependence assumption about state preparations—namely, theassumption that it is possible to prepare a pair of systems in such away that the probabilities for ontic states of the two systems areeffectively independent—then the answer is negative; anyontological model that reproduces quantum predictions and satisfiesthis Preparation Independence assumption must be a \(\psi\)-onticmodel.

The Pusey, Barrett and Rudolph (PBR) theorem does not close off alloptions for anti-realism about quantum states; an anti-realist aboutquantum states could reject the Preparation Independence assumption,or reject the framework within which the theorem is set; seediscussion in Spekkens (2015): 92–93. See Leifer (2014) for acareful and thorough overview of theorems relevant to quantum staterealism, and Myrvold (2020) for a presentation of a case for quantumstate realism based on theorems of this sort.

5.2 Ontological category of quantum states

The major realist approaches to the measurement problem are all, insome sense, realist about quantum states. Merely saying this isinsufficient to give an account of the ontology of a giveninterpretation. Among the questions to be addressed are: if quantumstates represent something physically real, what sort of thing is it?This is the question of the ontological construal of quantum states.Another question is the EPR question, whether a description in termsof quantum states can be taken as, in principle, complete, or whetherit must be supplemented by different ontology.

De Broglie’s original conception of the “pilot wave”was that it would be a field, analogous to an electromagnetic field.The original conception was that each particle would have its ownguiding wave. However, in quantum mechanics as it was developed at thehands of Schrödinger, for a system of two or more particles thereare not individual wave functions for each particle, but, rather, asingle wave function that is defined on \(n\)-tuples ofpoints in space, where \(n\) is the number ofparticles. This was taken, by de Broglie, Schrödinger and others,to militate against the conception of quantum wave functions asfields. If quantum states represent something in physical reality,they are unlike anything familiar in classical physics.

One response that has been taken is to insist that quantum wavefunctions are fields nonetheless, albeit fields on a space ofenormously high dimension, namely, \(3n\), where \(n\)is the number of elementary particles in the universe. On this view,this high-dimensional space is thought of as more fundamental than thefamiliar three-dimensional space (or four-dimensional spacetime) thatis usually taken to be the arena of physical events. See Albert (1996,2013), for the classic statement of the view; other proponents includeLoewer (1996), Lewis (2004), Ney (2012, 2013a,b, 2021), and North(2013). Most of the discussion of this proposal has taken place withinthe context of nonrelativistic quantum mechanics, which is not afundamental theory. It has been argued that considerations of how thewave functions of nonrelativistic quantum mechanics arise from aquantum field theory undermines the idea that wave functions arerelevantly like fields on configuration space, and also the idea thatconfiguration spaces can be thought of as more fundamental thanordinary spacetime (Myrvold 2015).

A view that takes a wave function as a field on a high-dimensionalspace must be distinguished from a view that takes it to be what Belot(2012) has called amulti-field, which assigns properties to\(n\)-tuples of points of ordinary three-dimensionalspace. These are distinct views; proponents of the \(3n\)-dimensionalconception make much of the fact that it restores Separability: onthis view, a complete specification of the way the world is, at sometime, is given by specification of local states of affairs at eachaddress in the fundamental (\(3n\)-dimensional) space. Taking a wavefunction to be a multi-field, on the other hand, involves acceptingnonseparability. Another difference between taking wave-functions asmulti-fields on ordinary space and taking them to be fields on ahigh-dimensional space is that, on the multi-field view, there is noquestion about the relation of ordinary three-dimensional space tosome more fundamental space. Hubert and Romano (2018) argue thatwave-functions are naturally and straightforwardly construed asmulti-fields.

It has been argued that, on the de Broglie-Bohm pilot wave theory andrelated pilot wave theories, the quantum state plays a role moresimilar to that of a law in classical mechanics; its role is toprovide dynamics for the Bohmian corpuscles, which, according to thetheory, compose ordinary objects. See Dürr, Goldstein, andZanghì (1997), Alloriet al. (2008), Allori(2021).

Dürr, Goldstein, and Zanghì (1992) introduced the term“primitive ontology” for what, according to a physicaltheory, makes up ordinary physical objects; on the de Broglie-Bohmtheory, this is the Bohmian corpuscles. The conception is extended tointerpretations of collapse theories by Alloriet al. (2008).Primitive ontology is to be distinguished from other ontology, such asthe quantum state, that is introduced into the theory to account forthe behavior of the primitive ontology. The distinction is meant to bea guide as to how to conceive of the nonprimitive ontology of thetheory.

6. Quantum computing and quantum information theory

Quantum mechanics has not only given rise to interpretationalconundrums; it has given rise to new concepts in computing and ininformation theory.Quantum information theory is the studyof the possibilities for information processing and transmissionopened up by quantum theory. This has given rise to a differentperspective on quantum theory, one on which, as Bub (2000, 597) putit, “the puzzling features of quantum mechanics are seen as aresource to be developed rather than a problem to be solved”(see the entries onquantum computing andquantum entanglement and information).

7. Reconstructions of quantum mechanics and beyond

Another area of active research in the foundations of quantummechanics is the attempt to gain deeper insight into the structure ofthe theory, and the ways in which it differs from both classicalphysics and other theories that one might construct, by characterizingthe structure of the theory in terms of very general principles, oftenwith an information-theoretic flavour.

This project has its roots in early work of Mackey (1957, 1963),Ludwig (1964), and Piron (1964) aiming to characterize quantummechanics in operational terms. This has led to the development of aframework of generalized probabilistic model. It also has connectionswith the investigations into quantum logic initiated by Birkhoff andvon Neumann (1936) (see the entryquantum logic and probability theory for an overview).

Interest in the project of deriving quantum theory from axioms withclear operational content was revived by the work of Hardy (2001[2008], Other Internet Resources). Significant results along theselines include the axiomatizations of Masanes and Müller (2011)and Chiribella, D’Ariano, and Perinotti (2011). See Chiribellaand Spekkens (2015) for an overview of this burgeoning researcharea.

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Feynman, R.,Lectures on Physics. These are introductory lectures aimed at physics undergraduates.
Hardy, Lucien, 2001 [2008], “Quantum Theory from Five Reasonable Axioms”, manuscript at arxiv.org originally submitted in 2001, but now islabeled version 4 (2008).
Lewis, Peter J., “Interpretations of Quantum Mechanics”,Internet Encyclopedia of Philosophy.
Norton, John, “Origins of Quantum Theory,” A good introduction to the history of quantum theory, about whichlittle is said in the present entry.
PhET Interactive Simulations project, University of Colorado Boulder; these pages contain usefulsimulations of classic quantum experiments.

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