On the reality of the quantum state Matthew F. Pusey,1, ∗ Jonathan Barrett,2 and Terry Rudolph1 1Department of Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom 2Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham TW20 0EX, United Kingdom (Dated: April 11, 2012) Quantum states are the key mathematical objects in quantum theory. It is therefore surprising that physicists have been unable to agree on what a quantum state truly represents. One possibility is that a pure quantum state corresponds directly to reality. However, there is a long history of suggestions that a quantum state (even a pure state) represents only knowledge or information about some aspect of reality. Here we show that any model in which a quantum state represents mere information about an underlying physical state of the system, and in which systems that are prepared independently have independent physical states, must make predictions which contradict those of quantum theory. At the heart of much debate concerning quantum the- we cannot know what we are talking about; it ory lies the quantum state. Does the wave function corre- is just that simple.[9] spond directly to some kind of physical wave? If so, it is an odd kind of wave, since it is defined on an abstract con- Here we present a no-go theorem: if the quantum state figuration space, rather than the three-dimensional space merely represents information about the real physical we live in. Nonetheless, quantum interference, as ex- state of a system, then experimental predictions are ob- hibited in the famous two-slit experiment, appears most tained which contradict those of quantum theory. The readily understood by the idea that it is a real wave that argument depends on few assumptions. One is that a is interfering. Many physicists and chemists concerned system has a \real physical state" { not necessarily com- with pragmatic applications of quantum theory success- pletely described by quantum theory, but objective and fully treat the quantum state in this way. independent of the observer. This assumption only needs Many others have suggested that the quantum state is to hold for systems that are isolated, and not entangled something less than real [1{8]. In particular, it is often with other systems. Nonetheless, this assumption, or argued that the quantum state does not correspond di- some part of it, would be denied by instrumentalist ap- rectly to reality, but represents an experimenter's knowl- proaches to quantum theory, wherein the quantum state edge or information about some aspect of reality. This is merely a calculational tool for making predictions con- view is motivated by, amongst other things, the collapse cerning macroscopic measurement outcomes. The other of the quantum state on measurement. If the quantum main assumption is that systems that are prepared inde- state is a real physical state, then collapse is a myste- pendently have independent physical states. rious physical process, whose precise time of occurrence In order to make some of these notions more precise, let is not well-defined. From the `state of knowledge' view, us begin by considering the classical mechanics of a point the argument goes, collapse need be no more mysterious particle moving in one dimension. At a given moment than the instantaneous Bayesian updating of a probabil- of time, the physical state of the particle is completely ity distribution upon obtaining new information. specified by its position x and momentum p, and hence The importance of these questions was eloquently corresponds to a point (x; p) in a two-dimensional phase stated by Jaynes: space. Other physical properties are either fixed, such as mass or charge, or are functions of the state, such as But our present [quantum mechanical] arXiv:1111.3328v3 [quant-ph] 18 Nov 2012 energy H(x; p). Viewing the fixed properties as constant formalism is not purely epistemological; it is functions, let us define \physical property" to mean some a peculiar mixture describing in part realities function of the physical state. of Nature, in part incomplete human infor- Sometimes, the exact physical state of the particle mation about Nature | all scrambled up by might be uncertain, but there is nonetheless a well- Heisenberg and Bohr into an omelette that defined probability distribution µ(x; p). Although µ(x; p) nobody has seen how to unscramble. Yet we evolves in a precise manner according to Liouville's equa- think that the unscrambling is a prerequisite tion, it does not directly represent reality. Rather, µ(x; p) for any further advance in basic physical the- is a state of knowledge: it represents an experimenter's ory. For, if we cannot separate the subjec- uncertainty about the physical state of the particle. tive and objective aspects of the formalism, Now consider a quantum system. The hypothesis is that the quantum state is a state of knowledge, repre- senting uncertainty about the real physical state of the ∗ [email protected] system. Hence assume some theory or model, perhaps 2 A m mL L′ l measure B mL m L′ l l 1 l2 D FIG. 1. Our definition of a physical property is illustrated. Consider a collection, labelled by L, of probability distribu- tions fµL(λ)g. λ denotes a system's physical state. If every pair of distributions are disjoint, as in a, then the label L is uniquely fixed by λ and we call it a physical property. If, how- ever, L is not a physical property, then there exists a pair of labels L; L0 with distributions that both assign positive prob- FIG. 2. Two systems are prepared independently. The quan- ability to some overlap region ∆, as in b.A λ from ∆ is tum state of each, determined by the preparation method, consistent with either label. is either j0i or j+i. The two systems are brought together and measured. The outcome of the measurement can only depend on the physical states of the two systems at the time of measurement. undiscovered, which associates a physical state λ to the system. If a measurement is performed, the probabilities for different outcomes are determined by λ. If a quantum system is prepared in a particular way, then quantum theory associates a quantum state (assume for simplic- Our main result is that for distinct quantum states ity that it is a pure state) . But the physical state λ and , if the distributions µ (λ) and µ (λ) over- j i 0 1 0 1 need not be fixed uniquely by the preparation { rather, lapj i (morej precisely:i if ∆, the intersection of their sup- the preparation results in a physical state λ according to ports, has non-zero measure) then there is a contradic- some probability distribution µ (λ). tion with the predictions of quantum theory. We present Given such a model, Harrigan and Spekkens[10] give a first a simple version of the argument, which works when precise meaning to the idea that a quantum state corre- 0 1 = 1=p2. Then the argument is extended to ar- j h j i j sponds directly to reality or represents only information. bitrary 0 and 1 . Finally, we present a more formal To explain this, the example of the classical particle is version ofj thei argumentj i which works even in the presence again useful. Here, if an experimenter knows only that of experimental error and noise. the system has energy E, and is otherwise completely uncertain, the experimenter's knowledge corresponds to Consider two methods of preparing a quantum sys- a distribution µ (x; p) uniform over all points in phase tem, corresponding to quantum states 0 and 1 , with E j i j i space with H(x; p) = E. Seeing as the energy is a physi- 0 1 = 1=p2. Choose a basis of the Hilbert space so j h j i j cal property of the system, different values of the energy that 0 = 0 and 1 = + = ( 0 + 1 )=p2. In order E and E0 correspond to disjoint regions of phase space, to derivej i a contradiction,j i j i supposej i j thati j thei distributions hence the distributions µE(x; p) and µE0 (x; p) have dis- µ0(λ) and µ1(λ) overlap. Then there exists q > 0 such joint supports. On the other hand, if two probability that preparation of either quantum state results in a λ distributions µL(x; p) and µL0 (x; p) have overlapping sup- from the overlap region ∆ with probability at least q. ports, i.e. there is some region ∆ of phase space where both distributions are non-zero, then the labels L and L0 Now consider two systems whose physical states are cannot refer to a physical property of the system. See uncorrelated. This can be achieved, for example, by con- Figure 1. structing and operating two copies of a preparation de- Similar considerations apply in the quantum case. vice independently. Each system can be prepared such Suppose that, for any pair of distinct quantum states 0 that its quantum state is either 0 or 1 , as illus- j i j i 2 j i and 1 , the distributions µ0(λ) and µ1(λ) do not over- trated in Figure 2. With probability q > 0 it happens j i lap: then, the quantum state can be inferred uniquely that the physical states λ1 and λ2 are both from the over- from the physical state of thej systemi and hence satisfies lap region ∆. This means that the physical state of the the above definition of a physical property. Informally, two systems is compatible with any of the four possible every detail of the quantum state is \written into" the quantum states 0 0 , 0 + , + 0 and + + .