A novel no-go theorem is presented which sets a bound upon the extent to
which '\Psi-epistemic' interpretations of quantum theory are able to explain
the overlap between non-orthogonal quantum states in terms of an experimenter's
ignorance of an underlying state of reality. The theorem applies to any Hilbert
space of dimension greater than two. In the limit of large Hilbert spaces, no
more than half of the overlap between quantum states can be accounted for.
Unlike other recent no-go theorems no additional assumptions, such as forms of
locality, invasiveness, or non-contextuality, are required.Comment: 5 pages. Noise tolerant calculation removed in favour of better
  calculation in forthcoming articl

Maroney, O. J. E.

English

arXiv

A novel no-go theorem is presented which sets a bound upon the extent to which '\Psi-epistemic' interpretations of quantum theory are able to explain the overlap between non-orthogonal quantum states in terms of an experimenter's ignorance of an underlying state of reality. The theorem applies to any Hilbert space of dimension greater than two. In the limit of large Hilbert spaces, no more than half of the overlap between quantum states can be accounted for. Unlike other recent no-go theorems no additional assumptions, such as forms of locality, invasiveness, or non-contextuality, are required

Maroney, OJE

Oxford University Research Archive (ORA)

How statistical are quantum states?

Maroney, Owen JE

ORA - Oxford University Research Archive

Oxford University Research Archive

arXiv:1207.6906v2  [quant-ph]  22 May 2013How statistical are quantum states?O. J. E. Maroney∗Faculty of Philosophy, University of Oxford, 10 Merton Street, Oxford, OX1 4JJ, UK†(Dated: May 23, 2013)A novel no-go theorem is presented which sets a bound upon the extent to which ‘Ψ-epistemic ’interpretations of quantum theory are able to explain the overlap between non-orthogonal quantumstates in terms of an experimenter’s ignorance of an underlying state of reality. The theorem appliesto any Hilbert space of dimension greater than two. In the limit of large Hilbert spaces, no morethan half of the overlap between quantum states can be accounted for. Unlike other recent no-gotheorems no additional assumptions, such as forms of locality, invasiveness, or non-contextuality,are required.One of the most remarkable features of a quantum sys-tem is that of non-orthogonality: it is possible to preparetwo different pure quantum states which cannot be per-fectly discriminated by a single ideal measurement. Nosuch property exists in classical systems, where two dif-ferent classical states can always by perfectly discrimi-nated by a suitably ideal measurement.Statistical interpretations of the quantum state (see [1–4] and references therein) present a natural way to under-stand this feature of non-orthogonality. Suppose that a‘quantum state’ is only a statistical state, associated witha probability distribution over some underlying physicalstates. Even an ideal quantum preparation procedurecannot uniquely fix this underlying state of the systembut instead prepares one of a number of possible suchstates, with a well defined probability. In such an inter-pretation, when two quantum states are non-orthogonal,the associated probability distributions will overlap: thepreparation procedures each have a non-zero probabilityof preparing the same physical states, and this is whythey cannot be perfectly distinguished. This has beencalled a Ψ-epistemic interpretation[2, 3], as the idea isthat the wavefunction does not itself represent a physi-cal state, but instead represents an epistemic uncertaintyabout the physical state.A number of recent no-go theorems[5–10] have ex-plored the consequences of attempting to construct suchΨ-epistemic interpretations of quantum theory. Thispaper contributes a new no-go theorem, demonstratingthat any attempt to explain non-orthogonality entirely interms of an epistemic overlap is impossible: there are no‘maximally epistemic’ interpretations of quantum theory.Using an empirically testable measure, it is shown thatin the limit of large Hilbert spaces, no more than half ofthe overlap between quantum states could be explainedin terms of an overlap of probability distributions. Un-like the other recent results, the theorem presented heremakes no additional assumptions beyond the definitionof a Ψ-epistemic theory. This means that this theorem is∗email:owen.maroney@philosophy.ox.ac.uk†Mailing address: Wolfson College, Linton Road, Oxford, OX26UD, UKthe only such theorem to apply, for example, to the typeof constructive Ψ-epistemic theories presented in [10, 11].Ontological models and Ψ-epistemic theories.- Twoprincipal assumptions are involved in analysing whethera theory is Ψ-epistemic :1. After a physical procedure prepares a quantumstate, |ψ〉, the system is actually in a physical stateλ. This physical state is not necessarily identifiedwith the quantum state;2. The probabilities of getting the results of a mea-surement procedure on the system are wholly de-termined by the physical state λ and the measure-ment procedure. The preparation procedure onlyinfluences the measurement outcomes indirectly,through the possible physical states prepared.These assumptions are developed into the ontologicalmodels formalism in[3, 12–15], where the physical stateλ, is referred to as an ontic state.• A preparation procedure, which prepares the quan-tum state |ψ〉, will prepare some ontic state λ withprobability µψ(λ):µψ(λ) ≥ 0,∫µψ(λ)dλ = 1 (1)• A measurement, M , has a number of possible out-comes {Q}, and a probability ξM (Q|λ) of obtaininga particular outcome Q, given the ontic state λ:ξM (Q|λ) ≥ 0,∑QξM (Q|λ) = 1 (2)• An ontological model will reproduce the results ofquantum theory if, and only if:∫µψ(λ)ξM (Q|λ)dλ = |〈Q |ψ〉|2(3)Additional structure is required to represent unitary op-erations, but this is not needed for this proof.Some immediate consequences of this formalism willbe useful. The set Λφ = {λ : µφ(λ) > 0} is the set of2all possible ontic states which may occur when preparingthe quantum state |φ〉.If one of the outcomes of the measurement Mis represented by the projection onto |φ〉, then∫µφ(λ)ξM (φ|λ)dλ = |〈φ |φ〉|2= 1. It follows that∀λ ∈ Λφ ξM (φ|λ) = 1 (4)If |ϕ〉 is orthogonal to |φ〉 then∫µϕ(λ)ξM (φ|λ)dλ =|〈φ |ϕ〉|2 = 0. It follows that∀λ ∈ Λϕ ξM (φ|λ) = 0 (5)which implies Λϕ ∩Λφ = ∅. (Sets of measure zero can beignored.)If the quantum two states |ψ〉 and |φ〉 are non-orthogonal, then:∫µψ(λ)ξM (φ|λ)dλ = |〈φ |ψ〉|2 6= 0 (6)This allows the possibility that Λψ ∩ Λφ is not empty.A Ψ-epistemic theory is defined[3] to be one in whichthere exist at least two distinct non-orthogonal quantumstates, |ψ〉 and |φ〉, for which Λψ ∩ Λφ 6= ∅. In sucha theory, the fact that the quantum states |ψ〉 and |φ〉cannot be discriminated by a single ideal measurementis, at least partially, explained by the fact that there issome λ ∈ Λψ∩Λφ. As all the results of any measurementM are determined by ξM (Q|λ), and both |ψ〉 and |φ〉 canprepare ontic states in Λψ ∩ Λφ, it follows that there isno possible measurement outcome which can occur withzero probability for |ψ〉 but with certainty for |φ〉 (andvice versa).The term∫Λφµψ(λ)dλ gives a measure of the epistemicoverlap between the quantum states |ψ〉 and |φ〉. This isbound by the measure of the quantum state overlap. As∀λ ∈ Λφ, ξM (φ|λ) = 1∫Λφµψ(λ)dλ =∫Λφµψ(λ)ξM (φ|λ)dλ≤∫µψ(λ)ξM (φ|λ)dλ = |〈φ |ψ〉|2 (7)The degree of epistemic overlap, Ω [φ, ψ], between twostates can now be defined via:∫Λφµψ(λ)dλ = Ω [φ, ψ] |〈φ |ψ〉|2(8)By definition, Ω [ψ, ψ] = 1. Where 〈φ |ψ〉 = 0, Ω [φ, ψ]may take any value.A maximally Ψ-epistemic theory would haveΩ [φ, ψ] = 1 for all pairs of quantum states. The quantumstate overlap |〈φ |ψ〉|2 would be accounted for entirely interms of the overlap in the ontic states Λψ ∩ Λφ.By contrast, a maximally Ψ-ontic theory1 is one inwhich Λψ ∩ Λφ = ∅ for all pairs of distinct quantum1 In much of the existing literature this is simply referred to as aΨ-ontic theory.states, and so Ω [φ, ψ] = 0 . In such a theory, no two dis-tinct quantum state preparations |ψ〉 and |φ〉 could pos-sibly produce the same ontic state λ. For a maximallyΨ-ontic theory, therefore, given the ontic state λ it is al-ways possible to identify the quantum state |ψ〉. The deBroglie-Bohm[16, 17], Everett[18] and CSL[19] interpre-tations of quantum theory are all examples of maximallyΨ-ontic theories.It will now be shown that, in Hilbert spaces of dimen-sion greater than two, there are no models of quantumtheory for which Ω [φ, ψ] = 1 for all states: there are nomaximally Ψ-epistemic interpretations. Making a sym-metrical assumption that Ω [φ, ψ] is a constant, Ω, be-tween all pairs of distinct states it will be shown thatΩ ≤ 910 for a 3 dimensional Hilbert space, falling to Ω ≤12in the limit of large Hilbert spaces. No more than 50%of the quantum state overlap can be accounted for by anepistemic overlap of probability distributions.Three dimensional Hilbert space.- Consider the follow-ing states of a three dimensional Hilbert space:|a〉 , |b〉 , |c〉|p〉 = 1√3(|a〉+ |b〉+ |c〉) , |m〉 = 1√3(|a〉+ |b〉 − |c〉)|a+〉 =1√2(|a〉+ |c〉) , |a−〉 =1√2(|a〉 − |c〉)|b+〉 =1√2(|b〉+ |c〉) , |b−〉 =1√2(|b〉 − |c〉)where |a〉 , |b〉 , |c〉 form an orthonormal basis, with thethree measurements:M1 : |a+〉 〈a+ | , |a−〉 〈a− | , |b〉 〈b |M2 : |b+〉 〈b+ | , |b−〉 〈b− | , |a〉 〈a |M3 : |a〉 〈a | , |b〉 〈b | , |c〉 〈c |(These are the states and measurements used inthe Clifton-Stairs proof of quantum contextuality[20,21]. For further connections to quantum contextuality,see[22].)The results are shown in Table I. From M1 it can beseen that∀λ ∈ Λa, ξM1(b|λ) = 0∀λ ∈ Λp, ξM1(a−|λ) = 0∀λ ∈ Λm, ξM1(a+|λ) = 0(9)For any given λ, it must be that ξM1(b|λ) + ξM1(a−|λ) +ξM1(a+|λ) = 1, so this means:Λa ∩ Λp ∩ Λm = ∅ (10)SimilarlyΛc ∩ Λp ∩ Λm = ∅ (11)M2 addsΛb ∩ Λp ∩ Λm = ∅ (12)leading to(Λa ∪ Λb ∪ Λc) ∩ Λp ∩ Λm = ∅ (13)3(a)M1|b〉 〈b | |a+〉 〈a+ | |a−〉 〈a− ||a〉 0 1212|b〉 1 0 0|c〉 0 1212|p〉 13230|m〉 130 23(b)M2|a〉 〈a | |b+〉 〈b+ | |b−〉 〈b− ||a〉 1 0 0|b〉 0 1212|c〉 0 1212|p〉 13230|m〉 130 23(c)M3|a〉 〈a | |b〉 〈b | |c〉 〈c ||a〉 1 0 0|b〉 0 1 0|c〉 0 0 1|p〉 131313|m〉 131313TABLE I: Measurement outcomesIf there is any overlap in Λp∩Λm then it must lie outsideΛa ∪ Λb ∪ Λc.Now M3 gives∫Λaµp(λ)dλ =Ω [a, p]3(14)and similarly for |b〉 and |c〉. As Λb ∩ Λa = ∅ etc.∫(Λa∪Λb∪Λc)µp(λ)dλ =(Ω [a, p] + Ω [b, p] + Ω [c, p])3(15)It follows that:∫Λmµp(λ)dλ ≤ 1−(Ω [a, p] + Ω [b, p] + Ω [c, p])3(16)but∫Λmµp(λ)dλ =Ω [m, p]9(17)givingΩ [m, p]9≤ 1−(Ω [a, p] + Ω [b, p] + Ω [c, p])3(18)If the ontic model has maximal epistemic overlaps withthe |a〉 , |b〉 , |c〉 states, then Ω [a, p] = Ω [b, p] = Ω [c, p] =1, but this requires Ω [m, p] = 0 while |〈m |p〉|2 = 19 .This proves that a maximally Ψ-epistemic theory is notpossible.Reducing the degree of epistemicness in the |a〉 , |b〉 , |c〉basis will allow some epistemic overlap between |p〉 and|m〉. For example, keeping Ω [a, p] = Ω [b, p] = 1 it ispossible to reach Ω [m, p] = 1 with Ω [c, p] = 23 . Moresymmetrically, Ω [m, p] = 1 is possible with Ω [a, p] =Ω [b, p] = Ω [c, p] = 89 .A basis independent measure of how epistemic a theorycan be and still reproduce quantum statistics is given byletting Ω [φ, ψ] = Ω be a constant for all non-orthogonalstates. This gives:Ω ≤910(19)Higher dimensional Hilbert spaces.- The above resultscan be generalised to Hilbert spaces of dimension d > 3.In the limit of large d, Ω ≤ 12 .Take the basis states|a1〉 , |a2〉 , . . . |ad〉the superpositions|ai+〉 =1√2(|ai〉+ |ad〉)|ai−〉 =1√2(|ai〉 − |ad〉)|pd〉 =1√d∑i=1,d |ai〉|md〉 =1√d(∑i=1,d−1 |ai〉)− 1√d|ad〉and the measurementsMi : |a1〉 〈a1 | , . . . , |ai−1〉 〈ai−1 | ,|ai+1〉 〈ai+1 | , . . . , |ad−1〉 〈ad−1 | ,|ai+〉 〈ai+ | , |ai−〉 〈ai− |Md : |a1〉 〈a1 | , . . . , |ad〉 〈ad |Each measurement Mi now shows:Λai ∩ Λpd ∩ Λmd = ∅Λad ∩ Λpd ∩ Λmd = ∅(20)leading to(∪iΛai) ∩ Λpd ∩ Λmd = ∅ (21)From the Md measurement:∫(∪iΛai)µpd(λ)dλ =∑iΩ [ai, pd] |〈ai |pd〉|2=1d∑iΩ [ai, pd](22)so∫Λmdµpd(λ)dλ ≤ 1−1d∑iΩ [ai, pd] (23)givingΩ [md, pd](1−2d)2≤ 1−1d∑iΩ [ai, pd] (24)4Once again, Ω [ai, pd] = 1 would require Ω [md, pd] = 0,even though |〈md |pd〉|2=(1− 2d)2→ 1. For a theory tobe maximally Ψ-epistemic in a specific basis, then as d→∞ it must become maximally Ψ-ontic between at leastsome states arbitrarily close to each other. To obtaina full overlap Ω [md, pd] = 1 by symmetrically reducingthe overlap with the |ai〉 basis would require Ω [ai, pd] =4d2(d− 1) → 0.The basis independent measure of how epistemic thisallows the theory overall sets Ω [φ, ψ] = Ω(d) as beforegivingΩ(d) ≤d22d2 − 4d+ 4(25)In the limit d gets large, Ω(d) ≤ 12 .Experimental Noise.- The calculations depend uponobtaining zero probability of obtaining a set of outcomesfor particular measurements. In any real experiment,there will be some noise, giving a value ǫ > 0. In suchcases the results calculated above become unstable.Following [5], the measure of overlap in the ontic statedistribution which can be compared to experimental datain the presence of noise is the quantum trace norm dis-tance, rather than the quantum state overlap:δQ(ρ, σ) =12Tr |ρ− σ|2 (26)This has an operational interpretation: if one of twoquantum states, ρ or σ is prepared with equal proba-bility, then using an optimal quantum measurement, theprobability of correctly identifying which is given byPQ =12(1 + δQ(ρ, σ)) (27)This should be compared to the classical trace norm dis-tanceδC(p, q) =12∫|p(x)− q(x)|2 dx (28)which also has an operational interpretation: if a randomvariable x is prepared by one of two methods, producingclassical distributions p(x) and q(x) respectively, the op-timal probability of guessing which method was used isgiven byPC =12(1 + δC(p, q)) (29)In any ontological model for quantum theory, the mea-suring device is presented with an ontic state λ, preparedwith the probability distribution µφ(λ) or µψ(λ). As thisis the most information the optimal measuring devicehas available, the probability of guessing which quantumstate φ or ψ was prepared, based only on the outputs ofthe measuring device, cannot be greater than the prob-ability of guessing based upon the ontic state itself, soPQ ≤ PC .The quantum trace distance for pure states isδQ(φ, ψ) =√1− |〈φ |ψ〉|2 (30)while the classical trace norm distance can be expressedasδC(φ, ψ) = 1−∫min[µφ(λ), µφ(λ)]dλ (31)leading to∫min[µφ(λ), µφ(λ)]dλ ≤ 1−√1− |〈φ |ψ〉|2 (32)In a maximally Ψ-ontic theory, the left hand side is zero,while in a maximally Ψ-epistemic theory the equalityshould be reached. A routine but lengthy calculationshows that for the experimental arrangements consideredhere, the inequality is strict between some pairs of statesin all Hilbert spaces of dimension d > 14.∫min[µφ(λ), µφ(λ)]dλ1−√1− |〈φ |ψ〉|2< 1 (33)A discussion of this measure, as well as an experimentalarrangement better suited to comparing the trace normdistances which extends the result to all Hilbert spacesof dimension d > 2 will be discussed in the forthcoming[23].Conclusions.- There exist constructive examples of Ψ-epistemic models, which can reproduce all the statisticsof quantum theory (such as the LJBR model[11] and theABCL model[10]). It is therefore not the intention of thisarticle to attempt to construct no-go theorems which aimto show that Ψ-epistemic theories are impossible sim-pliciter. Existing no-go theorems have effectively soughtto prove Ω [φ, ψ] = 0, but to do so must assume addi-tional properties, such as forms of locality[6], factoris-ability of product states[5] or non-invasiveness[8]. Thisshould not downplay the significance of these theorems:any attempt to construct Ψ-epistemic theories must nec-essarily reject these properties, in much the same waythat Bell’s theorem[24] requires realist interpretations ofquantum theory to reject local causality. They help tomap out the space of possible theories which can accountfor quantum phenomena.The no-go theorem presented here contributes to thistask. It does not make additional assumptions so applieseven to theories that reject all of these additional as-sumptions, such as the LJBR and ABCL models. Thesemodels demonstrate that Ω [φ, ψ] > 0 is indeed possible.This no-go theorem shows that Ω [φ, ψ] < 1, and this re-sult follows directly from the definition of a Ψ-epistemicmodel, even if all other assumptions are dropped.AcknowledgementsI would like to thank Chris Timpson, Richard George,Harvey Brown, George Knee, Andrew Briggs, ErikGauger, Simon Benjamin and Jon Barrett for helpfuldiscussions and comments. This research is supportedby the John Templeton Foundation.5[1] L. E. Ballentine, Reviews of Modern Physics 42, 358(1970).[2] R. W. Spekkens, Physical Review A 75, 032110 (2007).[3] N. Harrigan and R. W. Spekkens, Foundations of Physics40, 125 (2010).[4] S. D. Bartlett, T. Rudolph, and R.W. Spekkens, PhysicalReview A 86, 012103 (2012).[5] M. F. Pusey, J. Barrett, and T. Rudolph, Nature Physics8, 476 (2012).[6] R. Colbeck and R. Renner, Physical Review Letters 108,150402 (2012).[7] M. Schlosshauer and A. Fine, Physical Review Letters108, 260404 (2012).[8] L. 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How statistical are quantum states?

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