On the Reality of the Quantum State

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, representing 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 distributions {µL(λ)}. 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, however, 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 |0i or |+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) ψ . But the physical state λ ψ and ψ , if the distributions µ (λ) and µ (λ) over- | i 0 1 0 1 need not be fixed uniquely by the preparation – rather, |lapi (more| 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 µψ(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- ψ0 ψ1 = 1/√2. Then the argument is extended to ar- | h | i | sponds directly to reality or represents only information. bitrary ψ0 and ψ1 . Finally, we present a more formal To explain this, the example of the classical particle is version of| thei argument| 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 ψ0 and ψ1 , with E | i | i space with H(x, p) = E. Seeing as the energy is a physi- ψ0 ψ1 = 1/√2. Choose a basis of the Hilbert space so | h | i | cal property of the system, different values of the energy that ψ0 = 0 and ψ1 = + = ( 0 + 1 )/√2. In order E and E0 correspond to disjoint regions of phase space, to derive| i a contradiction,| i | i suppose| i | thati | 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 ψ0 that its quantum state is either ψ0 or ψ1 , as illus- | i | i 2 | i and ψ1 , the distributions µ0(λ) and µ1(λ) do not over- trated in Figure 2. With probability q > 0 it happens | i lap: then, the quantum state ψ can be inferred uniquely that the physical states λ1 and λ2 are both from the over- from the physical state of the| 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 + + . | i⊗| i | i⊗| i | i⊗| i | i⊗| i real physical state of affairs. But if µ0(λ) and µ1(λ) overlap for at least one pair of quantum states, then ψ can The two systems are brought together and measured. justifiably be regarded as “mere” information. | i The measurement is an entangled measurement, which 3 projects onto the four orthogonal states: 1  ξ1 = √ ( 0 1 + 1 0 ), x1 Zb H | i 2 | i ⊗ | i | i ⊗ | i 1 ξ2 = √ ( 0 + 1 + ), | i 2 | i ⊗ |−i | i ⊗ | i 1 ξ3 = √ ( + 1 + 0 ), | i 2 | i ⊗ | i |−i ⊗ | i 1  x Zb H ξ4 = √ ( + + + ), (1) 2

| i 2 | i ⊗ |−i |−i ⊗ | i

......

where = ( 0 1 )/√2. The ﬁrst outcome is orthog- ...

onal to|−i0 0| ,i hence − | i quantum theory predicts that this outcome| hasi⊗| probabilityi zero when the quantum state is 0 0 . Similarly, outcome ξ2 has probability zero  Z H | i ⊗ | i | i xn1 b if the state is 0 + , ξ3 if + 0 , and ξ4 if + + . This| i leads ⊗ | immediatelyi | i | i to ⊗ the | i desired| con-i tradiction.| i ⊗ | i At least q2 of the time, the measuring device is uncertain which of the four possible preparation meth-  Z Ra H ods was used, and on these occasions it runs the risk of xn b giving an outcome that quantum theory predicts should occur with probability 0. Importantly, we have needed to say nothing about the value of q per se to arrive at FIG. 3. The main argument requires a joint measurement this contradiction. on n qubits with the property that each outcome has probability zero on one of the input states. Such a measure- We have shown that the distributions for 0 and + | i | i ment can be performed by implementing the quantum cir- cannot overlap. If the same can be shown for any pair cuit shown, followed by a measurement of each qubit in of quantum states ψ0 and ψ1 , then the quantum state the computational basis. The single qubit gates are given | i | i iβ can be inferred uniquely from λ. In this case, the quan- by Zβ = |0i h0| + e |1i h1| and the Hadamard gate H = tum state is a physical property of the system. |+i h0| + |−i h1|. The entangling gate in the middle rotates iα For any pair of distinct non-orthogonal states ψ0 and the phase of only one state: Rα |00 ... 0i = e |00 ... 0i, leav- | i ψ1 , a basis of the Hilbert space can be chosen such that ing all other computational basis states unaffected. | i ψ0 = cos(θ/2) 0 + sin(θ/2) 1 | i | i | i ψ1 = cos(θ/2) 0 sin(θ/2) 1 , (2) A suitable measurement is most easily described as a | i | i − | i quantum circuit, followed by a measurement onto the with 0 < θ < π/2. These states span a two-dimensional 0 , 1 basis for each qubit. It is illustrated in Figure 3. subspace of the Hilbert space. We can restrict attention {| Thei | i} circuit is parameterized by two real numbers, α to this subspace and from hereon, without loss of gen- and β. In Appendix A it is shown that for any 0 < θ < erality, treat the systems as qubits. As above, suppose π/2, and for any n chosen large enough that 21/n 1 that there is a probability at least q > 0 that the physical tan(θ/2), it is possible to choose α and β such that− the≤ state of the system after preparation is compatible with measurement has the desired feature: each outcome has, either preparation method having been used, that is, the according to quantum theory, probability zero on one of resulting λ is in ∆. the states Ψ(x1 . . . xn) . A contradiction is obtained when n uncorrelated sys- The presentation| soi far has been somewhat heuristic. tems are prepared, where n will be fixed shortly. Depend- We turn to a more formal statement of the result, in- ing on which of the two preparation methods is used each cluding the possibility of experimental error. This is im- time, the n systems are prepared in one of the quantum portant because the argument so far uses the fact that states: quantum probabilities are sometimes exactly zero. It is important to have a version of the argument which is Ψ(x1 . . . xn) = ψx ψx ψx , (3) | i | 1 i ⊗ · · · ⊗ | n−1 i ⊗ | n i robust against small amounts of noise. Otherwise the where xi 0, 1 , for each i. Since the preparations are conclusion – that the quantum state is a physical prop- independent,∈ { there} is a probability at least qn that the erty of a quantum system – would be an artificial feature complete physical state of the systems emerging from the of the exact theory, but irrelevant to the real world and devices is compatible with any one of these 2n quantum experimental test would be impossible. states. The contradiction is obtained if there is a joint Let us restate our assumptions more mathematically. measurement on the n systems such that each outcome First, assume a measure space Λ, understood as the set of has probability zero on at least one of the Ψ(x1 . . . xn) . possible physical states λ that a system can be in. Prepa- | i (This type of measurement was first introduced in a dif- ration of the quantum state ψi is assumed to result a | i ferent context by Caves, Fuchs and Schack [11]; in their λ sampled from a probability distribution µi(λ) over Λ. terminology, the existence of such a measurement shows Second, assume that it is possible to prepare n systems the states are Post-Peierls-incompatible.) independently, with quantum states ψx ,..., ψx , re- | 1 i | n i 4 sulting in physical states λ1, . . . , λn distributed according assume the same thing for entangled systems. Neither to the product distribution theorem assumes underlying determinism. Bell’s theorem assumes that it is possible to make

µx1 (λ1)µx2 (λ2) µxn (λn). (4) independent choices of measurement, and since local ··· models which drop measurement independence can be Finally, assume that λ1, . . . , λn fixes the probability for constructed[14, 15], this assumption is necessary. Some- the outcome k of a measurement according to some what analogously, models where the quantum state is not probability distribution p(k λ1, . . . , λn). The operational a physical property can be constructed by dropping our | probabilities p (k Ψ(x1 . . . xn)) are given by assumption of preparation independence[16]. Since both | assumptions are very reasonable, it is not surprising that Z Z in both cases the models obtained by dropping them ap- p(k λ1, . . . , λn)µx1 (λ1) µxn (λn)dλ1 dλn. Λ ··· Λ | ··· ··· pear extremely contrived. (5) An important step towards the derivation of our re- If an experiment is performed, it will be possible to sult is the idea that the quantum state is physical if establish with high confidence that the probability for distinct quantum states correspond to non-overlapping each measurement outcome is within of the predicted distributions for λ. The precise formalisation of this quantum probability for some small > 0. The final idea appeared in Spekkens[17] and in Harrigan and result relates to the total variation distance [12] between Spekkens[10], and is also due to Hardy[18]. In the ter- µ0 and µ1, defined by minology of Harrigan and Spekkens, we have shown that ψ-epistemic models cannot reproduce the predictions of 1 Z D(µ0, µ1) = µ0(λ) µ1(λ) dλ. (6) quantum theory. The general notion that two distinct 2 Λ | − | quantum states may describe the same state of reality, however, has a long history. For example, in a letter It is a measure of how easy it is to distinguish two prob- to Schroedinger containing a variant of the famous EPR ability distributions. If D(µ0, µ1) = 1, then µ0 and µ1 (Einstein-Podolsky-Rosen) argument[1], Einstein argues are completely disjoint. In this case, the probability of from locality to the conclusion that λ being compatible with both preparations (q above) is zero. In Appendix B we show that if the probabilities ...for the same [real] state of [the system predicted by a model are within of the quantum prob- at] B there are two (in general arbitrarily abilities then many) equally justified ΨB, which contradicts the hypothesis of a one-to-one or complete de- n D(µ0, µ1) 1 2 √, (7) scription of the real states. [19] ≥ − 1/n for 2 1 tan(θ/2). For small , D(µ0, µ1) is close In this version of the argument, Einstein really is con- to 1. Hence− ≤ a successful experiment would show that λ cerned with the possibility that there are two distinct is normally closely associated with only one of the two quantum states for the same reality. He is not conclud- quantum states. ing that there are two different states of reality corre- Performing an experiment to implement the circuit in sponding to the same quantum state (which would be Figure 3 for small values of n is challenging but not un- the more commonly understood notion of incompleteness realistic given current technology. While all the gates associated with Einstein).” required have already been demonstrated at some point, Finally, what are the consequences if we simply accept our result requires such gates acting with high fidelity both the assumptions and the conclusion of the theorem? in a non post-selected fashion (this latter because other- If the quantum state is a physical property of a system wise the measuring device can use the extra freedom in then quantum collapse must correspond to a – problem- the postselection to escape the zero-probability outcomes atic and poorly defined – physical process. If there is those times it is unsure of the preparation procedure). no collapse, on the other hand, then after a measure- In conclusion, we have presented a no-go theorem, ment takes place, the joint quantum state of the sys- which – modulo assumptions – shows that models in tem and measuring apparatus is entangled and contains which the quantum state is interpreted as mere informa- a component corresponding to each possible macroscopic tion about an objective physical state of a system cannot measurement outcome. This would be unproblematic if reproduce the predictions of quantum theory. The re- the quantum state merely reflected a lack of information sult is in the same spirit as Bell’s theorem[13], which about which outcome occurred. But if the quantum state states that no local theory can reproduce the predictions is a physical property of the system and apparatus, it is of quantum theory. Both theorems need to assume that hard to avoid the conclusion that each macroscopically a system has a objective physical state λ such that prob- different component has a direct counterpart in reality. abilities for measurement outcomes depend only on λ. On a related, but more abstract note, the quantum But our theorem only assumes this for systems prepared state has the striking property that the number of real in isolation from the rest of the universe in a quantum parameters needed to specify it is exponential in the num- pure state. This is unlike Bell’s theorem, which needs to ber of systems n. This is to be expected if the quantum 5 state represents information but is – to us – very sur- ble measurement outcomes, and not about the objective prising if it has a direct image in reality. Note that in state of a system [24]. Another is to construct concrete previous work, Hardy has shown that the set Λ of physi- models of reality wherein one or more of our assumptions cal states must have infinite cardinality [20], and Montina fail. has shown that, given some assumptions about the underlying dynamics, the physical state must have at least as many real parameters as the quantum state [21, 22]. Similar conclusions can be drawn from ideas in commu- ACKNOWLEDGMENTS nication complexity [23]. For these reasons and others, many will continue to We thank Koenraad Audenaert for code, and Lucien view the quantum state as representing information. One Hardy, Matt Leifer, and Rob Spekkens for useful discus- approach is to take this to be information about possi- sions. All authors are supported by the EPSRC.

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⊗n ⊗n x1 . . . xn H RαZ ψx ψx h | β | 1 i ⊗ · · · ⊗ | n i ! 1 X ~x.~z ⊗n = ( 1) ~z RαZ ψx ψx √2n − h | β | 1 i ⊗ · · · ⊗ | n i ~z   1 iα X ~x.~z ⊗n = e 0 0 + ( 1) ~z  Z ψx1 ψxn √2n h ··· | − h | β | i ⊗ · · · ⊗ | i ~z6=00···0   n 1 iα X ~x.~z O θ x iβ θ = e 0 0 + ( 1) ~z  cos 0 + ( 1) i e sin 1 √2n h ··· | − h | 2 | i − 2 | i ~z6=00···0 i=1   n n−|~z| |~z| 1 θ iα X ~x.~z θ θ i|~z|β ~x.~z =  cos e + ( 1) cos sin e ( 1)  √2n 2 − 2 2 − ~z6=00···0 n n n−k k ! 1 θ X n θ θ = cos eiα + cos sin eikβ √ n 2 k 2 2 2 k=1 1 θ n θ n = cos eiα + 1 + eiβ tan 1 . (A7) √2n 2 2 −

P In the fifth line, ~z = i zi. within it. Consider Finally, we| | show that for any θ with 1 π n 2 arctan 2 n 1 θ , the angles α and β θ − ≤ ≤ 2 f(0) = 1 1 + tan . (A10) can be chosen so that − 2 θ n iα iβ 1 e + 1 + e tan 1 = 0, (A8) θ n 2 − Since tan 2 2 1, f(0) 1, hence it is outside (or on) the unit≥ circle.− On the other≤ − hand, and hence the probability is zero as required. Rearrang- ing, the required α will always exist (and be easy to find) n θ provided there exists a β with f(π) = 1 1 tan . (A11) − − 2 n iβ θ 1 1 + e tan = 1. (A9) − 2 θ Since 0 tan 2 1, 0 f(π) 1, hence it is inside (or on) the≤ unit circle.≤ This≤ concludes≤ the proof. Such a β exists if the curve of f(β) = 1 If the actual value of β for a particular θ and n is iβ θ n − 1 + e tan 2 in the complex plane intersects the unit required, it is not difficult to find it numerically. For circle, as in Figure 4. Since f is continuous, it suffices n = 2, (A9) can even be solved analytically to find β = to exhibit one point outside the unit circle and one point arccos (1 4t2 t4)/4t3 where t = tan θ . − − 2 7

ℑ µ0(λ) and µ1(λ) is 1 Z D(µ0, µ1) = µ0(λ) µ1(λ) dλ. 2 Λ | − | The aim is to show that if a model of the above form reproduces the predictions of quantum theory approxi- mately, so that for any measurement outcome, Eq. (B1) holds to within , then f(0) f(π) n D(µ0, µ1) 1 2 √. (B2) ℜ ≥ − Eq. (B2) holds for preparations of any pair of pure states ψ0 and ψ1 , as long as n is chosen to satisfy Eq. (A3). | Toi this| end,i consider n independent preparations of quantum systems, where each can be chosen such that the quantum state is either ψ0 or ψ1 . The joint quantum state is a direct product| giveni | byi Eq. (A6). These systems will be brought together so that the joint measurement illustrated in Figure 2 and described in Sec- tion A can be performed. π We have assumed that the behaviour of the measure- FIG. 4. Graph of f(β) (blue), with n = 2 and θ = 3 , and the unit circle (red). Suitable values for the parameters α and β ment device is determined by its own properties, and by ~ exist if the curves intersect. a complete list λ = (λ1, . . . , λn) of the real states of each one of the n systems. Seeing as the systems are prepared independently, the probability distribution for ~λ is given Appendix B: Formal, noise-tolerant version of the by argument ~ µ~x(λ) = µx (λ1) µx (λn). (B3) 1 × · · · × n This section proves Eq. (7). This is a lower bound on In order to prove Eq. (B2), it is useful to deﬁne a quan- the total variation distance between probability distri- tity which we call the overlap between µ0(λ) and µ1(λ): butions corresponding to distinct quantum states, which Z holds even in the presence of noise. In the speciﬁc case of ω(µ0, µ1) = min µ0(λ), µ1(λ) dλ. (B4) no noise ( = 0), this section provides a more mathemat- Λ { } ical version of the argument already given in the main Note that ω(µ0, µ1) = 1 D(µ0, µ1). For probability text. − distributions µ1, . . . , µk, the overlap can be generalised: Consider two methods of preparing a quantum system, Z such that quantum theory assigns the pure state ψ0 or ω(µ1, . . . , µk) = min µi(λ)dλ. (B5) | i i ψ1 . We assume that the quantum system after prepa- Λ | i ration has a real state λ. Each preparation method is as- Let Λn denote the n-fold Cartesian product of Λ, i.e. Λn sociated with a probability distribution µ (λ)(i = 0, 1). i is the space of possible values for ~λ. From Eq. (B3), This is to be thought of as the probability density for the system to be in the real state λ after preparation. min µ~x(λ1, . . . , λn) = min µ0(λ1), µ1(λ1) Another assumption is that when a measurement is per- ~x { } formed, the behaviour of the measurement device de- min µ0(λn), µ1(λn) . (B6) pends only on the physical properties of the system and × · · · × { } measuring device at the time of measurement. Formally, Integrating both sides gives for a given measurement procedure M, the probability of Z ~ ~ n outcome k is given by P (k M, λ) = ξM,k(λ), where ξM,k ω ( µ~x ) = min µ~x(λ) dλ = (ω(µ0, µ1)) . (B7) | { } Λn ~x is a function ξM,k :Λ [0, 1]. A model of this form → reproduces the predictions of quantum theory exactly if Now if the initial state is Ψ(~x) , and the measurement of Figure 2 of the main text is| performed,i Section A shows Z that the outcome corresponding to the basis state ~x has ξM,k(λ)µi(λ)dλ = ψi EM,k ψi , (B1) | i Λ h | | i probability zero according to quantum theory. If a model of the above form assigns probability to this outcome, for any ~x, then ≤ where EM,k is the positive operator which quantum theory assigns to outcome k. Z ~ ~ ~ ξM,~x(λ)µ~x(λ)dλ . (B8) The total variation distance between the distributions Λn ≤ 8

1 ω(µ0,µ1) which gives Eq. (B2).

0.8 Appendix C: Numerical results n =1

For a given ψ0 and ψ1 , the measurement described 0.6 in Section A requires| i the| usei of n systems, with n such that

1/n 0.4 2 arctan 2 1 arccos ψ0 ψ1 . (C1) − ≤ |h | i| It is natural to ask if there exists a measurement that 0.2 can make do with smaller values of n. We have checked n =2 for such a measurement by numerically solving [25, 26] n =3 the semi-definite program n =4 0 X 1 minimize σ := Tr(E Ψ(~x) Ψ(~x) ) 0 0.2 0.4 0.6√2 0.8 1 ~x Ei | i h | δ( ψ0 , ψ1 ) ~x | | subject to E~x 0, (C2) X FIG. 5. The overlap ω(µ0, µ1) (Equation (B4)), versus the E~x = I. q 2 ~x quantum trace distance δ(|ψ0i , |ψ1i) = 1 − |hψ1 | ψ0i| . The red region is ruled out by measurements on a single sys- Since all the terms in the definition of σ are non-negative, tem. The other regions can be ruled out by measurements a measurement described by the POVM operators E~x on 2, 3 and 4 systems. The content of the no-go theorem can be used to prove the no-go theorem if and only{ if} is that larger and larger n eventually fill the square, forcing ω(µ , µ ) = 0 for any pair of states. The boundaries of the σ = 0. A variety of values of θ and n were tested, and the 0 1 minimum value of σ was found to be 0 exactly when (C1) regions are not ruled out (except that ω(µ0, µ1) > 0 is ruled out for δ(|ψ0i , |ψ1i) = 1). is satisfied. Hence it appears that the measurement in Section A uses the smallest possible number of systems. Furthermore, when (C1) is not satisfied the optimal measurement is of the form described in Section A, but ~ ~ ~ ~ Since min~x µ~x(λ) µ~x(λ), and both ξM,~x(λ) and µ~x(λ) with α = π and β = 0. (For n = 1 this measurement is ≤ are non-negative, simply the standard minimum error discrimination measurement for ψ0 and ψ1 .) By a similar argument to the Z previous section,| i if the| quantumi theory predictions for ~ ~ ~ n ξM,~x(λ) min µ~x(λ)dλ . (B9) this measurement hold, then (ω(µ0, µ1)) σ. Hence, in n ~x ≤ Λ addition to our main result that there exists≤ a measurement showing ω(µ0, µ1) = 0 when (C1) is satisfied, this Finally, sum over ~x and use the normalization measurement can be used to place bounds on ω(µ0, µ1) P ~ ~x ξM,~x(λ) = 1 to obtain when it is not. The situation is depicted in Figure 5. Finally, we note that the problem (C2) has an unusual n operational interpretation. By considering each outcome ω ( µ~x ) 2 . (B10) { } ≤ E~x as the identification of Ψ(~x) , we have an error probability of 1 σ/2n, and so| thisi is the “maximum error” discrimination− problem for the quantum states Ψ(~x) Combining Eqs. (B7) and (B10) gives (with equal priors). For the special cases of two{| statesi} this becomes the minimum error problem under swapping n n (ω(µ0, µ1)) 2 , (B11) of the outcome labels. ≤