SUPPLEMENTARY INFORMATION

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1 DOI: /NPHYS309 O the reality of the quatum state Matthew F. Pusey, 1, Joatha Barrett, ad Terry Rudolph 1 1 Departmet of Physics, Imperial College Lodo, Price Cosort Road, Lodo SW7 AZ, Uited Kigdom Departmet of Mathematics, Royal Holloway, Uiversity of Lodo, Egham Hill, Egham TW0 0EX, Uited Kigdom I. THE MEASUREMENT CIRCUIT Cosider a preparatio device which ca produce a quatum system i either the state ψ 0, or the state ψ 1. Suppose that copies of this device are used idepedetly. The there are possible joit states of the systems, depedig o whether ψ 0 or ψ 1 was prepared each time. This sectio shows that for ay distict ψ 0 ad ψ 1, if the umber of systems is large eough, the there is a joit measuremet of the systems with the followig property: each outcome has zero probability give oe of the possible preparatios. Choose a basis { 0, 1 } such that ψ 0 = cos θ 0 + si θ 1, 1) ψ 1 = cos θ 0 si θ 1, ) where ψ 0 ψ 1 = cos θ). By restrictig attetio to the subspace spaed by ψ 0 ad ψ 1, we ca without loss of geerality take the quatum systems to be qubits. For reasos see below, choose large eough that arcta 1/ 1 θ. 3) The circuit cosists of a uitary rotatio Z applied to each qubit, followed by a etaglig gate R α, followed by a Hadamard gate applied to each qubit. The iitial rotatio is give by 1 0 Z = 0 e i. 4) The -qubit gate R α is defied via its actio o the computatioal basis states. Let R α 0 0 = e iα 0 0, ad let R α act as the idetity o all other computatioal basis states. Fially, the Hadamard gate correspods to the uitary operatio H = ) 1 1 The actio of the circuit is give by U α, = H R α Z. The measuremet procedure cosists of the uitary evolutio U α, for a particular choice of α ad discussed below), followed by a measuremet of each qubit i the { 0, 1 } basis. Electroic address: m@physics.org NATURE PHYSICS Macmilla Publishers Limited. All rights reserved.

2 DOI: /NPHYS309 Let x i be 0 1) if the ith system is prepared i the state ψ 0 ψ 1 ), ad write =x 1,...,x ). Before the circuit is applied, the joit state of the systems is a direct product Ψ ) = ψ x1 ψ x. 6) If the iitial preparatio is Ψx 1,...,x ), the the probability of the measuremet outcome correspodig to the basis state x 1...x is the squared absolute value of x 1...x H R α Z ψ x1 ψ x = 1 1). z z R α Z ψ x1 ψ x z = 1 e iα ). z z Z ψ x1 ψ x = 1 e iα ). z z cos θ 0 + 1)xi e i si θ ) 1 i=1 = 1 cos θ ) e iα + 1). z cos θ ) z si θ ) z e i z 1). z = 1 cos θ ) e iα + ) e iα + = 1 cos θ k=1 cos θ ) k si θ ) ) k e k ik 1+e i ta θ ) 1 ). 7) I the fifth lie, z = i z i. Fially, we show that for ay θ with arcta 1 1 θ π, the agles α ad ca be chose so that e iα + 1+e i ta ) θ 1=0, 8) ad hece the probability is zero as required. Rearragig, the required α will always exist ad be easy to fid) provided there exists a with 1+e 1 i ta ) θ =1. 9) Such a exists if the curve of f) =1 1+e i ta ) θ i the complex plae itersects the uit circle, as i Figure 1. Sice f is cotiuous, it suffices to exhibit oe poit outside the uit circle ad oe poit withi it. Cosider f0)=1 1 + ta ) θ. 10) Sice ta θ 1 1, f0) 1, hece it is outside or o) the uit circle. O the other had, fπ) =1 1 ta ) θ. 11) Sice 0 ta θ 1, 0 fπ) 1, hece it is iside or o) the uit circle. This cocludes the proof. If the actual value of for a particular θ ad is required, it is ot difficult to fid it umerically. For =, 9) ca eve be solved aalytically to fid = arccos 1 4t t 4 )/4t 3) where t = ta θ. NATURE PHYSICS 01 Macmilla Publishers Limited. All rights reserved.

3 DOI: /NPHYS309 SUPPLEMENTARY INFORMATION I f0) fπ) R FIG. 1: Graph of f) blue), with = ad θ = π, ad the uit circle red). Suitable values for the parameters α ad 3 exist if the curves itersect. II. FORMAL, NOISE-TOLERANT VERSION OF THE ARGUMENT This sectio proves Eq. 7) of the mai text. This is a lower boud o the total variatio distace betwee probability distributios correspodig to distict quatum states, which holds eve i the presece of oise. I the specific case of o oise ɛ = 0), this sectio provides a more mathematical versio of the argumet already give i the mai text. Cosider two methods of preparig a quatum system, such that quatum theory assigs the pure state ψ 0 or ψ 1. We assume that the quatum system after preparatio has a real state λ. Each preparatio method is associated with a probability distributio µ i λ) i =0, 1). This is to be thought of as the probability desity for the system to be i the real state λ after preparatio. Aother assumptio is that whe a measuremet is performed, the behaviour of the measuremet device depeds oly o the physical properties of the system ad measurig device at the time of measuremet. Formally, for a give measuremet procedure M, the probability of outcome k is give by P k M,λ) =ξ M,k λ), where ξ M,k is a fuctio ξ M,k : [0, 1]. A model of this form reproduces the predictios of quatum theory exactly if ξ M,k λ)µ i λ)dλ = ψ i E M,k ψ i, 1) where E M,k is the positive operator which quatum theory assigs to outcome k. The total variatio distace betwee the distributios µ 0 λ) ad µ 1 λ) is Dµ 0,µ 1 )= 1 µ 0 λ) µ 1 λ) dλ. The aim is to show that if a model of the above form reproduces the predictios of quatum theory approximately, so that for ay measuremet outcome, Eq. 1) holds to withi ɛ, the Dµ 0,µ 1 ) 1 ɛ. 13) Eq. 13) holds for preparatios of ay pair of pure states ψ 0 ad ψ 1, as log as is chose to satisfy Eq. 3). To this ed, cosider idepedet preparatios of quatum systems, where each ca be chose such that the quatum state is either ψ 0 or ψ 1. The joit quatum state is a direct product give by Eq. 6). These systems will be brought together so that the joit measuremet illustrated i Figure of the mai text ad described i Sectio I ca be performed. We have assumed that the behaviour of the measuremet device is determied by its ow properties, ad by a complete list λ =λ 1,...,λ ) of the real states of each oe of the systems. Seeig as the systems are prepared NATURE PHYSICS Macmilla Publishers Limited. All rights reserved.

4 DOI: /NPHYS309 idepedetly, the probability distributio for λ is give by µ λ)=µ x1 λ 1 ) µ x λ ). 14) I order to prove Eq. 13), it is useful to defie a quatity which we call the overlap betwee µ 0 λ) ad µ 1 λ): ωµ 0,µ 1 )= mi{µ 0 λ),µ 1 λ)}dλ. 15) Note that ωµ 0,µ 1 )=1 Dµ 0,µ 1 ). For probability distributios µ 1,...,µ k, the overlap ca be geeralised: ωµ 1,...,µ k )= mi µ i λ)dλ. 16) i Let deote the -fold Cartesia product of, i.e. is the space of possible values for λ. From Eq. 14), Itegratig both sides gives mi µ λ 1,...,λ ) = mi{µ 0 λ 1 ),µ 1 λ 1 )} mi{µ 0 λ ),µ 1 λ )}. 17) ω {µ })= mi µ λ)d λ =ωµ 0,µ 1 )). 18) Now if the iitial state is Ψ ), ad the measuremet of Figure of the mai text is performed, Sectio I shows that the outcome correspodig to the basis state has probability zero accordig to quatum theory. If a model of the above form assigs probability ɛ to this outcome, for ay, the ξ M, λ)µ λ)d λ ɛ. 19) Sice mi µ λ) µ λ), ad both ξ M, λ) ad µ λ) are o-egative, ξ M, λ) mi µ λ)d λ ɛ. 0) Fially, sum over ad use the ormalizatio ξ M, λ) = 1 to obtai Combiig Eqs. 18) ad 1) gives which gives Eq. 13). ω {µ }) ɛ. 1) ωµ 0,µ 1 )) ɛ, ) III. NUMERICAL RESULTS For a give ψ 0 ad ψ 1, the measuremet described i Sectio I requires the use of systems, with such that arcta 1/ 1 arccos ψ 0 ψ 1. 3) It is atural to ask if there exists a measuremet that ca make do with smaller values of. We have checked for such a measuremet by umerically solvig 1, the semi-defiite program miimize E i σ := TrE Ψ ) Ψ ) ) subject to E 0, E = I. 4) 4 NATURE PHYSICS 01 Macmilla Publishers Limited. All rights reserved.

5 DOI: /NPHYS309 SUPPLEMENTARY INFORMATION ωµ 0,µ 1 ) = =4 =3 = δ φ 0, φ 1 ) 1 FIG. : The overlap ωµ 0,µ 1) Equatio 15)), versus the quatum trace distace δ ψ 0, ψ 1 ) = 1 ψ 1 ψ 0. The red regio is ruled out by measuremets o a sigle system. The other regios ca be ruled out by measuremets o, 3 ad 4 systems. The cotet of the o-go theorem is that larger ad larger evetually fill the square, forcig ωµ 0,µ 1) = 0 for ay pair of states. The boudaries of the regios are ot ruled out except that ωµ 0,µ 1) > 0 is ruled out for δ ψ 0, ψ 1 ) = 1). Sice all the terms i the defiitio of σ are o-egative, a measuremet described by the POVM operators {E } ca be used to prove the o-go theorem if ad oly if σ = 0. A variety of values of θ ad were tested, ad the miimum value of σ was foud to be 0 exactly whe 3) is satisfied. Hece it appears that the measuremet i Sectio I uses the smallest possible umber of systems. Furthermore, whe 3) is ot satisfied the optimal measuremet is of the form described i Sectio I, but with α = π ad = 0. For = 1 this measuremet is simply the stadard miimum error discrimiatio measuremet for ψ 0 ad ψ 1.) By a similar argumet to the previous sectio, if the quatum theory predictios for this measuremet hold, the ωµ 0,µ 1 )) σ. Hece, i additio to our mai result that there exists a measuremet showig ωµ 0,µ 1 )=0 whe 3) is satisfied, this measuremet ca be used to place bouds o ωµ 0,µ 1 ) whe it is ot. The situatio is depicted i Figure. Fially, we ote that the problem 4) has a uusual operatioal iterpretatio. By cosiderig each outcome E as the idetificatio of Ψ ), we have a error probability of 1 σ/, ad so this is the maximum error discrimiatio problem for the quatum states { Ψ ) } with equal priors). For the special cases of two states this becomes the miimum error problem uder swappig of the outcome labels. 1 J. Löfberg. YALMIP : A toolbox for modelig ad optimizatio i MATLAB. I Proceedigs of CACSD, pages 84 89, Taipei, 004. K. C. Toh, M. J. Todd, ad R. H. Tütücü. SDPT3 - Matlab software package for semidefiite programmig, versio 1.3. Opt. Meth. Soft., 11: , NATURE PHYSICS Macmilla Publishers Limited. All rights reserved.

On Random Line Segments in the Unit Square

On Random Line Segments in the Unit Square O Radom Lie Segmets i the Uit Square Thomas A. Courtade Departmet of Electrical Egieerig Uiversity of Califoria Los Ageles, Califoria 90095 Email: tacourta@ee.ucla.edu I. INTRODUCTION Let Q = [0, 1] [0,