Efficient Hidden-Variable Simulation of Measurements in Quantum Experiments
Borivoje Dakic, Milovan Suvakov, Tomasz Paterek, Caslav Brukner
EEfficient Hidden-Variable Simulation of Measurements in Quantum Experiments
Borivoje Daki´c,
1, 2
Milovan ˇSuvakov, Tomasz Paterek, and ˇCaslav Brukner
1, 2 Institute for Quantum Optics and Quantum Information,Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria Institute of Physics, Pregrevica 118, 11080 Belgrade, Serbia (Dated: October 27, 2018)We prove that the results of a finite set of general quantum measurements on an arbitrary di-mensional quantum system can be simulated using a polynomial (in measurements) number ofhidden-variable states. In the limit of infinitely many measurements, our method gives models withthe minimal number of hidden-variable states, which scales linearly with the number of measure-ments. These results can find applications in foundations of quantum theory, complexity studiesand classical simulations of quantum systems.
PACS numbers: 03.65.Ta, 03.65.Ud, 03.67.Lx, 89.70.Eg
In classical physics, the position and momentum of aparticle determine the outcomes of all possible measure-ments that can be performed upon it. They define adeterministic classical state. If the state is not fully ac-cessible, a general probabilistic classical state is a mixtureof the deterministic states, arising from the inaccessibil-ity. Since quantum mechanics gives only probabilisticpredictions, it was puzzling already to the fathers of thetheory whether it can be completed with an underlyingclassical-like model [1]. The quantum probabilities wouldthen arise from an inaccessibility of some hidden variables (HV) describing analogs of deterministic classical states,the hidden-variable states, which determine the resultsof all quantum measurements.Since the seminal work of Kochen and Specker (KS), ithas been known that HV models must be contextual [2].On the operational level, the contextual HV models can-not be distinguished from quantum mechanics. However,one may ask how plausible these models are in terms ofresources, e.g., how many HV states (also called the “ on-tic states ” [3, 4, 5]) they require. In addition to the fun-damental question of the minimal HV model for a quan-tum system, this research is motivated by problems inquantum information theory. In particular, HV modelsallow a fair comparison between complexities of quantumand classical algorithms [6, 7], as a quantum algorithmcan now be represented by a classical circuit.For an infinite number of measurement settings, al-ready a single qubit requires infinitely many HV states,the result proved by Hardy [8] and, in a different con-text, by Montina [9, 10]. However, these authors did notconsider the scaling of the number of HV states with thenumber of measurements. Harrigan and Rudolph founda deterministic HV model that requires exponentiallymany HV states to simulate results of the finite set ofmeasurements on all quantum states [11]. Our construc-tion also provides such models and consumes at most a polynomial number of HV states, bringing exponentialimprovement. In the limit of infinitely many measure- ments, the number of HV states for an indeterministicmodel scales linearly with the number of measurements.Moreover, the number of real parameters that specifythese HV states saturates the lower bound derived byMontina [10] and, consequently, is the minimal numberpossible. Our method also allows a universal generaliza-tion of the Spekkens model [5].Consider a finite number, N , of projective measure-ments on a d -level quantum system in a state ρ . Theprobability to observe the r th result in the n th measure-ment is p ( n ) r (ˆ ρ ) = Tr[ˆ ρ ˆΠ ( n ) r ], where ˆΠ ( n ) r is a projectoron the r th orthogonal state of the n th measurement, i.e., r = 1 , ..., d and n = 1 , ..., N . We form a d -dimensionalvector, p ( n ) = ( p ( n )1 , . . . , p ( n ) d ) T , composed of the prob-abilities for distinct outcomes in the n th measurement.For the set of measurements, we build a dN -dimensional preparation vector , p = ( p (1) , ..., p ( N ) ) T [12]. The deter-ministic HV states predetermine the results of all mea-surements and can be represented as a dN -dimensionalvector O r ...r N = (0 , . . . , , . . . , | . . . | , . . . , , . . . , T , (1)where r n is the position of 1 in the n th sequence ( r n =0 , . . . , d − r n occurs in the n thmeasurement). The space of all HV states, Λ, is formedby classical mixtures of d N deterministic states O r ...r N .A set of κ quantum states ρ , . . . , ρ κ has a HVmodel for N measurements, if one can find L vectors O , . . . , O L ∈ Λ such that p ( ρ k ) = L (cid:88) l =1 α l ( k ) O l , for all k = 1 , ..., κ (2)where α l ( k ) ≥ (cid:80) l α l ( k ) = 1. The model iscalled deterministic if all O l are deterministic HV states;otherwise, it is called indeterministic . The model is preparation-universal , if the HV states simulate any phys-ical state ρ , and it is measurement-universal if they sim-ulate any measurement. a r X i v : . [ qu a n t - ph ] N ov Formally, the set Λ is a convex polytope in R dN havingthe states O l as vertices. Since all probabilities satisfy0 ≤ p ( n ) r ≤
1, any preparation vector p ( ρ ) lies inside thispolytope and has a HV model. We study the number ofHV states required for the model.We begin with a specific deterministic HV model fora two-level quantum system (qubit) which we shall oftenrefer to later on. An arbitrary state of a qubit can be rep-resented as ˆ ρ = ( (cid:80) i =1 x i ˆ σ i ), where ˆ σ i ’s are the Paulimatrices and x = ( x , x , x ) T is a Bloch vector, in a unitball | x | ≤
1. A set of N projective measurements, with2 N outcomes (states on which the qubit is projected),is described by 2 N unit vectors ± m , . . . , ± m N on theBloch sphere. The preparation vector for these directionsis p ( x ) = ( ± m x , . . . , ± m N x ). Since the probabilityfor the measurement − m is fully determined by the onefor the + m , one can reduce (”compress”) preparationvector to p ( x ) = ( m x , . . . , m N x ). Similarly, thedeterministic HV states are reduced to N -dimensionalvectors O r ...r N = ( r , . . . , r N ) T , where r n = 0 ,
1. The(reduced) space Λ is a hypercube in N dimensions, with2 N vertices defined by these states. By Carath´eodory’stheorem [15] for each vector p ( x ) = ( p , . . . , p N ) T , onecan identify N + 1 HV states the convex hull of whichcontains p ( x ). For a given x , the vector p ( x ) can be writ-ten as a permutation of a reordered preparation vector p ↓ ( x ) wherein the probabilities appear in increasing or-der, p ↓ ≤ p ↓ ≤ · · · ≤ p ↓ N , and the latter can be expressedin terms of N + 1 HV states as p ↓ ( x ) = · · ·
00 1 1 0 · · ·
00 1 1 1 · · · · · · α α α ... α N − α N , (3)where the columns of the displayed matrix are the HVstates. The expansion coefficients are α = 1 − p ↓ N , α = p ↓ ,α n = p ↓ n − p ↓ n − for n = 2 , ..., N, (4)and, due to the ordering of probabilities, the coefficientsare all positive and sum up to 1. One can suitably per-mute the rows in matrix given by (3) to bring the proba-bilities in order given by p ( x ). Thus, p ( x ) can be writtenas a convex combination of N +1 columns (HV states) of areordered matrix. The number of N +1 states can be fur-ther reduced. E.g., for two equal probabilities, p = p ,the number of HV states is decreased because α = 0. If,say, p = 1 − p , one can exchange m → − m , such thatthe probabilities become equal, leading to another reduc-tion. Importantly, different quantum states are generallymodeled by different sets of N + 1 HV states.As an illustrative example, consider a model for threecomplementary measurements along m x , m y , m z . We show a nonuniversal model, only for the eigenstatesof these measurements: ± m x , ± m y , ± m z . The cor-responding preparation vectors are: p (+) x = (1 , , ), p ( − ) x = (0 , , ), p (+) y = ( , , ), p ( − ) y = ( , , ), and p (+) z = ( , , p ( − ) z = ( , , L = 4 HV states are sufficient for the simulation: O = (1 , , T , O = (1 , , T , O = (0 , , T , and O = (0 , , T . These four states, together with theirdecomposition of the preparation vectors, p (+) x = 12 O + 12 O , p ( − ) x = 12 O + 12 O , p (+) y = 12 O + 12 O , p ( − ) y = 12 O + 12 O , p (+) z = 12 O + 12 O , p ( − ) z = 12 O + 12 O . (5)are equivalent to the toy model of Spekkens [5].We give a constructive proof that a preparation-universal simulation of N quantum measurements on aqubit can be achieved with the number of HV statesthat is polynomial in N . Let M denote a polytopeformed as a convex hull of the measurement settings, M = conv {± m , . . . , ± m N } . Its dual polytope is a set[16], D M = { y ∈ R | − ≤ m n y ≤ , n = 1 . . . N } . (6)The polytope M lies inside the Bloch sphere and itsdual contains the sphere. Therefore, every Bloch vec-tor can be written as a convex combination of the ver-tices, y l , of the dual polytope, x = (cid:80) l α l ( x ) y l . Thecomponents of the measurement vector can now be de-composed as p n ( x ) = (cid:80) l α l ( x ) (1 + m n y l ). Accord-ing to the definition of the dual polytope, the quantity (1 + m n y l ) ∈ [0 ,
1] and can be interpreted as the n thcomponent (probability) of the l th HV state. Since theBloch vectors corresponding to projections onto orthog-onal states sum up to the zero vector, the correspond-ing probabilities assigned by a HV state sum up to 1,as it should be. Thus, the set of HV states correspond-ing to vertices of the dual polytope is sufficient for apreparation-universal HV model. Note that this modelcan in general be indeterministic. In such a case, eachindeterministic HV state can be further reduced into atmost N − N −
2, and not N + 1, states stemsfrom the observation that a vertex of the dual polytopesaturates at least three of the inequalities defining thepolytope (at least three facets have to meet at each ver-tex), i.e., the corresponding probability is 1 or 0, andreduces the number of required deterministic HV states.Finally, the total number of HV states required for anindeterministic model is L ≤ F , and for a deterministicmodel is L ≤ ( N − F , where F is the number of ver-tices of the dual polytope or, equivalently, the number offacets of the measurement polytope. A convex polytopewith 2 N vertices (in three-dimensional space) can have N + 2 ≤ F ≤ N −
1) facets [13], which implies thatindeterministic HV models require at most a number ofHV states that is linear in N , and deterministic onesrequire quadratic number of HV states.Using the dual polytope approach, we generalizeSpekkens’ model [5], originally formulated to explainthe measurement results on the eigenstates of the threecomplementary directions, to the preparation-universalmodel. For these directions, the measurement polytopeis an octahedron, see Fig. 1(a). The dual polytope is acube, whose interior forms the whole space of HV states,with the vertices being the deterministic states. Anotherinteresting example is illustrated in Fig. 1(b).The dual polytope approach can be applied to arbi-trary preparation vectors. However, efficient simulationsare only expected for highly symmetric polytopes. Forthis reason, we move to more complicated Platonic solidsand general symmetry considerations.Consider a set of measurement directions ± m , . . . , ± m N , which is generated by a group; e.g., an octahe-dron and a cube can be generated via the chiral octa-hedral group O with 24 rotations. Generally, if G is asymmetry of the measurement polytope, M , it is also asymmetry of its dual, D M ; i.e., the dual polytope canalso be generated by G . The group action permutes thevectors ± m n as well as vertices of the dual polytope.Since the last are related to the HV states, we can define FIG. 1: Preparation-universal HV models and dual poly-topes. (a)
The vertices of the octahedron inside the Blochsphere define the three complementary qubit measurements.A preparation-universal
HV model for these measurementsrequires eight HV states, which are written near their repre-sentative vertices of the cube containing the sphere. It gen-eralizes the Spekkens model [5], which is not universal andutilizes only four out of eight states (see main text). Theircorresponding vertices span a tetrahedron inside the cube,which does not contain the whole Bloch sphere. (b)
Here,the measurement directions form a cube inside the sphere. Al-though more measurements are to be simulated, the universalHV model requires only six HV states, which are written neartheir representative vertices of the octahedron containing thesphere. the permutation representation of the group in the HVspace, D P ( G ). The HV state, h ( y (cid:48) ), corresponding to avertex of a dual polytope, y (cid:48) = g y , which is generated by g ∈ G acting on an initial vertex, y , can be found usingthe group representation: h ( g y ) = D P ( g ) h ( y ) . (7)Decomposing h ( y ) into deterministic HV states brings(7) to the form h ( g y ) = (cid:80) l =1 α l D P ( g ) O l . There-fore, the set of deterministic HV states required for thepreparation-universal model is the union of a number ofgroup orbits { D P ( g ) O l | g ∈ G } . Because of the symme-tries involved, the minimal number of HV states cannotbe smaller than the number of elements in the smallestorbit.Let us consider two other Platonic solids, the icosahe-dron and the dodecahedron [17]. Both of them posses thesame symmetry, the chiral icosahedral group I , with 60rotations. Consider the icosahedron as the measurementpolytope, N = 6. Its dual, the dodecahedron, has 20 ver-tices corresponding to indeterministic HV states that canbe further reduced to deterministic HV states. The totalnumber of possible deterministic HV states is 2 = 64 inthis case. We have found four different orbits of action of I with 12 , , ,
20 different elements, respectively. Onlyone orbit, with 20 elements, gives deterministic states foruniversal simulation. For N = 10 measurement settings,the dodecahedron is the measurement polytope. Its dual,the icosahedron, has 12 vertices. The total number ofpossible deterministic HV states is 2 = 1024, which ispartitioned into 24 different orbits: 2 with 12 elements,8 with 20, and 14 with 60 elements. The two lowestorbits are suitable for the universal model. Thus, theminimal deterministic model, among all HV models ob-tained through the dual polytope construction, requiresonly 24 HV states, twice the number of vertices of thedual polytope.The presentation so far was limited to qubits. How-ever, a similar line of reasoning applies to any d -levelquantum system. In the general case, Pauli operatorshave to be replaced by generalized Gell-Mann operators,ˆ λ i , which naturally leads to the generalized, D ≡ d − ρ = d [ d − (cid:80) Di =1 x i ˆ λ i ], is now repre-sented by a generalized Bloch vector, x , with components x i = Tr(ˆ ρ ˆ λ i ). We normalize the Gell-Mann operatorsas Tr(ˆ λ i ˆ λ j ) = dd − δ ij , such that pure quantum statesare represented by normalized generalized Bloch vectors.Contrary to the qubit case, not every unit vector corre-sponds to a physical state. The probability of an out-come associated with a projector on a state representedby m n , in a measurement on a state represented by x , is p n ( x ) = d [1+( d − m n x )]. The requirement of positiveprobabilities reveals that, e.g., the vector x = − m n doesnot represent a physical state.In analogy to the dual polytope, for a set of dN preparaion vectors, representing N d -valued observables,we introduce a convex polytope the interior of which in-cludes all vectors y leading to physically allowed proba-bilities p n ( y ) ∈ [0 , P M = { y ∈ R D | − d − ≤ m n y ≤ , n = 1 , ..., dN } . (8)Among others, this polytope contains all the vectors ofquantum states. The generalized Bloch vectors corre-sponding to a complete set of orthogonal quantum statessum up to the zero vector, implying the probabilities as-signed by a HV state for different outcomes of any mea-surement sum up to 1, as it should be. Again, the vectorsof quantum states can be expressed as a convex combi-nation of vertices of P M , and their number gives theupper bound on the amount of HV states sufficient forpreparation-universal simulation. The polytope P M isspecified by q = 2 dN linear inequalities, two inequalitiesfor each vector m n , and its maximal number of verticesis given by L ≤ (cid:0) q − δq − D (cid:1) + (cid:0) q − δ (cid:48) q − D (cid:1) , where δ ≡ (cid:98) ( D + 1) / (cid:99) , δ (cid:48) ≡ (cid:98) ( D + 2) / (cid:99) , and (cid:98) x (cid:99) is the integer part of x [13]. Inthe special case of a qubit, the dual polytope is definedby 2 N , and not 4 N , inequalities because the two boundsof Eq. (6) are the same for the vectors ± m n . Since thebinomial coefficient (cid:0) ab (cid:1) increases with a , L ≤ (cid:0) q − δq − D (cid:1) .Using (cid:0) ab (cid:1) = (cid:0) aa − b (cid:1) , we have L ≤ (cid:0) q − δD − δ (cid:1) , and since (cid:0) ab (cid:1) ≤ a b /b !, the maximal number of vertices is polyno-mial in N , L ∼ (2 dN − δ ) D − δ . The related HV statescan in general be indeterministic, and each of them canbe decomposed to O ( N ) deterministic HV states, usingdecomposition (3) in the dN dimensional space Λ. There-fore, for any system, the number of (in)deterministic HVstates required for a preparation-universal simulation ispolynomial in N .In the limit of infinitely many measurements, ourmethod gives (preparation and measurement) universalmodels with the minimal number of HV states. As provedby Montina, in this limit the optimal model requires2( d −
1) real parameters to describe the HV states [10].We show that for an infinite number of settings the set ofuniversal HV states converges to the set of pure quantumstates, which is known to be parameterized by 2( d − n with n = 1 , ..., dN , and the corresponding polytope (8)in the Hilbert-Schmidt space of Hermitian operators withunit trace. The operators of its vertices, ˆ y l , correspondto the HV states, i.e., for all n , Tr(ˆ y l ˆΠ n ) gives the prob-ability that is assigned by the HV state, of the outcomeassociated with projector ˆΠ n . For other projectors, notwithin the set of dN , the trace does not have to representa probability and therefore the set of operators ˆ y l is largerthan the set of quantum states [18]. However, in the limitof infinitely many measurements, Tr(ˆ y l ˆΠ n ) ∈ [0 ,
1] for allpossible projectors; therefore, the eigenvalues of ˆ y l ’s liewithin the [0 ,
1] interval. Since Tr(ˆ y l ) = 1, the operators ˆ y l are just quantum states and the HV states correspond-ing to pure quantum states are universal. Their numberscales linearly with N , because N measurements corre-spond to dN projectors and each of them represents oneHV state (and also one pure quantum state).Regarding the polytope P M in the space of Hermi-tian operators allows for an easy generalization of our ap-proach to POVM measurements. POVM elements, ˆ E n ,are positive operators being vertices of a measurementpolytope. The polytope P M includes all the unit-traceoperators ˆ y for which Tr(ˆ y ˆ E n ) ∈ [0 , ρ ˆ E n ) ∈ [0 , P M containsall of them and, as before, its vertices define HV states.For a d -level system the KS argument disqualifies non-contextual HV theories [2], and one might wonder howcontextuality enters our models. Consider the KS ar-gument of Peres [14]. It involves 33 different vectors in R , which belong to 16 different orthogonal triads. Non-contextuality requires a value associated with a singlevector to be the same irrespectively of other vectors in thetriad. In the present models, the results of 16 differentmeasurements are described by HV states with 3 ·
16 = 48components; i.e., a value assigned to the same vector candepend on the other vectors in the triad.In conclusion, we proved that a preparaion-universalHV model of the results of N quantum measurements re-quires at most a number of HV states which is polynomialin N . In the limit of infinitely many measurements, ourmethod gives optimal preparation- and measurement-universal HV models, with the minimal number of realparameters describing the HV states. There is no HVmodel that would require less HV states than the modelin which every quantum state is associated with a HVstate [10]. Furthermore, since there are infinitely manymeasurements that can be performed on a quantum sys-tem, its HV description requires infinitely many HVstates. This “ontological baggage” [8] can be seen asan argument against the HV approach because it is ex-tremely resource demanding already for a single qubit. Acknowledgments . This work is supported by theFWF Project No. P19570-N16, EC Project QAP (No.015846), the FWF project CoQuS (No. W1210-N16),and by the Foundational Questions Institute (FQXi). [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. ,777 (1935).[2] S. Kochen and E. Specker, J. Math. Mech. , 59 (1967).[3] T. Rudolph, arXiv:quant-ph/0608120.[4] R. W. Spekkens, Phys. Rev. A , 052108 (2005).[5] R. W. Spekkens, Phys. Rev. A , 032110 (2007).[6] S. Aaronson, Phys. Rev. A , 032325 (2005).[7] N. Harrigan and T. Rudolph, arXiv:0709.4266.[8] L. Hardy, Stud. Hist. Philos. Mod. Phys. , 267 (2004).[9] A. Montina, Phys. Rev. Lett. , 180401 (2006). [10] A. Montina, Phys. Rev. A , 022104 (2008).[11] N. Harrigan and T. Rudolph, arXiv:0709.1149.[12] L. Hardy, arXiv:quant-ph/0101012.[13] P. McMullen, Mathematika , 179 (1970)[14] A. Peres, J. Phys. A , L175 (1991).[15] The Carath´eodory’s theorem states that a point, x , in aconvex polytope in R n can be written as a convex com-bination of n + 1 vertices. [16] In the special case of measurement settings within aplane, we consider the dual polygon lying in that plane.[17] Similar analysis applies to cube and octahedron.[18] E.g., if the preparation vector of a qubit involves pro-jectors on | z ±(cid:105) and | x ±(cid:105) , it is valid to consider ˆ y l = + ˆ σ x + ˆ σ zz