Analysis of Bell inequality violation in superconducting qubits
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Analysis of Bell inequality violation in superconducting phase qubits
Abraham G. Kofman and Alexander N. Korotkov
Department of Electrical Engineering, University of California, Riverside, California 92521 (Dated: November 1, 2018)We analyze conditions for violation of the Bell inequality in the Clauser-Horne-Shimony-Holtform, focusing on the Josephson phase qubits. We start the analysis with maximum violation in theideal case, and then take into account the effects of the local measurement errors and decoherence.A special attention is paid to configurations of the qubit measurement directions in the pseudospinspace lying within either horizontal or vertical planes; these configurations are optimal in certaincases. Besides local measurement errors and decoherence, we also discuss the effect of measurementcrosstalk, which affects both the classical inequality and the quantum result. In particular, wepropose a version of the Bell inequality which is insensitive to the crosstalk.
PACS numbers: 03.65.Ud, 85.25.Cp, 03.67.Lx
I. INTRODUCTION
In 1935 Einstein, Podolsky, and Rosen (EPR) haveshown in the classical paper that quantum mechanicscontradicts the natural assumption (the “local realism”)that a measurement of one of two spatially separated ob-jects does not affect the other one. This “spooky actionat a distance” – known as an entanglement – is now rec-ognized as a major resource in the field of quantum infor-mation and quantum computing. The paradox has ledEPR to conclude that quantum mechanics is an incom-plete description of physical reality, thus implying thatsome local hidden variables are needed.The EPR paradox remained at the level of semi-philosophical discussions until 1964, when John Bell con-tributed an inequality for results of a spin-correlationexperiment, which should hold for any theory involvinglocal hidden variables, but is violated by quantum me-chanics. Inspired by Bell’s idea, in 1969, Clauser, Horne,Shimony, and Holt (CHSH) proposed a version of theBell inequality (BI, the generic name for a family of in-equalities), which made the experimental testing of localhidden-variable theories possible. The main advantageof the CHSH inequality in comparison with the originalBI is that it does not rely on an experimentally unre-alistic assumption of a perfect anticorrelation betweenthe measurement results when two spin-1/2 particles (inthe spin-0 state) are measured along the same direction.Many interesting experiments have been donesince then. The results of these experiments clearly showa violation of the Bell inequalities, in accordance withquantum mechanical predictions.The BI violation has been mostly demonstrated inthe experiments with photons; it has been alsoshown in the experiments with ions in traps and withan atom-photon system; a Bell-like inequality viola-tion has been also demonstrated in an experiment withsingle neutrons. Experimental violation of the BI insolid-state qubits would be an important step towardspractical quantum information processing by solid-statedevices.
Experiments on observation of theBI violation in Josephson phase qubits are currently underway.
Theoretical study related to the BI viola-tion in solid-state systems has also attracted significantattention in recent years.
In this paper we discuss the Bell inequality (in theCHSH form) for solid-state systems, focusing on experi-ments with superconducting phase qubits. We study ef-fects of various factors detrimental for observation of theBI violation, including local measurement errors, deco-herence, and interaction between qubits (crosstalk), andanalyze optimal conditions in presence of these nonideal-ities.The paper is organized as follows. In Sec. II we reviewthe CHSH type of the BI, and also discuss tomography-type measurements using qubit rotations. In Sec. IIIwe consider the ideal case and describe all situations forwhich the BI is violated maximally. Sections IV-VI aredevoted to the effects of various nonidealities on the ob-servation of the BI violation. In Sec. IV we discuss theeffect of local measurement errors, using a more generalerror model than in the previous approaches.
Analytical results for maximally entangled states and nu-merical results for general two-qubit states are presented.In Sec. V we consider the effect of local decoherence ofthe qubits. In Sec. VI we discuss measurement crosstalk,which affects both the BI (since crosstalk is a classicalmechanism of communication between qubits) and thequantum result. We also propose a version of the BI,which is not affected by the crosstalk. Section VII pro-vides concluding remarks.
II. PRELIMINARIESA. CHSH inequality
We begin with a brief review of the CHSHinequality, a type of the Bell inequality usuallyused in experiments. Let us consider a pair of two-level systems (qubits) a and b . Assuming that a real-istic (classical) theory based on local observables holdsand there is no communication between the qubits (i.e.,no crosstalk), the two-qubit measurement results shouldsatisfy the CHSH inequality − ≤ S ≤ , (1)where S = E ( ~a,~b ) − E ( ~a,~b ′ ) + E ( ~a ′ ,~b ) + E ( ~a ′ ,~b ′ ) . (2)Here ~a and ~a ′ ( ~b and ~b ′ ) are the unit radius-vectors on theBloch sphere along the measurement axes for qubit a ( b )and E ( ~a,~b ) is the correlator of the measurement results: E ( ~a,~b ) = p ++ ( ~a,~b ) + p −− ( ~a,~b ) − p + − ( ~a,~b ) − p − + ( ~a,~b ) , (3)where p ij ( ~a,~b ) ( i, j = ± ) is the joint probability of mea-surement results i and j for qubits a and b , respectively.The sum of the probabilities in Eq. (3) equals one. Thiscan be used to recast Eq. (2) in the form S = 4 T + 2 , (4)where T = p ( ~a,~b ) − p ( ~a,~b ′ ) + p ( ~a ′ ,~b ) + p ( ~a ′ ,~b ′ ) − p a ( ~a ′ ) − p b ( ~b ) . (5)Here p ( ~a,~b ) = p ++ ( ~a,~b ), whereas p a ( ~a ′ ) = p ++ ( ~a ′ ,~b ) + p + − ( ~a ′ ,~b ) [or p b ( ~b ) = p ++ ( ~a,~b ) + p − + ( ~a,~b )] is the prob-ability of the measurement result “+” for qubit a (or b ) irrespective of the measurement result for the otherqubit [in the classical theory in absence of communica-tion between qubits, p a ( ~a ′ ) is obviously independent ofthe direction ~b and even independent of the very fact ofthe qubit b measurement; similarly for p b ( ~b )]. Thus, in-stead of the probabilities p ij , one can equivalently usethe probabilities p , p a , and p b , and the inequality (1) canbe recast in an equivalent form − ≤ T ≤ . (6)Notice that both inequalities (1) and (6) can havesomewhat different meanings in different physical sit-uations. In particular, the results “+” and “ − ” maycorrespond to the presence or absence of the detector“click” (so called “one-channel measurement” ); in thiscase a low-efficiency detector significantly increases thechance of the result “ − ”. Another possibility is the so-called “two-channel measurement”, in which the qubitstates“+” and “ − ” are supposed to produce clicks in dif-ferent detectors; in this case inefficient detection leadsto three possible results: “+”, “ − ”, and “no result”. Sig-nificant inefficiency of the optical detectors leads to theso-called “detector loophole,” which arises becauseeffectively not the whole ensemble of the qubit pairs isbeing measured. This problem is often discussed in termsof contrasting the Clauser-Horne (CH) and CHSH in-terpretations of the inequalities, which differ by consid-ering either the whole ensemble or a subensemble of thequbit pairs. It is important to mention that in the caseof the Josephson phase qubits (which formally belongs to the class of one-channel measurements) the whole en-semble of qubit pairs is being measured, and thereforethere is no detector loophole (if one avoids correctionsfor measurement errors), as well as there is no differencebetween CHSH and CH interpretations. B. Tomographic measurements
In some cases, as for Josephson phase qubits, the mea-surement (detector) axis cannot be physically rotated.However, instead of the detector rotation, one can rotatethe qubit state.
Let us show the equivalence of thetwo methods explicitly, using the example of the phasequbit and assuming ideal (orthodox) measurement.The Hamiltonian of the phase qubit in a microwavefield in the subspace of the two lowest states in the qubitpotential well | ψ i and | ψ i is H q = (¯ hω q / | ψ ih ψ | − | ψ ih ψ | )+¯ h Ω( t ) sin( ωt + φ )( | ψ ih ψ | + | ψ ih ψ | ) , (7)where ¯ h is the Planck constant, ω q is the qubit reso-nance frequency, ω is the microwave frequency, Ω( t ) = d E ( t ) / ¯ h is the time-dependent Rabi frequency, d is the dipole-moment matrix element, and E ( t ) isthe electric-strength amplitude of the microwave field.Transforming to the qubit basis (the rotating frame) | i = | ψ i , | i = e iωt | ψ i , neglecting fast oscillatingterms in the Hamiltonian (the rotating-wave approxima-tion) and assuming the resonance condition ω = ω q , wearrive at the Hamiltonian H ( t ) = ¯ h Ω( t )2 ~n · ~σ, (8)where the unit vector ~n = (sin φ, − cos φ,
0) lies in the xy plane making the angle φ − π/ x axis, whereas ~σ = ( σ x , σ y , σ z ) is the vector of the Pauli matrices. Hereand below we associate state | i ( | i ) with the measure-ment result “+” (“ − ”) and with the eigenvalue 1 ( −
1) of σ z , so that σ z = | ih | − | ih | .As follows from Eq. (8), the microwave pulse rotatesthe qubit state such that the initial density matrix ρ q becomes ˜ ρ q = U R ρ q U † R , where U R = e − iθ~n · ~σ/ = cos( θ/ − i~n · ~σ sin( θ/ (cid:18) cos( θ/ e − iφ sin( θ/ − e iφ sin( θ/
2) cos( θ/ (cid:19) . (9)Here θ = R t t dt Ω( t ), where t and t are the pulse start-ing and ending time moments.The probability of the qubit to be found in state | i i is p i = Tr( | i ih i | ˜ ρ q ) = Tr( P i ρ q ) ( i = 0 , . (10)Here P i is the projection operator P i = U † R | i ih i | U R , i.e., P ( ~a ) = 12 (cid:18) θ e − iφ sin θe iφ sin θ − cos θ (cid:19) = 12 ( I + ~a · ~σ ) ,P ( ~a ) = I − P ( ~a ) = 12 ( I − ~a · ~σ ) , (11)where I is the identity matrix and ~a is the unit vector ~a = (cos φ sin θ, sin φ sin θ, cos θ ) (12)defining the measurement axis.Equations (10) and (11) show explicitly the equiva-lence between the qubit and detector rotations. Namely,one can interpret p ( p ) as the probability of the qubitto be found in the state with the pseudospin parallel (an-tiparallel) to the measurement axis ~a .Notice that the microwave phase φ is naturally definedmodulo 2 π , while the Rabi rotation angle θ can be alwaysreduced to a 2 π range (we will assume − π < θ ≤ π ).Nevertheless, with this restriction there are still two setsof angles ( θ, φ ) corresponding to the same measurementdirection ~a , because ~a is invariant under the change( θ, φ ) ↔ ( − θ, φ + π ) . (13)It is easy to make a one-to-one correspondence betweenthe measurement direction ~a and angles ( θ, φ ) by limitingeither θ or φ to a π -range (instead of 2 π ). However, weprefer not to do that because it is convenient and naturalphysically to have a 2 π range for one angle when the otherangle is fixed. So, as follows from Eqs. (12) and (13), thepolar (zenith) and azimuth spherical coordinates of ~a areequal to, respectively, θ and φ when θ ≥
0, and − θ and φ + π when θ < p ij ( ~a,~b ) = Tr[ P ai ( ~a ) P bj ( ~b ) ρ ] , (14)where ρ is the two-qubit density matrix, P ai = P i ⊗ I ,and P bi = I ⊗ P i . III. MAXIMUM BI VIOLATION: IDEAL CASE
The purpose of this paper is to analyze conditionsneeded to observe the BI violation in experiment. Sinceit is usually easier to observe an effect when it is maximal,we start the analysis with the situations where violationof the BI is maximal.
A. Bell operator and Cirel’son’s bounds
Equations (3) and (14) yield E ( ~a,~b ) = Tr( ABρ ), where A = P a ( ~a ) − P a ( ~a ) = ~a · ~σ a (15)and similarly B = ~b · ~σ b . Here ~σ a = ~σ ⊗ I and ~σ b = I ⊗ ~σ ,the eigenvalues of A and B being ±
1. Correspondingly,as follows from Eq. (2), S = Tr( B ρ ) , (16)where the Bell operator B is B = AB − AB ′ + A ′ B + A ′ B ′ (17) (here A ′ = ~a ′ · ~σ a and B ′ = ~b ′ · ~σ b ).The maximum and minimum values of S (so-calledCirel’son’s bounds ) S ± = ± √ Since the Bell operator B is Hermitian, S ± are equalto the maximum and minimum eigenvalues of B . Theseeigenvalues can be found analyzing the eigenvalues of B : B = 4+[ A, A ′ ][ B, B ′ ] = 4 − ~a × ~a ′ · ~σ a ) ( ~b × ~b ′ · ~σ b ) (19)(here the vector product is taken before the scalar prod-uct). Since the eigenvalues of the Pauli matrices are equalto ±
1, the largest eigenvalue of B is 8, achieved when ~a ⊥ ~a ′ and ~b ⊥ ~b ′ . In this case the maximum and mini-mum eigenvalues of B are ±√
8, thus leading to Eq. (18).(Both values ±√ B is 0, and the sum of all four eigen-values of B should be equal to 0 since Tr B = 0.)Consider some useful properties of S . As follows fromEqs. (16) and (17), the value of S is invariant under ar-bitrary local unitary transformations U a and U b , ρ → ( U a ⊗ U b ) ρ ( U † a ⊗ U † b ) , (20a)if simultaneously A → U a AU † a , A ′ → U a A ′ U † a , B → U b BU † b , B ′ → U b B ′ U † b or, equivalently, ~a → R a ~a, ~a ′ → R a ~a ′ , ~b → R b ~b, ~b ′ → R b ~b ′ , (20b)where R a ( R b ) is the rotation matrix corresponding to U a ( U b ), so that, e.g., U a ( ~a · ~σ ) U † a = ( R a ~a ) · ~σ . Thisinvariance is an obvious consequence of the equivalencebetween the qubit and detector rotations, discussed inthe previous section. As a result of the invariance, if somestate is known to violate the BI for a given configurationof the detectors, one can obtain many other states andthe corresponding detector configurations providing thesame BI violation, by using Eqs. (20) with all possiblelocal rotations.Note also that B inverts the sign if the pair of vectors ~a, ~a ′ (or ~b,~b ′ ) inverts the sign. Correspondingly, for agiven state S → − S if ~a → − ~a, ~a ′ → − ~a ′ (or ~b → − ~b, ~b ′ → − ~b ′ ) . (21)As follows from Eq. (21), there is a one-to-one corre-spondence between the classes of detector configurationsmaximizing and minimizing S for a given state. B. Optimal detector configurations for maximallyentangled states
For any given detector configuration satisfying the con-dition ~a ⊥ ~a ′ and ~b ⊥ ~b ′ , (22)each of the Cirel’son’s bounds (18) is achieved for aunique maximally entangled state. In contrast, fora given maximally entangled state there can be manyoptimal detector configurations giving the maximal BIviolation. To the best of our knowledge, only the config-urations with the detector axes lying in one plane havebeen usually considered in the literature, though gener-ally the detector axes for different qubits may lie in twodifferent planes. Moreover, the BI violation has beenstudied mainly for one of the Bell states, | Ψ ± i = ( | i ± | i ) / √ , (23) | Φ ± i = ( | i ± | i ) / √ , (24)while an arbitrary maximally entangled state can be writ-ten as | Ψ me i = ( | χ a χ b i + | χ a χ b i ) / √ , (25)where {| χ k i , | χ k i} is an orthonormal basis for qubit k .Our purpose here is to determine all optimal detectorconfigurations for any maximally entangled state.
1. Singlet state
Let us start, assuming that the qubits are in the singletstate | Ψ − i . For this state E ( ~a,~b ) = h Ψ − | ( ~a · ~σ ) ⊗ ( ~b · ~σ ) | Ψ − i = − ~a · ~b , so that [see Eq. (2)] S = ~a · ( ~b ′ − ~b ) − ~a ′ · ( ~b + ~b ′ ) . (26)Maximizing this formula over ~a and ~a ′ , we should choose ~a to be parallel to ~b ′ − ~b , while ~a ′ should be antiparallelto ~b ′ + ~b . Therefore ~a ⊥ ~a ′ , since ~b ′ − ~b and ~b + ~b ′ are mutually orthogonal. Similarly, rewriting S as S = ~b ′ · ( ~a + ~a ′ ) − ~b · ( ~a − ~a ′ ), we can show that maximizationof S requires ~b ⊥ ~b ′ . In this way we easily show that the necessary and sufficient condition for reaching the upperbound S + = 2 √ ~a ⊥ ~a ′ , ~b = − ( ~a + ~a ′ ) / √ , ~b ′ = ( ~a ′ − ~a ) / √ . (27)Because of the symmetry (21), the necessary and suffi-cient condition for reaching the lower bound S − = − √ ~a ⊥ ~a ′ , ~b = ( ~a + ~a ′ ) / √ , ~b ′ = ( ~a − ~a ′ ) / √ . (28)Equations (27) and (28) show that for the singlet statethe maximum BI violation requires that the detector axesfor both qubits lie in the same plane. However, the orien-tation of this plane is arbitrary, since Eqs. (27) and (28)determine only the angles between the detector axes.All configurations maximizing (or minimizing) S forthe singlet state can be obtained from one maximiz-ing (minimizing) configuration by all possible rotations of the plane containing the detector axes. As the ini-tial maximizing case we can choose the most standardconfiguration when all detector axes are within xz plane, φ a = φ ′ a = φ b = φ ′ b = 0 , (29)and the polar (zenith) angles of the detector directions ~a , ~a ′ , ~b , ~b ′ are θ a = 0 , θ ′ a = π/ , θ b = − π/ , θ ′ b = − π/ . (30)Then all detector configurations with S = 2 √ κ , κ , and κ (0 ≤ κ , ≤ π , 0 ≤ κ ≤ π ), which describe an arbitraryrotation of the configuration (29)–(30).Similarly, all minimizing configurations ( S = − √ xz case [Eq. (29)]with θ a = 0 , θ ′ a = π/ , θ b = π/ , θ ′ b = 3 π/ κ , , .
2. General maximally entangled state
Any maximally entangled two-qubit state can be ob-tained from the singlet state by a unitary transformationof the basis of one of the qubits (i.e. a one-qubit rota-tion). Therefore, an arbitrary case corresponding to thebounds S ± = ± √ U b of thequbit b basis and simultaneous corresponding rotation ofthe detector axes ~b and ~b ′ for the second qubit. Sincethe transformation U b can also be characterized by threeEuler angles κ b , κ b , and κ b , an arbitrary situation with S = 2 √ κ , κ , κ , κ b , κ b , κ b ), using the standard con-figuration (29)–(30) as a starting point. Similarly, anysituation with S = − √ xz configuration (31).Since these six parameters can describe arbitrary direc-tions of four measurement axes ( ~a, ~a ′ ,~b,~b ′ ) still satisfyingthe conditions ~a ⊥ ~a ′ and ~b ⊥ ~b ′ , it is obvious that anysuch four-axes configuration produces S = 2 √ S = − √ S can obviously be flipped by a π -rotation of qubit a (or b )around the axis ~a × ~a ′ ( ~b × ~b ′ ) instead of the π -rotation(21) of its detector axes. Also notice that six indepen-dent parameters for an optimal configuration can be al-ternatively chosen as any parameters characterizing thefour measurement axes, which are pair-wise orthogonal: ~a ⊥ ~a ′ and ~b ⊥ ~b ′ .
3. Odd states
An important special case is the class of “odd” maxi-mally entangled states | Ψ i = ( | i + e iα | i ) / √ ≤ α < π ) , (32)which is of relevance for experiments with Josephsonphase qubits. Such states can be obtained (with anaccuracy up to an overall phase factor) from the sin-glet state | Ψ − i by unequal rotations of the two qubitsaround the z axis. Indeed, since U z ( ϕ ) = e − iϕσ z / ro-tates a spin 1/2 around the z axis by angle ϕ , we obtain[ U z ( α ) ⊗ U z ( α + π − α )] | Ψ − i = − ie − iα/ | Ψ i , where α is arbitrary. Thus, in view of Eq. (20), the optimaldetector configurations for the odd state (32) can be ob-tained from those for | Ψ − i by rotating the detectors forthe qubit b around the z axis by the angle π − α [noticethat the state (32) reduces to the singlet for α = π ]. Interms of the parameters θ and φ , this is equivalent to thechange φ b → φ b + π − α, φ ′ b → φ ′ b + π − α. (33)Thus, for the odd states the class of optimal configura-tions maximizing S (as well as the class minimizing S ) ischaracterized by four parameters: κ , κ , κ , and α .Now let us focus on the optimal configurations with thedetector axes lying either in a “vertical” plane for eachqubit (i.e., a plane containing the z axis) or the “horizon-tal” ( xy ) plane; such configurations will be important inthe study of effects of errors (Sec. IV) and decoherence(Sec. V).To obtain all “vertical” cases with the maximum BIviolation S = 2 √
2, we start with the standard configura-tion (29)–(30) for the singlet, then apply a rotation in the xz plane by an arbitrary angle C (we can also apply themirror reflection), then rotate the resulting configurationaround the z axis by an arbitrary angle φ , and finallyapply the α -rotation (33) determined by the phase of theodd state (32). As a result, the optimal measurementdirections for the qubits a and b generally lie in differentvertical planes, φ a = φ ′ a = φ , φ b = φ ′ b = φ + π − α, (34)while the polar angles corresponding to S = 2 √ θ a , θ ′ a , θ b , θ ′ b ) = ± (0 , π/ , − π/ , − π/
4) + C, (35)where φ and C are arbitrary angles, while α is deter-mined by the state (32). Notice that the signs ± corre-spond to the possibility of the mirror reflection, whichwe did not have to consider in the previous subsectionsbecause it can be reproduced using 3D rotations, whileit is a necessary extra transformation in the 2D case.Similarly, the minimum S = − √ θ a , θ ′ a , θ b , θ ′ b ) = ± (0 , π/ , π/ , π/
4) + C. (36) Recall that we define both θ and φ modulo 2 π , andtherefore each measurement direction corresponds to twosets of ( θ, φ ) [see Eq. (13)]. Consequently, the optimalconfigurations described by Eqs. (34)–(36) can be alsodescribed in several equivalent forms by applying thetransformation (13) to some of the four measurement di-rections.As follows from Eq. (34), the only odd states for whichthe optimal vertical configurations lie in the same planeare the Bell states | Ψ − i (corresponding to α = π ) and | Ψ + i (corresponding to α = 0). The optimal verticalconfigurations for the singlet state | Ψ − i are given by φ a = φ ′ a = φ b = φ ′ b = φ (37)and Eqs. (35)–(36). To describe the optimal vertical con-figurations for the state | Ψ + i , it is natural to apply theequivalence (13) to the qubit b measurement directions,so that the angles φ are still all equal as in Eq. (37), whilethe angles θ are given by Eqs. (35)–(36) with flipped signsfor the qubit b , i.e. θ b → − θ b and θ ′ b → − θ ′ b .Now let us consider the optimal detector configurationsin the horizontal ( xy ) plane: θ a = θ ′ a = θ b = θ ′ b = π/ . (38)All configurations for S = 2 √ xy plane (so that the angles θ are essentially replacedby the angles φ ), then applying an arbitrary rotationwithin xy plane and possibly the mirror reflection, andfinally applying the transformation (33) with the state-dependent parameter α , so that( φ a , φ ′ a , φ b + α, φ ′ b + α ) = ± (0 , π/ , π/ , π/
4) + C (39)with arbitrary C (the signs ± correspond again to thepossibility of the mirror reflection).Similarly, all horizontal configurations correspondingto S = − √ φ a , φ ′ a , φ b + α, φ ′ b + α ) = ± (0 , π/ , − π/ , − π/
4) + C. (40)Notice that the application of the equivalence (13) toall four measurement directions changes π/ − π/ π -shift of angles φ can be absorbed bythe arbitrary parameter C . IV. LOCAL MEASUREMENT ERRORS
In this section we consider the effects of local (inde-pendent) measurement errors on the BI violation.
A. Error model
The probabilities of the measurement results for a sin-gle qubit can be written in the form p Mi = X m =0 F im p m = Tr( Q i ρ q ) , (41)where p m are the probabilities which would be obtainedby ideal measurements, F im is the probability to find thequbit in the state | i i when it is actually in the state | m i ,and operator Q i = F i P + F i P contains the projectoroperators P , [see Eq. (10)]. The operators Q i satisfy thesame condition as the POVM measurement operators, namely, Q i are positive and Q + Q = 1. The condition p M + p M = 1 implies that F m + F m = 1. Hence, thematrix F has two independent parameters, which canbe chosen as the measurement fidelities F ≡ F and F ≡ F for the states | i and | i , so that p M = F ˜ ρ + (1 − F )˜ ρ , p M = (1 − F )˜ ρ + F ˜ ρ (42)(here ˜ ρ ij are the components of the one-qubit densitymatrix after the tomographic rotation and 0 ≤ F , ≤ F + F ≥
1, since inthe opposite case the measurement results can be simplyrenamed: 0 ↔
1; as a consequence, max( F , F ) ≥ / p Mij = X m,n =0 F aim F bjn p mn = Tr( Q ai Q bj ρ ) , (43)where F kim is the matrix F im for qubit k and Q ki = F ki P k + F ki P k [see Eq. (14)].In this section we will discuss the condition for the BIviolation as a function of measurement fidelities F k and F k . Sometimes we will limit the analysis to the case ofequal measurement fidelities for both qubits, F ai = F bi = F i , (44)however, the case of different measurement fidelities forthe two qubits is also of interest. (Different fidelitiesare especially of interest when the qubits have differentphysical implementations. For instance, in the case of anatom-photon qubit pair the detection efficiency for theatom is nearly 100%, whereas the photon-detector effi-ciency is significantly less than 100%.) Notice that for theJosephson phase qubits the trade-off between the fideli-ties F k and F k can be controlled in the experiment for each qubit individually by changing the measurementpulse strength.Several special cases of our error model have beenpreviously discussed in the literature, starting with theCHSH paper. For example, in the problem of the detec-tor loophole the CH inequality with F a = F b = 1 is often considered; then F k is called the detector efficiency;both the cases F a = F b (Refs. 26,29) and F b = F a (Ref.30) have been considered. Let us also mention the effectof nonidealities on the BI violation considered for the ex-periments on two-photon interference. The situationsof Refs. 27,28 formally correspond to the special case ofour model with F a = F a = F a , F b = F b = F b . (45)Then the product (2 F a − F b −
1) equals either thevisibility or the product of the visibility and the squareof the signal acceptance probability. B. General relations for S The Bell operator (17) can be generalized to the caseof measurement errors. Inserting Eq. (43) into Eq. (3)yields E ( ~a,~b ) = Tr( ˜ A ˜ Bρ ), where˜ A = Q a − Q a = F a − F a + ( F a + F a − ~a · ~σ a , (46a)˜ B = F b − F b + ( F b + F b − ~b · ~σ b . (46b)Therefore S can be expressed as S = Tr( ˜ B ρ ) (47)via the modified Bell operator˜ B = ˜ A ˜ B − ˜ A ˜ B ′ + ˜ A ′ ˜ B + ˜ A ′ ˜ B ′ , (48)where ˜ A ′ and ˜ B ′ are obtained from ˜ A and ˜ B by replac-ing ~a and ~b with ~a ′ and ~b ′ , respectively. Notice that ˜ B is a Hermitian operator and therefore in some cases it isuseful to think about the measurement of S as a measure-ment of a physical quantity corresponding to the operator˜ B (even though this analogy works only for averages).It is rather trivial to show that in the presenceof local measurement errors the Cirel’son’s inequality | S | ≤ √ forany POVM-type measurement). Moreover, a stricter in-equality for | S | [see Eq. (50) below] can be obtained, us-ing the method similar to that of Ref. 51. We will provethis inequality for all pure two-qubit states, ρ = | ψ ih ψ | ,which automatically means that it is also valid for anymixed state ρ . Using notation h O i = Tr( Oρ ) = h ψ | O | ψ i for any operator O , we start with the obvious relation | S | = |h ˜ A ( ˜ B − ˜ B ′ ) i + h ˜ A ′ ( ˜ B + ˜ B ′ ) i| ≤ |h ˜ A ( ˜ B − ˜ B ′ ) i| + |h ˜ A ′ ( ˜ B + ˜ B ′ ) i| . The next step is to apply the generalinequality |h O O i| ≤ h O O † ih O † O i to both terms inthe sum (this inequality is the direct consequence of theCauchy-Schwartz inequality |h ψ | ψ i| ≤ h ψ | ψ ih ψ | ψ i for the vectors | ψ i = O † | ψ i and | ψ i = O | ψ i ). In thisway we obtain | S | ≤ q h ˜ A ih ( ˜ B − ˜ B ′ ) i + q h ˜ A ′ ih ( ˜ B + ˜ B ′ ) i (49)(notice that operators ˜ A , ˜ A ′ , ˜ B , and ˜ B ′ are Hermi-tian). As the next step, we notice that the eigenval-ues of ˜ A (as well as eigenvalues of ˜ A ′ ) are 2 F a − − F a , which follows from Eq. (46) and thefact that the eigenvalues of ~a · ~σ a are ±
1. Therefore, h ˜ A i ≤ (2 F a max − and h ˜ A ′ i ≤ (2 F a max − , where F k max = max( F k , F k ); and so from Eq. (49) we ob-tain | S | ≤ (2 F a max − (cid:20)q h ( ˜ B − ˜ B ′ ) i + q h ( ˜ B + ˜ B ′ ) i (cid:21) .Next, since √ x + √ x ≤ p x + x ) for any positivenumbers x and x , and using the relation h ( ˜ B − ˜ B ′ ) i + h ( ˜ B + ˜ B ′ ) i = 2 h ˜ B + ˜ B ′ i , we obtain the inequality | S | ≤ F a max − q h ˜ B + ˜ B ′ i . Finally, using the re-lations h ˜ B i ≤ (2 F b max − and h ˜ B ′ i ≤ (2 F b max − ,derived in a similar way as above, we obtain the upperbound | S | ≤ √ F a max − F b max − . (50)This upper bound is generally not exact and can bereached only in the case when the errors are symmetric inboth qubits [Eq. (45)], leading to Eq. (55) below. Whilethe bound (50) depends only on the largest measurementfidelity for each qubit, our numerical results show thatthe exact bounds S ± shrink monotonously with the de-crease of all fidelities, if the errors are small enough toallow the BI violation (see below).A useful expression for S can be obtained from Eqs.(46)–(47) by separating the terms for the ideal case: S = 2 ξ a − ξ b − + 2 ξ a + ξ b − ~a ′ · ~s a + 2 ξ a − ξ b + ~b · ~s b + ξ a + ξ b + S , (51)where ξ k + = F k + F k − , ξ k − = F k − F k , S is the value of S in the absence of errors [Eq. (16)], and ~s k is the Blochvector characterizing the reduced density matrix for thequbit k , i.e. ρ k = Tr k ′ = k ρ = ( I + ~s k · ~σ ) /
2. Notice that ~s a = ~s b = 0 for a maximally entangled state and there-fore the second and third terms in Eq. (51) may increase | S | for nonmaximally entangled optimal states. That iswhy in the presence of errors the states maximizing andminimizing S are usually nonmaximally entangled (seebelow).Notice that in the presence of errors, S still preservesthe invariance with respect to the local transformations ofqubits and simultaneous rotation of measurement direc-tions described by Eqs. (20). The symmetry described byEq. (21) (sign flip of S for the reversal of one-qubit mea-surement directions) is no longer valid; however, it canbe easily modified by adding simultaneous interchange F ↔ F of one-qubit fidelities: S → − S if ~a → − ~a, ~a ′ → − ~a ′ , F a ↔ F a ; (52a) S → − S if ~b → − ~b, ~b ′ → − ~b ′ , F b ↔ F b . (52b)Obviously, S does not change if both transformations(52) are made simultaneously. As a consequence, themaximum S + and minimum S − (optimized over the qubit states and over measurement directions) are invariantwith respect to simultaneous interchange of measurementfidelities F a ↔ F a , F b ↔ F b , (53)while the extremum values change as S + → − S − , S − →− S + if only one-qubit fidelity interchange ( F a ↔ F a or F b ↔ F b ) is made. (The corresponding optimal statesobviously do not change.)In the presence of measurement errors the magnitudesof the maximum and minimum of S generally differ, S + = | S − | . However, as follows from the latter sym-metry, S + = | S − | (as in the ideal case) if the two mea-surement fidelities are symmetric (equal) at least for onequbit: F a = F a or F b = F b . (54)If the fidelities are symmetric for both qubits [the sit-uation described by Eq. (45)], then the expression for S given by Eq. (51) becomes simple: S = (2 F a − F b − S and directly related to the value S without mea-surement errors. Then the extremum values S ± = ± √ F a − F b −
1) (55)are obviously achieved for any maximally entangled stateunder the same conditions as in Sec. III. Correspond-ingly, the requirement for the fidelities for a violation ofthe BI is (2 F a − F b − > − / ≈ . . (56)Notice that when the measurement fidelities for the bothqubits are the same, F a = F b = F , Eq. (56) reduces tothe threshold fidelity F > . − / ≈ . , (57)while if the measurement for one of the qubits is ideal(for example, F a = 1), then the BI violation requires F b > . − / ≈ . . (58) C. Analytical results for maximally entangledstates
Let us first analyze the extremum values of S for theclass of maximally entangled states (25). Since in thiscase ~s a = ~s b = 0, we obtain from Eq. (51) that for maxi-mally entangled states S = 2( F a − F a )( F b − F b ) + ( F a + F a − F b + F b − S (59)is directly related to the corresponding quantity S in theabsence of errors. Therefore the extremum values of S for maximally entangles states are S ± = 2( F a − F a )( F b − F b ) ± √ F a + F a − F b + F b − , (60)and they are achieved under the same conditions as dis-cussed in Sec. III.When the asymmetry of measurement fidelities is sim-ilar for both qubits ( F a > F a and F b > F b or bothinequalities with “ < ” sign), the first term in Eq. (60) ispositive, and therefore the BI | S | ≤ S than for negative S . Similarly, ifthe asymmetries are opposite (for example, F a > F a and F b < F b ), then it is easier to violate the BI for negative S . If fidelities are symmetric at least for one qubit [Eq.(54)], then the first term in Eq. (60) vanishes, and there-fore S − = − S + , as discussed in the previous subsection.If the fidelities are the same for both qubits, F a = F b = F and F a = F b = F , then the positive S ispreferable and S + = 2( F − F ) + 2 √ F + F − (61)(see the dashed lines in Fig. 1). This value of S + reachesthe maximum 2 √ F = F =1) and decreases with the decrease of each fidelity in theinteresting region S + > S + > − √ ≈ . F − F ) + √ F + F − > . (62)This threshold of the BI violation on the F - F plane isshown by the lowest dashed line in Fig. 1. It is an arc ofthe ellipse (corresponding to S + = 2), which is symmetricwith respect to the line F = F and is centered at F = F = 0 .
5. However, as seen from Fig. 1, this thresholdlooks quite close to a straight line on the F - F plane.Notice that in the case F = 1 the threshold (62) reducesto the most well-known condition F > √ − ≈ . , (63)while in the case of symmetric error, F = F = F , werecover Eq. (57). D. Numerical results
To optimize the CHSH inequality violation in presenceof the measurement errors over all two-qubit states, in-cluding non-maximally entangled states, we have usednumerical calculations. The analysis has been performedin two different ways (with coinciding results). First, wehave searched for the maximum violation by finding ex-trema of the eigenvalues of the modified Bell operator ˜ B defined by Eq. (48). Since ˜ B is a Hermitian operator and S = T r ( ˜ B ρ ), for fixed measurement directions ( ~a, ~a ′ ,~b,~b ′ )the maximum and minimum eigenvalues of ˜ B are equalto the maximum and minimum values of S , optimizedover the two-qubit states. Therefore, optimization of theeigenvalues over the measurement directions gives the ex-trema of S . Similar method has been previously used for the case of identical local errors (44) with F = 1, F F S + = FIG. 1: Solid lines: contour plot of S + (the maximum quan-tum value of S ) versus the measurement fidelities F and F (assumed equal for both qubits), optimized over all two-qubitstates. The dashed lines show the result of S + maximizationover the maximally-entangled states only [Eq. (61)]. The Bell(CHSH) inequality can be violated when S + > while we apply this method to our more general errormodel.The numerical maximization (minimization) of thelargest (smallest) eigenvalue of the Bell operator ˜ B hasbeen performed using the software package Mathemat-ica. The full optimization should be over all four mea-surement directions ( ~a, ~a ′ ,~b,~b ′ ), described by 8 angles to-tal. However, because of the invariance of S under localtransformations, it is sufficient to optimize ˜ B over onlytwo angles: the angle between ~a and ~a ′ and the anglebetween ~b and ~b ′ , while the other angles are kept fixed.Our numerical results show that in general the optimalvalues of these two angles are different from each other;however, for equal fidelity matrices [Eq. (44)] these anglesare equal, so that ~a · ~a ′ = ~b · ~b ′ . This result has beenobtained previously for the special case F = 1. Ournumerical results also show that S + > | S − | for positivevalues of the product ( F a − F a )( F b − F b ) and S + < | S − | when this product is negative, similar to the result forthe maximally entangled states [see discussion after Eq.(60)].We have checked that the numerical results for S + and | S − | obtained via optimization of the eigenvalues of theBell operator ˜ B coincide with the results (see Fig. 1) ob-tained by our second numerical method based on the di-rect optimization of S . The second method happenedto be more efficient numerically; as another advantage,it provides the optimal measurement directions togetherwith optimal values S + and S − , while the Bell-operatormethod gives only S + and S − .In principle, direct optimization of S (for fixed mea-surement fidelities) implies optimization over the two-qubit density matrix and over 8 measurement directions.However, there is a simplification. It is obviously suf-ficient to consider only pure states, since probabilisticmixtures of pure states cannot extend the range of S .Moreover, it is sufficient to consider only the states ofthe form | Ψ i = cos( β/ | i + sin( β/ | i , (64)since any pure two-qubit states can be reduced to thisform by local rotations of the qubits (which are equiv-alent to rotations of the measurement directions); thisfact is a direct consequence of the Schmidt decompositiontheorem. The angle β can be limited within the range0 ≤ β ≤ π because the coefficients of the Schmidt de-composition are non-negative. This range can be furtherreduced to 0 ≤ β ≤ π/ π -rotation of both qubitsabout x -axis (or any horizontal axis) exchanges states | i ↔ | i and therefore corresponds to the transforma-tion β → π − β .Our numerical optimization of S within the class oftwo-qubit states (64) has shown that for non-zero mea-surement errors (we considered 2 / ≤ F ki ≤
1) the opti-mal measurement directions ( ~a, ~a ′ ,~b,~b ′ ) always lie in thesame vertical plane [such configuration is described byEq. (37)]. This vertical plane can be rotated by an ar-bitrary angle about z -axis [such rotation is equivalentto an overall phase factor in Eq. (64)]; therefore we canassume φ = 0 in Eq. (37). Notice that for the state(64) the vectors ~s a and ~s b in Eq. (51) are along the z -axis, ~s a = − ~s b = ~z cos β . These vectors are zero for themaximally entangled state ( β = π/ S has the form S = 2 ξ a − ξ b − − ξ a + ξ b + ( g − h sin β )+2 cos β ( ξ a + ξ b − cos θ ′ a − ξ a − ξ b + cos θ b ) , (65)where g = cos θ a cos θ b − cos θ a cos θ ′ b + cos θ ′ a cos θ b + cos θ ′ a cos θ ′ b ,h = sin θ a sin θ b − sin θ a sin θ ′ b + sin θ ′ a sin θ b + sin θ ′ a sin θ ′ b . (66)The numerical maximization and minimization of S inthis case involves optimization over 5 parameters: β , θ a , θ ′ a , θ b , and θ ′ b . Nevertheless, in our calculations this pro-cedure happened to be faster than optimization over onlytwo parameters in the method based on the Bell operatoreigenvalues (we used Mathematica in both methods).The solid lines in Fig. 1 show the contour plot of max-imum value S + on the plane F - F for the case whenthe measurement fidelities for two qubits are equal [Eq.(44); in this case S + ≥ | S − | ]. Notice that the line for S + = 2 ends at the points F = 1 , F = 2 / F = 2 / , F = 1 (strictly speaking, this line correspondsto S + = 2 + 0 since S = 2 can be easily realized with-out entanglement). The dashed lines, which correspondto the optimization over the maximally entangled statesonly [Eq. (61)], coincide with the solid lines at the points F = F , because in this case the optimum is achievedat the maximally entangled states, as follows from thediscussion after Eq. (54). When F = F , the use ofnon-maximally entangled states gives a wider range ofmeasurement fidelities allowing the BI violation. How-ever, as seen from Fig. 1, the difference between the solidand dashed lines significantly shrinks with the increase of S + , so that there is practically no benefit of using non-maximally entangled states for the BI violation strongerthan S + > .
4. Notice that the solid and dashed lines inFig. 1 are symmetric about the line F = F since the in-terchange F ↔ F does not change S + , as was discussedafter Eq. (52).Numerical calculations show that in the case of equalfidelity matrices, Eq. (44), the optimal detector configu-rations for a nonmaximally entangled state (64) have a“tilted-X” shape: ~a = − ~b ′ and ~a ′ = − ~b . In this case thenumber of parameters to be optimized in (65) reducedfrom 5 to 3, significantly speeding up the numerical pro-cedure.Notice that each optimal configuration within the classof states (64) corresponds to a 6-dimensional manifoldof optimal configurations, obtained by simultaneous lo-cal rotations of the measurement axes and the two-qubitstate (see discussion in Sec. III B 2). V. DECOHERENCE
The detailed analysis of the effects of decoherence willbe presented elsewhere. In this sections we discuss onlysome results of this analysis, and also discuss the com-bined effect of local measurement errors and decoherence.To study effects of decoherence we assume for simplic-ity that the qubit rotations are infinitely fast. Thus, weassume that after a fast preparation of a two-qubit state ρ there is a decoherence during time t resulting in the state ρ ′ , which is followed by fast measurement of ρ ′ (includingtomographic rotations). Now S is given by Eq. (16) (inthe absence of errors) or (47) and (51) (in the presenceof errors) where ρ should be substituted by ρ ′ . To obtain ρ ′ we assume independent (local) decoherence of eachqubit due to zero-temperature environment, described bythe parameters γ k = exp( − t/T k ) and λ k = exp( − t/T k )(here k = a, b ) where T k and T k are the usual relaxationtimes for the qubit k ( T k ≤ T k ).As the initial state we still assume the state of theform (64) (even though in presence of decoherence thisstate actually does not always provide the extrema of S ). It can be shown analytically that in the absenceof measurement errors the maximum violation of the BIfor the state (64) can be achieved when the detector0axes lie in either a horizontal [Eq. (38)] or vertical [Eq.(37)] plane (any other detector configuration cannot givestronger violation). In the case of only population relax-ation ( T k = 2 T k ) the horizontal configuration is better,while in the case of only T -effect ( T k = ∞ ) the verticalconfiguration is better.When local measurement errors are considered to-gether with decoherence, the optimal detector configu-rations may be neither vertical nor horizontal. To eluci-date this fact, note that in Eq. (51) with ρ replaced by ρ ′ the vectors ~s a and ~s b remain vertical in the presence ofdecoherence. As a result, when in the absence of errorsthe optimal configuration is horizontal, measurement er-rors may make the optimal detector axes to go out of thehorizontal plane. Note, however, that for some parame-ter ranges the vertical and horizontal configurations arestill optimal.In numerical calculations we should optimize S over 8parameters: β and 7 detector angles (one of the angles φ can be fixed because of the invariance of S under identi-cal rotations of the qubits around the z axis). We haveperformed such optimization for a few hundred param-eter points, choosing the measurement fidelities F ki anddecoherence parameters γ k and λ k randomly from therange (0.7, 1). For many (more than half of) parame-ter points the optimal configuration was still found to beeither vertical or (in much smaller number of cases) hori-zontal. Even when the optimal configuration was neithervertical nor horizontal, we found that restricting opti-mization to only the vertical and horizontal configura-tions gives a very good approximation of the extrema S ± (within 0.01 for all calculated parameter points). Suchrestriction significantly speeds up the calculations, sincewe need to optimize over only 5 parameters instead of 8parameters.Assuming initial state (64) and replacing ρ by ρ ′ in Eq.(47) we obtain S = 2 ξ a − ξ b − + ξ a + ξ b + { [1 − γ a − γ b − ( γ a − γ b ) cos β ] g + λ a λ b h sin β } + 2 ξ a + ξ b − ( γ a + γ a cos β −
1) cos θ ′ a +2 ξ a − ξ b + ( γ b − γ b cos β −
1) cos θ b , (67)when the detector axes are in a vertical plane [ g and h are defined in Eq. (66)], and S = 2 ξ a − ξ b − + ξ a + ξ b + λ a λ b sin β [cos( φ a − φ b ) − cos( φ a − φ ′ b ) + cos( φ ′ a − φ b ) + cos( φ ′ a − φ ′ b )] (68)for a horizontal detector configuration.To find the extrema of S within the class of verticalconfigurations, Eq. (67) should be numerically optimizedover the parameter β and four angles θ . The optimiza-tion of S within the class of horizontal configurations ismuch simpler, because the term in the square bracketsin Eq. (68) can be optimized independently of β . Thisoptimization is exactly the same as in the ideal case [seeEqs. (39) and (40) with α = 0], therefore the term in thesquare brackets has extrema ± √
2, and therefore the extrema of Eq. (68) are reached at β = π/ S ± = 2 ξ a − ξ b − ± √ ξ a + ξ b + λ a λ b , (69)which depends only on the T -relaxation and measure-ment fidelities. F F S + = FIG. 2: Contour plot of the maximum value S + versus themeasurement fidelities F and F for the initial state (64) inpresence of decoherence with γ a = γ b = 0 .
96 and λ a = λ b =0 .
94, in absence (solid lines) or presence (dashed lines) of thesymmetric crosstalk with p c = 0 .
1. Solid and dashed linescoincide for S + = 2. For the numerical example shown by solid lines in Fig.2 we assume that decoherence is identical for the twoqubits and choose γ a = γ b = 0 .
96 and λ a = λ b = 0 . T ≃
450 ns, T ≃
300 ns, and t ≃
20 ns. We alsoassume identical errors for both qubits, Eq. (44), whichimplies S + ≥ | S − | (as in the absence of decoherence), soin Fig. 2 we show the contour plot only for S + . We havefound that for these decoherence parameters the verticaldetector configuration is better than any other configu-ration [assuming initial state (64)] for any measurementfidelities in the analyzed range (0 . ≤ F , ≤ S + = 2 . F > .
947 (for F = F = F ), that shouldbe compared to the threshold F > .
920 in absence ofdecoherence.Let us mention that the error model previously dis-cussed in relation to the BI violation in the two-photoninterference can be shown to be formally equivalentto the special case of our model with pure dephasing( T k = ∞ ) and identical errors with F = 1. Then ourquantities F and λ a λ b correspond, respectively, to thedetector efficiency and visibility in Ref. 53. The case ofpure dephasing in the absence of errors has been also con-1sidered in connection with the BI violation in mesoscopicconductors. VI. MEASUREMENT CROSSTALK
The nonidealities discussed above are common formany types of qubits. Now let us discuss a more specifictype of error: the measurement crosstalk for Josephsonphase qubits.
The crosstalk error originates from thefixed (capacitive) coupling between the qubits, which isstill on in the process of measurement. The mechanismof the crosstalk is the following. If the measurementoutcome for the phase qubit a is “1”, then this qubit isphysically switched to a highly excited state (outside ofthe qubit Hilbert space), and its dissipative oscillatingevolution after the switching affects the qubit b . As aresult, the extra excitation of the qubit b may lead toits erroneous switching in the process of measurement,so that instead of the measurement outcome “1,0” wemay get “1,1” with some probability p ac . Similarly, be-cause of the crosstalk from the qubit b to the qubit a ,we may obtain the measurement result “1,1” instead of“0,1” with some probability p bc . The values of p ac and p bc significantly depend on the timing of the measurementpulses applied to the qubits. If the qubit a is measuredfew nanoseconds earlier than the qubit b , then p ac ≫ p bc ;if the qubit b is measured first, then p ac ≪ p bc . In the casewhen the measurement pulses are practically simultane-ous, the crosstalk probability becomes significantly lowerand p ac ≈ p bc .Let us model the crosstalk in the following simple way.Even though physically the crosstalk develops at thesame time as the measurement process and its descrip-tion is quite nontrivial, we will assume (for simplicity)that the crosstalk effect happens after the “actual” mea-surement (characterized by measurement fidelities as inSections IV and V), so that the only effect of the crosstalkis the change of the outcome “1,0” into “1,1” with prob-ability p ac and the change of the outcome “0,1” into “1,1”with probability p bc . Moreover, we assume that the prob-abilities p ac and p bc do not depend on the measurementaxes ( ~a , ~b , etc.).Notice that the measurement crosstalk obviously vio-lates the fundamental assumption of locality, on whichthe BI is based (so, strictly speaking the BI approach isnot applicable in such situation). In this section we dis-cuss the modification of the classical bound for S , tak-ing the crosstalk into account (this bound is now model-dependent, in contrast to the usual BI), and we also an-alyze the effect of the crosstalk on the quantum resultfor S ± . Besides that, we discuss a simple modification ofthe experimental procedure, which eliminates the effectof crosstalk by using only the “negative result” outcomes. A. Modified Bell (CHSH) inequality
First, let us briefly review the derivation of the CHSHinequality, presented in Ref. 32. In a local realistic theory S = Z s (Λ) F (Λ) d Λ , (70)where F (Λ) is a distribution of the hidden variable Λ and s (Λ) = A (Λ , ~a ) B (Λ ,~b ) − A (Λ , ~a ) B (Λ ,~b ′ )+ A (Λ , ~a ′ ) B (Λ ,~b ) + A (Λ , ~a ′ ) B (Λ ,~b ′ ) . (71)Here the measurement outcomes A (Λ , ~a ) and B (Λ ,~b ) cantake only the values ±
1, depending on the hidden vari-able Λ and the detector orientations for the qubits a and b . (Notice that in Sections III and IV the notation A and B has been used for operators; now they are clas-sical quantities. Also notice that the outcome value − s (Λ) = ± locality assumption : the result A (Λ , ~a ) does not depend on the orientations ~b of the qubit b and vice versa; similarly, F (Λ) does not depend on ~a and ~b . After integration (70), this leads to the CHSHinequality (1).In our model the measurement crosstalk cannot changethe positive product of outcomes AB = 1; however, itchanges AB = − AB = 1 with the probabil-ity p ac (Λ) if A = − B = 1 or with probability p bc (Λ) if A = − B = −
1. (The locality assumption is obviouslyviolated, since the value of A now depends not only onΛ and ~a but also on B and thus implicitly on ~b ; similarlyfor B .) Notice that we assume that the crosstalk proba-bilities p ac and p bc may in principle depend on Λ (this is aslight generalization of the more natural Λ-independentmodel for p ac and p bc ).The random change from AB = − AB = 1 dueto the crosstalk leads to the modification of the CHSHinequality (1). Here we mention only the main pointsof the derivation of the modified CHSH inequality; thedetails are in the Appendix. We start with fixing Λ andconsidering all possible changes of the quantity s (Λ) inEq. (71) due to the crosstalk, thus obtaining the new val-ues of s together with their probabilities for each of 16realizations of the vector C = ( A, A ′ , B, B ′ ). It is easyto see that due to the crosstalk s (Λ) can get the values ±
4, which are outside of the limits ±
2. Averaging thevalue s (Λ) over the crosstalk scenarios and then maxi-mizing and minimizing the result over 16 realizations ofthe vector C , we get − { p ac (Λ) , p bc (Λ) } ≤ h s (Λ) i ≤ | p ac (Λ) − p bc (Λ) | . (72)Finally averaging this result over Λ, we obtain the mod-ified CHSH inequality: − { p ac , p bc } ≤ S ≤ | p ac − p bc | , (73)which is the main result of this subsection.2Let us consider two special cases. For a symmetriccrosstalk, p ac = p bc = p c , the inequality (73) becomes − p c ≤ S ≤ , (74)while for a fully asymmetric crosstalk, p bc = 0, it becomes − ≤ S ≤ p ac . (75)(similarly for p ac = 0).It is interesting to notice that the inequality in the sym-metric case is more restrictive than the BI (1) (“easier”negative bound and no change of the positive bound).This is actually quite expected because in the limitingcase p c = 1 the crosstalk makes all AB products equal 1,so that S = 2 always, as also follows from Eq. (74). Incontrast, for the fully asymmetric crosstalk the inequal-ity (75) is less restrictive than (1) (“harder to violate”positive bound and no change of the negative bound). Inthe case of a finite crosstalk asymmetry, both the positiveand negative bounds change [Eq. (73)].Let us emphasize that in contrast to the derivation ofthe original CHSH inequality, s (Λ) may be significantlyoutside of the range ( − , s (Λ) = ± S never decreases and theupper bound increases only slightly for small crosstalk,is due to the statistical averaging of the random increaseand decrease of s (Λ) due to the crosstalk. B. Quantum calculation of S In the quantum case the crosstalk changes the mea-surement probabilities p ij → p Cij as p C = p , p C = (1 − p ac ) p , p C = (1 − p bc ) p ,p C = p + p ac p + p bc p . (76)Then using the definitions (2) and (3) we obtain that themeasured value of S becomes S C = 2˜ p c + (1 − ˜ p c ) S + 2( p ac − p bc )[ p b ( ~b ) − p a ( ~a ′ )] , (77)where ˜ p c = ( p ac + p bc ) /
2, while S , p a ( ~a ′ ), and p b ( ~b ) are thequantities obtained in the absence of crosstalk [ p a ( ~a ′ ) and p b ( ~b ) are defined after Eq. (5)].In a general case the maxima and minima S C ± of Eq.(77) can be found numerically. To estimate the effect ofthe crosstalk, let us calculate Eq. (77) for a maximallyentangled state in the absence of errors and decoherence.Then p a ( ~a ′ ) = p b ( ~b ) = 1 /
2, and using S ± = ± √ S C + = 2 √ − (2 √ − p c , S C − = − √ √ p c . (78)As we see, both extrema are affected by the crosstalk,making the range narrower from both sides; however, thelower boundary is affected much stronger than the upper boundary. Comparing Eq. (78) with the modified CHSHinequality (73), we see that the lower bound shifts up forthe quantum result always faster than for the classicalbound; therefore the gap between the quantum and clas-sical bounds always shrinks due to the crosstalk. Theclassical-quantum gap at positive S shrinks from bothsides due to the crosstalk.In the case of symmetric crosstalk, p ac = p bc = p c , wecan easily consider non-maximally entangled states, mea-surement errors and decoherence, since Eq. (77) in thiscase reduces to S C = 2 p c + (1 − p c ) S . Therefore, S C ± aresimply related to the values S ± without crosstalk (butwith measurement errors and decoherence): S C ± = 2 p c + (1 − p c ) S ± (79)[similar dependence was used in Eq. (78)]. A violation ofthe upper bound of the modified CHSH inequality (74)can be observed when S C + = 2 p c + (1 − p c ) S + >
2, whichyields S + >
2, while a violation of the lower limit inEq. (74) requires S C − = 2 p c + (1 − p c ) S − < − p c ,which yields S − < −
2. Quite surprisingly, the symmet-ric crosstalk does not change the conditions for the BIviolation. (Of course, the violation of the increased lowerbound is not as convincing psychologically as the viola-tion on the increased upper bound.)Let us discuss the combined effect of local errors, deco-herence, and symmetric crosstalk for the numerical exam-ple considered in Sec. V, assuming symmetric crosstalkwith p c = 0 .
1. Now for the state (64) in the absence of lo-cal measurement errors ( F = F = 1) we get S C + = 2 . S + = 2 .
50 obtainedin the absence of the crosstalk. The dependence of S C + on the measurement fidelities F and F in this case isillustrated by the dashed lines in Fig. 2. In accordancewith the above discussion, the solid line for S + = 2 andthe dashed line for S C + = 2 coincide (the BI violationboundary is not affected), while the comparison of thesolid and dashed lines for S + = 2 . . S + more difficult.Figure 3 shows a similar contour plot on the F - F plane for the lowest quantum value S C − assuming thesame parameters as in Fig. 2. Comparing Figs. 2 and 3we see that it is more difficult to violate the lower classicalbound (even though it is now only − .
6) than the upperclassical bound of 2. This is because we assumed thesame measurement fidelities for both qubits, that gener-ally shifts the quantum result up [as in Eq. (60)]. Noticethat in Fig. 3 the reference bound of − C. Elimination of the crosstalk effect
Slight modification of the CHSH inequality (1) canmake it insensitive to the crosstalk. The main idea isto use only experimental outcomes with the result “0”,when a qubit does not switch, and therefore the crosstalk3 F F S - C = - - - FIG. 3: Contour plot of the quantum minimum S C − versus themeasurement fidelities F and F (same for both qubits) op-timized over the states (64). We assume symmetric crosstalkwith p c = 0 . γ a = γ b = 0 . λ a = λ b = 0 .
94 (same as for Fig. 2). The classical boundis shifted from − − . does not occur. Such “negative-result” (“null-result”) ex-periments with the Josephson phase qubits are very in-teresting from the quantum point of view; however,here we are interested only in the classical consequence(or rather absence of it) for a measurement with a nullresult.Instead of the inequality (1) let us use the equivalentinequality (6), and let us change the definition of T inEq. (5) by interchanging the measurement outcomes “1”and “0”. Then by symmetry the inequality (6) is stillvalid, so we get the classical bounds − ≤ ˜ T ≤ T = p ( ~a,~b ) − p ( ~a, ~b ′ ) + p ( ~a ′ ,~b ) + p ( ~a ′ , ~b ′ ) − p ( ~a ′ ) − p ( ~b ) , (81)where the probability p ( ~a ′ ) is for measuring the qubit a only (without measuring the qubit b ), while p ( ~b ) is formeasuring only the qubit b .With this simple modification, the CHSH inequalitybecomes insensitive to the mechanism of the measure-ment crosstalk considered in this section. Notice,however, that in performing experiment in this way it isstill important to check the absence of a direct crosstalk(due to the measurement pulse itself). This can be doneby applying a measurement pulse to the well-detunedqubit b (so that it cannot switch) and checking that thisdoes not affect switching probabilities for the qubit a (and similarly for a interchanged with b ). VII. CONCLUSION
In this paper we have considered the conditions for theviolation of the Bell inequality in the CHSH form for theentangled pairs of solid-state qubits, when instead of therotation of optical polarizers (detectors) we have to rotatethe states of two qubits before the measurement, whichitself is always performed in the logical z -basis ( | i , | i )for each qubit. While most of our results are applicable tomany types of qubits, we have focused on the experimentswith the Josephson phase qubits. We have analyzed theBI violation for the ideal case as well as in presence ofvarious nonidealities, including local measurement errors,local decoherence, and measurement crosstalk.In the ideal case the maximum violation of the BI( S ± = ± √
2, while the classical bound is | S | ≤
2) can berealized for any maximally entangled state. The optimalconfiguration of the measurement directions in this casecan be realized with three degrees of freedom for eachmaximally-entangled state (the measurement directionin our terminology actually refers to the qubit rotationbefore the measurement). However, in presence of non-idealities there is typically less freedom in choosing theoptimal configuration. For the “odd” two-qubit statesinvolving superpositions (64) of the states | i and | i ,we have focused on the “vertical” measurement configu-rations, for which all measurement directions ( ~a, ~a ′ ,~b,~b ′ )are within the same vertical plane of the Bloch sphere,and the “horizontal” configuration, for which the fourmeasurement axes are within the x - y plane.The qubit measurement with finite local errors (charac-terized by the fidelities F and F for each qubit) shrinksthe quantum range for S . We have found that for a max-imally entangled state the BI violation is still possiblewhen the classical bounds ± F + F ) / > .
92. A significantlysofter violation condition can be obtained when allow-ing the two-qubit state to be non-maximally entangled; this condition is shown by the lowest solid line in Fig. 1.However, the trick of using a non-maximally-entangledstate does not help much when we need a BI violationwith a significant margin (not just barely); this can beseen by comparing the solid and dashed lines in Fig. 1.For non-maximally entangled odd two-qubit states (64)in presence of local measurement errors, the vertical mea-surement configuration is found to be preferable in com-parison with other configurations.Analyzing the effect of local decoherence of the qubitsfor the odd two-qubit states (64), we have found that ei-ther vertical or horizontal configuration of the measure-ment directions is optimal, depending on the parame-ters. In particular, in the case of population (energy)relaxation in z -basis, the horizontal configuration is opti-mal, while for pure dephasing the vertical configuration4is optimal. In presence of both the decoherence and lo-cal measurement errors, the optimal configuration can beneither horizontal nor vertical; however, restricting opti-mization to only these two classes of configurations givesa very good approximation of the extrema S ± . Obvi-ously, both the decoherence and the measurement errorsmake the observation of the BI violation more difficult.We have also analyzed the effect of the measurementcrosstalk which plays an important role in measure-ment of the capacitively-coupled phase qubits. Since thecrosstalk is a mechanism of classical communication be-tween the qubits, strictly speaking the BI is inapplica-ble. However, for a particular model of the crosstalk itis possible to derive a modified CHSH inequality [see Eq.(73)]. In particular, we have found that the symmetriccrosstalk does not change the upper classical bound butincreases the lower classical bound. The crosstalk alsoaffects the quantum bounds, which are given by Eq. (78)for the maximally entangled state in the otherwise idealcase with arbitrary crosstalk and by Eq. (79) for an ar-bitrary case but assuming a symmetric crosstalk. Quiteunexpectedly, the symmetric crosstalk does not changethe threshold condition for the observation of the BI vi-olation. However, the crosstalk always reduces the gapbetween the classical and quantum bounds and makes anobservation of the BI violation with a finite margin moredifficult. It is important to mention that the detrimentaleffect of the crosstalk can be eliminated by a slight changeof the CHSH inequality (by using only negative-resultoutcomes), which makes it insensitive to the crosstalk[see Eqs. (80) and (81)].We have performed the numerical simulations with theparameters similar to the experimental values for thebest present-day experiments with the Josephson phasequbits. Our results (see Fig. 2) show the possibilityof the CHSH inequality violation with a significant mar-gin even without further improvement of the phase qubittechnology.
Acknowledgments
We thank Qin Zhang, John Martinis, Nadav Katz, andMarkus Ansmann for useful discussions. The work wassupported by NSA and DTO under ARO grant W911NF-04-1-0204.
APPENDIX: DERIVATION OF THEINEQUALITY (73)
To derive the inequality (72), we fix Λ and use theabbreviated notation s = s (Λ), p ac = p ac (Λ), p bc = p bc (Λ), A = A (Λ , ~a ) , A ′ = A (Λ , ~a ′ ) and similarly for B and B ′ . Let us introduce the vectors C = ( A, A ′ , B, B ′ ) and A = ( AB ′ , AB, A ′ B, A ′ B ′ ), so that s = A · ( − , , , C can assume 16 values, whereas without thecrosstalk A can take 8 values, since the number of pluses or minuses in A is even. Each pair of C and −C yieldsone value of A . Generally, the crosstalk effect differs for C and −C , except for the symmetric case, p ac = p bc = p c ,when crosstalk depends only on A . We start the analysiswith the symmetric case and then consider the generalasymmetric case.
1. Symmetric crosstalk
It is easy to check that without crosstalk s = 2 or −
2. To obtain the upper bound of the modified BI,we consider four values of A corresponding to s = 2.The crosstalk cannot change A = (1 , , , − , − , , , ( − , , − , − , , , − s to take the values 0, 2, and 4 with the probabili-ties p c q c , q c + p c , and p c q c , respectively, where q c = 1 − p c .Indeed, the change of any − A occurs with theprobability p c q c , yielding s = 0 for the change of thefirst − A and s = 4 for the change of any other − q c + p c is the probability of no change or changeto A = (1 , , , s = 2. Thus,though now s can achieve the maximal mathematicallypossible value 4, it is easy to see that the maximal valuefor the average of s over crosstalk is still h s i max = 2.To obtain the lower limit of the inequality, we considerfour values of A corresponding to s = − A = ( − , − , − , − s = − , − , , , p c q c , q c + 3 p c q c , p c q c + 3 p c q c , p c q c + p c , p c q c , re-spectively (note that the sum of the above probabilitiesequals 1). The above probabilities can be easily obtainedif one takes into account that s = − − A , s = − A have not changed or the first and one of thelast 3 components have changed, s = 0 occurs when thecrosstalk results in A with the first and two other com-ponents equal to 1 or − s = 2 occurs when two of thelast three components or all the components of A changethe sign, and s = 4 results from the changes of all thelast three components in A . As a result, in the case whenonly A = ( − , − , − , −
1) is realized, we obtain h s i = − p c . (A.1)Finally, let us consider the values A = (1 , − , − , , , − , − , − , , − s to take the values −
2, 0, 2 withthe probabilities q c , p c q c , and p c , respectively. Theseprobabilities follow from the fact that s = − A is not changed, s = 0 results from the change of onlyone component −
1, and s = 2 results from the change ofthe both negative components. It is easy to check thatfor any of the three above vectors, we again obtain Eq.(A.1).5Combining the results for the upper and lower bounds,we obtain the inequality − p c ≤ h s i ≤
2, the averageof which over Λ yields the modified BI for the symmetriccrosstalk given by Eq. (74).
2. Asymmetric crosstalk
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