Finite-temperature Bell test for quasiparticle entanglement in the Fermi sea
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Finite-temperature Bell test for quasiparticle entanglement in the Fermi sea
W.-R. Hannes
Department of Physics, University of Konstanz, D–78457 Konstanz, Germany
M. Titov
School of Engineering & Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK (Dated: February 2008)We demonstrate that the Bell test cannot be realized at finite temperatures in the vast majority of electronicsetups proposed previously for quantum entanglement generation. This fundamental difficulty is shown tooriginate in a finite probability of quasiparticle emission from Fermi-sea detectors. In order to overcome thefeedback problem we suggest a detection strategy, which takes advantage of a resonant coupling to the quasi-particle drains. Unlike other proposals the designed Bell test provides a possibility to determine the criticaltemperature for entanglement production in the solid state.
PACS numbers: 03.67.Mn, 05.30.Fk, 05.60.Gg, 73.23.-b
It is well-known that, unlike photons, quasiparticles in theFermi sea injected from reservoirs, which are kept at thermalequilibrium, can be entangled by just a tunnel barrier. This al-lows for particularly simple proposals for quantum quasipar-ticle entanglement, which do not involve interactions.
Theoretical results for the entanglement production in differ-ent electronic setups have been summarized in Refs. 6 and 7,while yet no experimental evidence of the quasiparticle entan-glement in the Fermi sea has become available.The quantum entanglement of two particles with respect toa spin-like degree of freedom can be accessed experimentallyby measuring the spin correlator C ( a , b ) = h ( a · σ ) ⊗ ( b · σ ) i , (1)where σ = ( σ x , σ y , σ z ) is the vector of Pauli matrices. Thespin projection of the particles in the detectors and is mea-sured with respect to the unit vectors a and b , correspond-ingly. If the correlation between the particles is of a classicalorigin the following Bell inequality holds, B = |C ( a , b ) + C ( a ′ , b ) + C ( a , b ′ ) − C ( a ′ , b ′ ) | ≤ , (2)for arbitrary choice of the unit vectors a , b , a ′ , b ′ . The vi-olation of the inequality (2) is, therefore, sufficient but notnecessary condition for quantum entanglement.In solid-state electronics we deal with elementary excita-tions in the Fermi gas, which are referred to as quasiparti-cles. Even though the pairwise quasiparticle entanglement is believed to be generated in many devices, its ex-perimental observation is obscured by the nature of electronicdetectors. Those, unlike the photodetectors in optical setups,contain a number of quasiparticles in the ground state, whichfill up available quantum levels below the Fermi energy. Ifa part of the device is at finite temperature the electron andhole excitations are spontaneously created near the Fermi sur-face resulting in a finite probability for a Fermi-sea detectorto emit. Such processes are harmful for any sensible Bell test.The problem of quasiparticle entanglement detection hasbeen put forward in Refs. 15 and 16, where the possibility toconstruct a Bell-type inequality with current cross-correlatorsis discussed. It has been suggested to take advantage of the generalized spin-correlator C M ( a , b ) = h ( N ↑ − N ↓ ) ( N ↑ − N ↓ ) ih ( N ↑ + N ↓ ) ( N ↑ + N ↓ ) i , (3)where N nσ is a number of particles with a spin projection σ registered by the detector n . (In solid state the role of spincan be played by other quantum degrees of freedom such asorbital momentum or isospin). Similarly to Eq. (1) the spinprojection in Eq. (3) is measured with respect to the direction a in the first detector and b in the second one. Both definitions(1) and (3) are equivalent in the original Bell setup, if no morethan two particles are received within the detection time andthe detectors do not emit particles. In electronic circuits thenumber of quasiparticles N nσ is given by the time integral of acurrent I nσ flowing to the corresponding Fermi-sea reservoir N nσ ∝ Z t det dt I nσ ( t ) , (4)which is not restricted. For large detection times t det one typ-ically observes | N n ↑ − N n ↓ | ≪ | N n ↑ + N n ↓ | , hence the Bellinequality (2) cannot be violated and the corresponding mea-surement is useless for entanglement detection. The difficultyhas been discussed in Ref. 16 for zero temperature.An essential problem occurs in the opposite limit t det → ,because N nσ defined by Eq. (4) can take on negative values.This leads to fluctuations with | N n ↑ − N n ↓ | > | N n ↑ + N n ↓ | ,which are explicitly forbidden in the Bell test. This situation isrealized at finite temperatures. Then, the violation of Eq. (2)has no relation to the entanglement detection and the corre-sponding measurement is not of a Bell type.Thus, the violation of the inequality (2) with the corre-lator C substituted by C M does not provide a conclusiveevidence for quantum entanglement generation at any fi-nite temperature. This difficulty clearly applies to the de-tection of electron-hole entanglement produced by tun-neling events or by time-dependent gating. But even inmore sophisticated setups where zero-temperature detectorsand finite-temperature sources are represented by differentmetallic leads (in close resemblance to the original Bell pro-posal) the Bell test based on Eq. (3) is flawed. Exam-ples include three-terminal fork geometries and four-terminalbeam-splitter geometries with grounded detectors. We focuson the latter (see Fig. 1) due to a number of previously pro-posed realizations, which are mostly based on thedirected transport along quantum-Hall edge channels. Minormodifications, such as lowering chemical potential in one ofthe detectors or increasing detection time, can suppress theprobability of detector emission but lead, instead, to uselessmeasurement with | N n ↑ − N n ↓ | ≪ | N n ↑ + N n ↓ | . The genericsituation is illustrated in Fig. 2 for the case of electronic beamsplitter.For quantum particles the spin correlator from Eq. (3) isexpressed through the expectation value h N σ N σ ′ i ∝ K σσ ′ , (5) K σσ ′ = t (cid:20) h I σ ih I σ ′ i + Z dω π P σσ ′ ( ω ) F ( ωt det / (cid:21) . where N and I are regarded as operators. We introduce thefunction F ( x ) = (sin x ) /x and the frequency-dependentcross-correlator P σσ ′ ( ω ) = Z dt e iωt h δI σ ( t ) I σ ′ (0) i , (6)with δI nσ ( t ) = I nσ ( t ) − h I nσ i . In Figs. 2,3 the correlator C M defined by experimentally measurable quantities (5,6) iscompared with the exact result of the density matrix analysisof the final state. We restrict ourselves to an important class of systems whichdo not involve spin-dependent scattering, because the chancesto generate quantum entanglement with respect to the spin de-gree of freedom are obviously maximized in such setups. Thevalues of P σσ ′ in Eq. (6) are related to the cross-correlator P of the corresponding spin-independent problem as P ↑↑ = P ↓↓ = (1 + ab ) P , (7) P ↑↓ = P ↓↑ = (1 − ab ) P . (8)This symmetry holds even for interacting electronic systemsprovided the absence of spin dephasing. It follows fromEqs. (7,8) that a neglection of the mean currents in Eq. (5)is equivalent to C M ( a , b ) = ab , hence the inequality (2) isviolated with B max = 2 √ irrespective of voltages, tempera-ture or other setup characteristics. Clearly, such violation hasnothing to do with pairwise quantum entanglement. We willsee that the problem persists even if the exact expression for K σσ ′ is used.In the absence of spin-dependent scattering the mean cur-rents measured by the detectors do not depend on the direc-tions a and b , h I nσ i = h I n i . From Eqs. (5,6,7,8) we obtain C M ( a , b ) = γ γ ab , (9)with a parameter γ given by the ratio γ = R dω P ( ω ) F ( ωt det / π h I ih I i , (10)where both the cross-correlator P ( ω ) and the product of themean currents have to be calculated for the correspondingspin-independent problem. D 1 S 2D 2 −/+ +/− D S 1
V V
FIG. 1: A generic beam splitter for entanglement production in thesolid state. The voltage bias applied between the sources S and S generates an entangled outgoing state at the detectors D , D provided the temperature in the sources T is smaller than a criticaltemperature T c . An example of such a calculation can be performed withinthe Landauer-B¨uttiker scattering approach, which is validas far as inelastic processes in between the reservoirs can bedisregarded. Within the scattering approach the mean currentto the reservoir α is given by h I α i = eh Z dE X β (cid:0) δ αβ − | S αβ ( E ) | (cid:1) f β ( E ) , (11)where f α ( E ) = (1 + exp [( E − eV α ) /k B T α ]) − is the Fermidistribution function, which depends on the temperature of thecorresponding reservoir T α and the voltage bias V α applied.The frequency-dependent correlator (6) of the currents flow-ing to the reservoirs α and α ′ reads, P αα ′ ( ω ) = e h Z dE X ββ ′ M αα ′ ,ββ ′ ( E, ~ ω ) F ββ ′ ( E, ~ ω ) ,F ββ ′ ( E, Ω) = f β ( E ) ˜ f β ′ ( E + Ω) + ˜ f β ( E ) f β ′ ( E + Ω) ,M αα ′ ; ββ ′ ( E, Ω) = (cid:0) δ αβ δ αβ ′ − S ∗ αβ ( E ) S αβ ′ ( E + Ω) (cid:1) × (cid:0) δ α ′ β δ α ′ β ′ − S ∗ α ′ β ′ ( E + Ω) S α ′ β ( E ) (cid:1) , (12) ˜ f ( E ) ≡ − f ( E ) . (13)Let us consider a generic beam splitter with no spin-dependent scattering depicted schematically in Fig. 1. Sucha setup is characterized by an energy-independent S -matrix S = (cid:18) s ′ s (cid:19) , (14)where × unitary matrices s and s ′ describe the transportfrom sources to detectors and from detectors to sources, cor-respondingly. We parameterize s = (cid:18) e iφ e iφ ′ (cid:19) (cid:18) √ − τ i √ τi √ τ √ − τ (cid:19) (cid:18) e iθ e iθ ′ (cid:19) , (15)where τ ∈ [0 , is the beam-splitter transparency and thespin index is omitted. Following the majority of propos-als both detectors and the second source are grounded, i.e. √ τ = 0 . τ = 0 . k B T /eV − C ( M ) ( a , b ) / a b FIG. 2: The spin cross-correlator C obtained from the density matrixof the final scattering state (solid lines; cf. Eq. 20), and its general-ization C M , evaluated numerically from Eqs. (9-15) for different val-ues of the detection time eV t det /h = 0 . • ) , . N ) , (cid:4) ) , (cid:7) ) (dashed lines; see Eqs. 21,22). V D ≡ V D = V D = 0 , V S = 0 , while V S = V is thevoltage applied between the sources.At zero temperature the beam splitter acts as a source ofspin-entangled Bell pairs | Ψ B i = 1 √ | ↑ ↓ − ↓ ↑ i (16)where the index n = 1 , refers to the detector number.Such an entanglement generation is due to the Pauli princi-ple, which guarantees that a filled state with E ∈ (0 , eV ) inthe first source contains exactly two quasiparticles with theopposite spins.The Bell pairs can be accessed at zero temperature by per-forming a time coincidence detection. For finite temperature T in the sources the density matrix projection, which corre-sponds to a single particle in each detector, is derived in theAppendix A ρ out11 = (1 − ξ ) + ξ | Ψ B ih Ψ B | , (17)where is the unit matrix in the two-particle Hilbert spaceand ξ is an energy-independent weight factor ξ = τ (1 − τ )( f S − f S ) τ (1 − τ )( f S − f S ) + 2 f S ˜ f S f S ˜ f S , (18)where f Sn is the Fermi distribution function in n -th source.The result (17) describes the mixed Werner state, whichis entangled as far as ξ > / according to the Woottersformula. In the present case this condition is equivalent to
T < T c with the critical temperature T c determined by theequation τ (1 − τ ) sinh ( eV / k B T c ) = 1 / . (19)From Eqs. (1,17) one obtains the exact spin correlator C ( a , b ) = − ξ ab , (20) which is plotted in Fig. 2 with the solid line for different val-ues of the transparency parameter. The corresponding Bellinequality (2) can be violated for ξ > / √ , which is, in-deed, a sufficient condition for the entanglement. Whether ornot such a Bell test can be performed by measuring currentcross-correlator (3) is, however, an open question.In order to answer this question we substitute the expression(14) for the S -matrix to Eqs. (11,12), where the summationruns over the index α = { S , S , D , D } . The correlator C M is, then, obtained from Eqs. (9,10) with I ≡ I D , I ≡ I D , and P ≡ P D ,D .For t det ≫ min { h/eV, h/k B T } we obtain γ = − heV t det (cid:18) coth (cid:18) eV k B T (cid:19) − k B TeV (cid:19) ≪ , (21)i.e. the corresponding measurement is useless for an entangle-ment detection. Indeed, such a long-time measurement is notprojective, therefore it does not single out the state with onequasiparticle in each detector. In the opposite limit we, however, find γ = − , t det ≪ min { h/eV, h/k B T } , (22)hence the inequality (2) is violated for any temperature of thesource. Thus, according to the density matrix analysis (17,20),the corresponding measurement is not of a Bell type. Bothresults (21) and (22) formally hold for any temperature of thedetectors.The transition from non Bell-type measurement to the use-less measurement with the increase of t det is illustrated inFig. 2. The Bell parameter defined with the correlator C M does not depend on the beam-splitter transparency τ and caneasily exceed even in the absence of any entanglement.The result of Eq. (22) is equivalent to h I D ( t ) I D ( t ) i = 0 . (23)At T = 0 the currents I Dn ( t ) are sign-definite, hence Eq. (23)is exact for every single time-coincidence measurement inagreement with the prediction of the density matrix approach.For rising temperatures T > the correlation (23) holds onlyon average and is not sensitive to vanishing quantum entangle-ment in the final state of the beam splitter (17). Consequently,the inequality (2) with C substituted by C M can be violated forarbitrarily high temperatures. The absence of critical temper-ature indicates once again that such a violation has nothingto do with the entanglement detection. Instead, the decay of C M ( a , b ) with the temperature in Fig. 2 (dashed lines) is de-termined by the detection time t det .Thus, the measurement of C M ( a , b ) cannot be used for theentanglement test in the beam-splitter setup and the value of T c cannot be inferred from such a measurement as the matterof principle.We propose a way to rescue the Bell measurement by cou-pling detectors via the energy filters, which are described byenergy dependent scattering amplitudes: r n , r ′ n , t n , t ′ n , where n = 1 , is the number of the detector. The use of energy-filters in the context of Bell measurement at zero temperature √ E = 0 . eVτ = 0 . k B T /eV − C ( M ) ( a , b ) / a b FIG. 3: The case of resonant detector coupling. The short-dashedline shows C M from Eq. (26), while the long-dashed lines are nu-merical results for Breit-Wigner resonances (24) with finite width Γ = 0 . eV , detector voltage V D = − V , and different detectiontimes Γ t det /h = 0 . • ) , . N ) . The measurement is useless for t det & . h/ Γ . The solid line shows the correlator C from Eq. (20). has been discussed in Ref. 24. Let us illustrate our results forthe case of identical filters with the Breit-Wigner form of thetransmission amplitude t n ( E ) = e iδ n (Γ / E − E − i Γ / − . (24)The S -matrix of the full setup including the filters is given by S ( E ) = (cid:18) s ′ r ( E ) s s ′ t ′ ( E ) t ( E ) s r ′ ( E ) (cid:19) (25)where t = diag( t , t ) , r = diag( r , r ) , etc. The condi-tion for time-coincidence detection now reads t det ≪ h/ Γ .The currents I Dn ( t ) can be made sign-definite by applying anadditional voltage bias V D , as shown in Fig. 1. The currentfluctuations due to temperature are not harmful for the Belltest as far as | eV D | ≫ k B T , which is the only restriction onthe value of V D . In this case there is no requirement for anadditional cooling of the detectors, so that a whole setup canbe kept in temperature equilibrium. Moreover, for Γ ≪ eV the dependence on t det vanishes, meaning that C M ( a , b ) canbe obtained experimentally from zero-frequency noise mea-surements. The feasibility of such a Bell test is illustrated inFig. 3 for realistic values of the parameters.For Γ → we obtain from Eqs. (9,10,12,24,25) C M = − τ (1 − τ )( f S − f S ) ab τ (1 − τ )( f S − f S ) + 2 f S f S (cid:12)(cid:12)(cid:12)(cid:12) E = E . (26)The result is plotted with the short-dashed line in Fig. 3. It isevident from the comparison with Eqs. (18,20) that the pro-posed measurement is always of the Bell type. The corre-lator (26) tends to the exact one (20) for E ≫ eV . Thesetup efficiency is, however, exponentially low in this limit.The numerical results in the case of finite resonance width Γ = 0 . eV are plotted in Fig. 3 with the dashed lines. Thetest provides the lower estimate for the critical temperature. In conclusion we point out the fundamental restrictions forthe Bell test in electronic setups due to the quasiparticle emis-sion from Fermi-sea detectors. We propose a way to rescuethe Bell measurement by a resonant coupling to the detectors.We show that the lower estimate of the critical temperaturefor entanglement production can be experimentally obtainedin the proposed setup.This research was supported by the DFG Priority Pro-gramm 1285. The discussions with C. W. J. Beenakker,W. Belzig, and Yu. V. Nazarov are gratefully acknowledged. APPENDIX A: DENSITY MATRIX PROJECTION
Following Ref. 7 we review the derivation of the densitymatrix projection (17) for final scattering state in the case ofthe setup depicted in Fig. 1. The density matrix of the incom-ing state is given by ρ in = Y n,E,σ (cid:16) ˜ f Sn ( E ) | ih | + f Sn ( E ) a † nσE | ih | a nσE (cid:17) , (A1)where f Sn is the Fermi distribution function in the source S n , ˜ f Sn ≡ − f Sn , and a nσE is the fermion annihilation operatorfor an incoming scattering state at the channel n and energy E .The annihilation operators for the outgoing scattering states, b nσE , are obtained from the relation, b nσE = X m s nm ( E ) a mσE , (A2)where s nm are the components of a unitary scattering matrix.Thus, the density matrix of the final state is ρ out = Y n,E,σ n ˜ f Sn ( E ) | ih | + f Sn ( E ) c † nσE | ih | c nσE o , (A3)where c † nσE = X m b † mσE s mn ( E ) . (A4)In order to quantify the two-particle entanglement for thepartition H D ⊗ H D of Hilbert space with respect to thedetectors, the state ρ out has to be projected onto the sec-tors N E N ,E N of the Fock space with the energies E , E and particle numbers N , N in the corresponding detectors D , D . The density matrix ρ out N ,N of the projection factor-izes into a product state in all sectors except for the sector N E ,E with E = E = E and N = N = 1 . Projectiononto this sector is found from Eq. (A3) as w ρ = f S ˜ f S f S ˜ f S + 2 τ (1 − τ )( f S − f S ) | Ψ B ih Ψ B | , (A5)where is the unit matrix in the two-particle Hilbert space, | Ψ B i is the Bell state (16), and the weight factor w is deter-mined from the condition Tr ρ = 1 as w = 4 f S ˜ f S f S ˜ f S + 2 τ (1 − τ )( f S − f S ) . (A6)From Eqs. (A5,A6) we obtain Eqs. (17,18). By substituting f Sn = (1 + exp[( E − eV n ) /k B T ]) with V = V , V = 0 inEq. (18) we can further simplify the parameter ξ as ξ ( T ) = 1 − (cid:20) τ (1 − τ ) sinh eV k B T (cid:21) − . (A7)The critical temperature T c is determined from the equation ξ ( T c ) = 1 / , which is equivalent to Eq. (19). APPENDIX B: EVALUATION OF THE CORRELATOR C M
1. Plain beam splitter
We evaluate the generalized spin-correlator C M ( a , b ) givenby Eqs. (9,10) in the framework of the scattering approach.By substituting the scattering matrix (14, 15), into Eq. (11)we calculate the mean currents, which are measured in thedetectors D , D , as h I D i = − eh (1 − τ ) eV, h I D i = − eh τ eV. (B1)The cross-correlator (12) is found as P D ,D ( ω ) = e h τ (1 − τ ) (cid:20) ~ ω coth (cid:18) ~ ω k B T (cid:19) (B2) − X ζ = ± ( eV + ζ ~ ω ) coth (cid:18) eV + ζ ~ ω k B T (cid:19)(cid:21) . The parameter γ given by Eq. (10) can be calculated analyti-cally in two opposite limits:(i) For large detection times, t det ≫ min { h/eV, h/k B T } ,one can replace t det F ( ωt det / with πδ ( ω ) , hence γ = P D ,D (0) t det h I D ih I D i . (B3)This leads to the result (21).(ii) For short detection times t det ≪ min { h/eV, h/k B T } one can approximate F ( ωt det / ≈ in the relevant fre-quency range | ~ ω | . eV . In this limit the integral in Eq. (10)does not depend on temperature Z dω P D ,D ( ω ) = − π (cid:18) e Vh (cid:19) τ (1 − τ ) , (B4) which leads to the simple result (22).
2. Beam splitter with energy filters
We repeat the calculation in a more general case of anenergy-dependent scattering matrix (25). From Eq. (11) weobtain the mean currents h I D i = eh Z dE | t ( E ) | × [ f D ( E ) − (1 − τ ) f S ( E ) − τ f S ( E )] , (B5) h I D i = eh Z dE | t ( E ) | × [ f D ( E ) − τ f S ( E ) − (1 − τ ) f S ( E )] , (B6)where f D ( E ) = (1 + exp [( E − eV D ) /k B T ]) is the Fermidistribution function in the detectors. From Eq. (12) we cal-culate the cross-correlator P D ,D ( ω ) = e h τ (1 − τ ) Z dE × t ∗ ( E ) t ( E + ~ ω ) t ∗ ( E + ~ ω ) t ( E ) × (cid:2) F S ,S ( E, ~ ω ) + F S ,S ( E, ~ ω ) − F S ,S ( E, ~ ω ) − F S ,S ( E, ~ ω ) (cid:3) , (B7)where the function F αβ is defined in Eq. (12). These expres-sions allow for the numerical evaluation of C M ( a , b ) for arbi-trary energy-dependent scattering matrix (25).In the case of sharp resonances, such as those of the Breit-Wigner form (24) with Γ → , we have lim Γ → Γ − | t n ( E ) | = π δ ( E − E ) , (B8)and obtain from Eqs. (B5,B6,B7,B8) γ = − τ (1 − τ ) ( f S − f S ) [ f D − (1 − τ ) f S − τ f S ][ f D − τ f S − (1 − τ ) f S ] , (B9)where the Fermi functions are evaluated at the position of theresonance E = E > . For | eV D | ≫ k B T the value f D ( E ) is exponentially small, hence Eq. (26) is justified. In generalthe setup is functional provided the detector voltage V D is suf-ficiently large to ensure that f D is much smaller than both f S and f S within the energy window of the filters. C. W. J. Beenakker, C. Emary, M. Kindermann, andJ. L. van Velsen, Phys. Rev. Lett. , 147901 (2003). C. W. J. Beenakker and M. Kindermann, Phys. Rev. Lett. ,056801 (2004). P. Samuelsson, E. V. Sukhorukov, and M. B¨uttiker,Phys. Rev. Lett. , 026805 (2004). A. V. Lebedev, G. B. Lesovik, and G. Blatter, Phys. Rev. B ,045306 (2005). A. Di Lorenzo and Y. V. Nazarov, Phys. Rev. Lett. , 210601 (2005). P. Samuelsson, E. V. Sukhorukov, and M. B¨uttiker, Phys. Rev. B , 115330 (2004). C. W. J. Beenakker, Proc. Int. School Phys. E. Fermi, Vol. (IOS Press, Amsterdam, 2006). J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt,Phys. Rev. Lett. , 880 (1969). G. Burkard, D. Loss, and E. V. Sukhorukov, Phys. Rev. B ,R16303 (2000). G. Burkard and D. Loss, Phys. Rev. Lett. , 087903 (2003). P. Samuelsson, E. V. Sukhorukov, and M. B¨uttiker,Phys. Rev. Lett. , 157002 (2003). V. Scarani, N. Gisin, and S. Popescu, Phys. Rev. Lett. , 167901(2004). C. W. J. Beenakker, M. Titov, and B. Trauzettel, Phys. Rev. Lett. , 186804 (2005). P. Samuelsson and M. B¨uttiker, Phys. Rev. B , 245317 (2005). S. Kawabata, J. Phys. Soc. Jpn. , 1210 (2001). N. M. Chtchelkatchev, G. Blatter, G. B. Lesovik, and T. Martin,Phys. Rev. B , 161320(R) (2002). V. Giovannetti, D. Frustaglia, F. Taddei, and R. Fazio, Phys. Rev. B , 115315 (2006); , 241305(R) (2007). L. Faoro and F. Taddei, Phys. Rev. B , 165327 (2007). Ya. M. Blanter and M. B¨uttiker, Phys. Rep. , 1 (2000). R. F. Werner, Phys. Rev. A , 4277 (1989). W. K. Wootters, Phys. Rev. Lett. , 2245 (1998). K. V. Bayandin, G. B. Lesovik, and T. Martin, Phys. Rev. B ,085326 (2006). B. V. Fine, F. Mintert, and A. Buchleitner, Phys. Rev. B ,153105 (2005). G. B. Lesovik, T. Martin, and G. Blatter, Eur. Phys. J. B24