On the possibility to detect quantum correlation regions with the variable optimal measurement angle
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On the possibility to detect quantum correlation regions withthe variable optimal measurement angle
Ekaterina V. Moreva , Marco Gramegna , and Mikhail A. Yurischev Istituto Nazionale di Ricerca Metrologica, strada delle Cacce 91, 10135 Torino, Italy Institute of Problems of Chemical Physics of the Russian Academy of Sciences, 142432 Chernogolovka, Moscow Region, RussiaRevised version: date 14/01/2019
Abstract.
Quantum correlations described by quantum discord and one-way quantum deficit can containordinary regions with constant (i.e., universal) optimal measurement angle 0 or π/ z -axis and regions with a variable (state-dependent) angle of the optimal measurement. The latter regionswhich are absent in the Bell-diagonal states are very tiny for the quantum discord and cannot be observedexperimentally due to various imperfections on the preparation and measurement steps of the experiment.On the contrary, for the one-way quantum deficit we succeeded in getting the special two-qubit X stateswhich seem to allow one to reach all regions of quantum correlation exploiting available quantum op-tical techniques. These states give possibility to deep investigation of quantum correlations and relatedoptimization problems at new region and its boundaries. In the paper, explicit theoretical calculations ap-plicable to one-way deficit are reported, together with the design of the experimental setup for generatingsuch selected family of states; moreover, there are presented numerical simulations showing that the mostinaccessible region with the intermediate optimal measurement angle may be resolved experimentally. PACS.
Quantum correlations lie at the heart of quantum infor-mation science and technology. Many kinds of quantumcorrelations have been introduced so far and now theirproperties are scrupulously analyzed both theoreticallyand experimentally. Among quantumness quantifiers be-yond quantum entanglement, relevant places in the scaleof importance are occupied respectively by quantum dis-cord and quantum work (information) deficit [1,2,3,4,5].The quantum discord Q for a bipartite system AB is defined as the minimum difference between the quan-tum generalizations of symmetric ( I ) and asymmetric ( J )forms of classical mutual information: Q = min { Π k } ( I − J ),where { Π k } is the measurement performed on one of thetwo subsystems [6,7] (see also [8] in this regard). Thequantum discord is always non-negative, equals zero forthe classically correlated states, and coincides with thequantum entanglement for the pure states. However, dis-cord and entanglement exhibit essentially different behav-ior even for the simplest mixed states — the Werner andBell-diagonal ones (see, e.g, [9,10]). Note that discord isnot a symmetric quantity and in general it depends onwhich subsystem the local measurement was performed. a corresponding author: [email protected] A value of quantum discord for two-qubit systems canvary from zero to one bit.The quantum work deficit is a measure of quantum cor-relation based on thermodynamics. It was defined firstlyby Oppenheim et al. [11] as the difference between thework W which can be extracted from a heat bath usingoperations on the entire quantum system and the largestamount of work W drawn from the same heat bath bymanipulating only the local parts of composite system;in other words, the work deficit ∆ is the amount of po-tential work which cannot be extracted under local op-erations and classical communication (LOCC) because ofquantum correlations [11,12,13]. Several forms of deficitexist, depending on the type of communication allowedbetween parts A (Alice) and B (Bob). For example, letus consider the case in which the bipartite state ρ AB isshared by Alice and Bob: if Bob performs a single vonNeumann measurement on his local subsystem and usesclassical communication to send the resulting state to Al-ice, when she extracts the maximum amount of work W from the new entire state, then the dimensionless quantity∆ = min { Π k } ( W − W ) /k B T is called the one-way quantumdeficit ( T is the temperature of the common bath and k B is Boltzmann’s constant). In spite of quite different con-ceptual sources, the one-way deficit and discord coincide inconsiderably more general cases than entanglement. They E. V. Moreva et al. : Quantum correlation regions with the variable optimal measurement angle are the same for the Bell-diagonal states and even for thetwo-qubit X states with zero Bloch vector for one qubitif the local measurement is performed on this qubit [14].On the other hand, these quantum correlations exhibit,generally speaking, different quantitative and qualitativebehavior in more general cases.Due to the optimization procedure entering in quan-tum correlation definitions, evaluation of quantum discordand deficit entails great difficulties even for the two-qubitsystems. Up to the present, closed analytical formula forthe quantum discord has been derived only for a particularclass of X states, namely for the Bell-diagonal states [15].Since the one-way deficit is identical to the discord in thiscase, one automatically possesses the closed analytical ex-pression for the former quantity.An attempt to extend the success of Luo [15] to the ar-bitrary X states was undertaken in 2010 by Ali, Rau, andAlber [16]. Unfortunately, the authors decided that theextreme values of parameters which characterize the vonNeumann measurement were attained for discord only attheir endpoints. Shortly after, however, the counterexam-ples of X density matrices have been given which demon-strate a measurement-dependent discord minimum insidethe interval of measurement parameters [17,18]. Thus, theanalytic formula of Ref. [16] is incorrect in general.At that time it was also established that for the generaltwo-qubit X states the optimization of discord over theprojectors { Π k ( θ, ϕ ) } can be worked out exactly over theazimuthal angle ϕ but one optimization procedure, in thepolar angle θ ∈ [0 , π/ Q = min { Q , Q θ ∗ , Q π/ } . (1)Here the subfunctions (branches) Q and Q π/ are theanalytical expressions (corresponding to the discord withoptimal measurement angles 0 and π/
2, respectively) andonly the third branch Q θ ∗ requires one-dimensional search-ing of the optimal state-dependent measurement angle θ ∗ ∈ (0 , π/
2) if, of course, the interior global extremumexists.Thus, the total domain of definition for the discordfunction consists of subdomains each one correspondingto the own branch (phase or fraction - in physical lan-guage) separated by strong boundaries. Equations for suchboundaries have been proposed in Refs. [24,25,26]. Theequations for 0- and π/ Q and Q π/ regions with the Q θ ∗ one are writ-ten as Q ′′ (0) = 0 , Q ′′ ( π/
2) = 0 . (2) A matrix having non-zero entries only along the diagonaland anti-diagonal is called the X one because it looks like theletter “X”.
Here Q ′′ (0) and Q ′′ ( π/
2) are the second derivatives ofthe measurement-dependent discord function Q ( θ ) withrespect to θ at the endpoints θ = 0 and π/
2, correspond-ingly. The equations (2) are based on the unimodality hy-pothesis for the function Q ( θ ) and bifurcation mechanismof appearance of the extremum inside the interval (0 , π/ { ∆ , ∆ ϑ , ∆ π/ } , (3)where the branches ∆ and ∆ π/ are again known in theanalytical form while the third branch ∆ ϑ requires to per-form numerical minimization to obtain state-dependentminimizing polar angle ϑ ∈ (0 , π/ ∆ ( θ ) can exhibit now the bimodal be-havior that in turns can lead additionally to the new mech-anism of formation of a boundary between the phases,namely via finite jumps of optimal measured angle fromthe endpoint to the interior minimum or vice versa [29,30].The analysis performed shows that the discordant re-gion Q θ ∗ is very narrow. It is characterized by the linearsizes of order 10 − , leading to the fantastically high fi-delity between the boundary states: F = 99 . Q θ ∗ -region is 0 .
08% of totalvolume of the domain of definition [27]. The latter agreesapproximately with the estimation obtained by Monte-Carlo simulations [31] equivalent to 0 . Q θ ∗ -region experimentally.On the other hand, the analogous regions ∆ ϑ of one-way quantum deficit can achieve the sizes comparable tothose of the regions ∆ and ∆ π/ , therefore inspiring thepossibility that an insight into the considered region canbe obtained experimentally.The aim of present paper is to select suitable quantumstates showing the widest possible regions with the vari-able intermediate optimal measurement angle, to performfor them numerical simulations, and to give a responseabout the possibility to resolve such regions using con-temporary optical apparatus.Our aspiration to experimentally detect the new re-gions (phases) of quantum correlations is motivated by thefollowing. First, this is a study of properties of quantumcorrelations which are absent in the Bell-diagonal states.In particular, the observation of continuous and smoothtransitions between the phases which manifest in higherderivatives with respect to the state parameters. Secondly,the fact that the state lies in the region of the variableoptimal measurement allows us to estimate the value ofquantum correlation via the shift of the angle.Third, the regions of quantum states with the inter-mediate optimal measurement angle are rather surprising The fidelity of two quantum states, F , leads to the Buresdistance, d B , between the same states through the relation d B = [2(1 − √ F )] / .. V. Moreva et al. : Quantum correlation regions with the variable optimal measurement angle 3 because the most practical constrained optimization prob-lems in the natural sciences have an optimal solution atthe boundary.(See, e.g., [32]: “Real life optimization prob-lems often involves one or more constraints and in mostcases, the optimal solutions to such problems lie on con-straint boundaries.”) However this expectation can leadto incorrect results like in [16].In the following sections, a suitable candidate of quan-tum state is given, its properties are described in detail,a scheme of optical setup is considered and the expectedresults are discussed. Finally, in the last section, a briefconclusion is given. We begin with the theoretical description of the problemunder question.
The maximum amount of useful work that can be ex-tracted from a system in the state ρ is given as [11,12,13] (see also [1,2,3]) w = k B T (log d − S ( ρ )) , (4)where S ( ρ ) = − tr( ρ log ρ ) is the entropy of state ρ and d the dimension of Hilbert space in which the density op-erator ρ acts. Applying this general relation to the statesbefore and after Bob’s measurement it is possible to obtainthe following equation for the one-way deficit∆ = min { Π k } S (˜ ρ AB ) − S ( ρ AB ) , (5)where˜ ρ AB ≡ X k p k ρ kAB = X k (I ⊗ Π k ) ρ AB (I ⊗ Π k ) + (6)is the weighted average of post-measured states ρ kAB = 1 p k (I ⊗ Π k ) ρ AB (I ⊗ Π k ) + (7)with the probabilities p k = Tr(I ⊗ Π k ) ρ AB (I ⊗ Π k ) + . (8)Thus, the one-way quantum deficit equals the minimalincrease of entropy after a von Neumann measurement onone party of the bipartite system ρ AB .It is clear from Eq. (5) that the main problem is to findthe post-measurement entropy, because the pre-measurementone, i.e. S ( ρ AB ), does not depend on the measuring angleand hence plays a role of a trivial constant shift. Therefore,below we will stress attention mainly on the S (˜ ρ AB ).Generally the one-way deficit, as the quantum discord,is asymmetric quantity under replacement of the measuredsubsystem, however we avoid such cases in our work. Notice that the map ρ AB ˜ ρ AB defined by Eq. (6) canbe interpreted as non-selective measurement (see, e.g., thetextbook [33]) because not the individual measurementoutcomes are recorded but only the statistics of outcomesis known. (Note in passing that the quantum discord isbased on selective measurements.) Moreover, this map hasa form of quantum operation and therefore the one-waydeficit has the operational significance beyond entangle-ment.It arises a question: how to determine the entropy ofsome quantum state ρ experimentally? In order to get thethermodynamic entropy one measures the heat capacityof the given sample, takes the ratio of heat capacity tothe temperature and then integrates this ratio with re-spect to the temperature. In the quantum case, in linewith the measurement propositions [34,35], direct way isto take the entropy operator − log ρ [36] or the densityone ρ as an observable. However, it is not known how toexperimentally realize the projectors | λ k ih λ k | , where | λ k i are the eigenvectors of above operators. Instead, one canfirst restore the quantum state ρ through a tomographicreconstruction in the computational basis (i.e., find thenumerical values for all entries of the density matrix),solve eigenvalue problem for this matrix on a computer,and then calculate the quantum entropy via the relation S ( ρ ) = − P i λ i log λ i with λ i being the eigenvalues of ρ [37]. It is the way that we use in the present paper.There is, of course, the radical way to obtain the one-way deficit without performing any local measurements atall (as it is usually made to get the quantum entanglement;say, [38]). Indeed, since the full tomography is needed tofind the entropy of pre-measurement state, one can usethe digital representation of ρ AB to numerically performthe required local measurement, compute the minimizedpost-measurement entropy, and finally arrive at the valueof one-way deficit ∆. We keep in mind this possibility andwill compare both approaches in the real experimentalwork. Both possibilities have their pluses and minuses. Ifthe local measurement is performed in analog way, thenthe numerical calculations are simplified, moreover we pre-fer to consider ”real” measurements with their imperfec-tions for the simulation of the experiment.To continue our consideration one should specify thequantum state. ρ AB Focussing on two-qubit systems, to this date, phase dia-grams for the quantum discord and one-way deficit havebeen studied in detail for the three-parameter subclass oftwo-qubit X states [27,30]. This allows us to choose thesuitable state to examine it in an experiment.From the available variety of states, we consider herethe maximally simple (but non-trivial) one-parameter state ρ AB = q | Φ + ih Φ + | + (1 − q ) | ih | , (9)where | Φ + i = ( | i + | i ) / √
2. This state in a Bloch formis written as ρ AB = 4 − [I ⊗ I + (1 − q )( σ z ⊗ I − I ⊗ σ z ) E. V. Moreva et al. : Quantum correlation regions with the variable optimal measurement angle + q ( σ x ⊗ σ x − σ y ⊗ σ y ) + (2 q − σ z ⊗ σ z ] , (10)where σ α ( α = x, y, z ) is the vector of the Pauli matri-ces. The given state will show the obvious symmetry un-der permutations of particles ( A ↔ B ) after performingthe local unitary (orthogonal) transformation U = I ⊗ σ x which does not change any of the quantum correlations.Lastly, the density matrix of chosen state ρ AB in explicitform is given by ρ AB = q/ q/
20 1 − q q/ q/ . (11)Let us consider now the class of states (9) showing themaximum amount of entanglement investigated in Refs. [39,40,41]. It is remarkable that the authors [39] were able toachieve the fidelity F ≥ θ ∗ = π/ q and hence the discord has here noregions with the variable optimal measurement angles.In this paper we focus on the one-way deficit. Accord-ing to Eq. (5), to find this quantity, one should first calcu-late the pre- and post-measurement entropies — S ( ρ AB )and S (˜ ρ AB ), respectively. For this purpose, we find thecorresponding eigenvalues.Eigenvalues of matrix (11) equal λ = q, λ = 1 − q, λ = λ = 0 . (12)Owing to the non-negativity requirement for any densitymatrix, one obtains that the domain of definition for theparameter (argument) q is restricted by the condition0 ≤ q ≤ . (13)The quantity q may be interpreted as a concentration ofBell-diagonal state in the two-component mixture (9).Using Eq. (12) one gets the pre-measured entropy func-tion S ( q ) ≡ S ( ρ AB ) = − q log q − (1 − q ) log (1 − q ) . (14)This is exactly the binary entropy and its value can varyfrom zero to one bit. ˜ ρ AB Since ρ AB is the two-qubit state, then Π k in Eq. (6) arethe two projectors ( k = 0 , k = V π k V + , (15)where π k = | k ih k | and transformations { V } belong to thespecial unitary group SU . Rotations V are parametrizedby two angles θ and ϕ (polar and azimuthal, respectively): V = (cid:18) cos( θ/ e iϕ sin( θ/ θ/ − e iϕ cos( θ/ (cid:19) (16) with 0 ≤ θ ≤ π and 0 ≤ ϕ < π .Performing the necessary calculations it is possible toget the eigenvalues of the density matrix ˜ ρ AB : Λ , = 14 [[1 + (1 − q ) cos θ ± { [1 − q + (1 − q ) cos θ ] + q sin θ } / ]] (17) Λ , = 14 [[1 − (1 − q ) cos θ ± { [1 − q − (1 − q ) cos θ ] + q sin θ } / ]] . It is seen that the azimuthal angle ϕ has dropped outfrom the given expressions. This is due to the fact thatone pair of anti-diagonal entries of the density matrix (11)vanishes. Using Eqs. (17) we arrive at the post-measuredentropy (entropy after measurement)˜ S ( θ ; q ) ≡ S (˜ ρ AB ) = h ( Λ , Λ , Λ , Λ ) , (18)where h ( x , x , x , x ) = − P i =1 x i log x i , with the addi-tional condition x + x + x + x = 1, is the quaternaryentropy function. Notice that function ˜ S of argument θ isinvariant under the transformation θ → π − θ therefore itis enough to consider the values for which θ ∈ [0 , π/ V from the measurement operators (projectors)on the state ρ AB :˜ ρ AB ˜ ρ ′ AB = (I ⊗ π ) · [(I ⊗ V + ) ρ AB (I ⊗ V )] · (I ⊗ π )+(I ⊗ π ) · [(I ⊗ V + ) ρ AB (I ⊗ V )] · (I ⊗ π ) (19)and in this case S (˜ ρ AB ) = S (˜ ρ ′ AB ). It means that we mayfirstly rotate the state (e.g., with a half-wave plate) andthen perform two orthogonal projections of the rotatedstate in the initial computational basis. We describe here specific properties of post-measuremententropy which will be needed for performing the experi-ment. In other words, we shall try to supply experimen-talists with technological maps suitable in the work.Equations (14), (17), and (18) define the measurement-dependent (non-optimized) one-way deficit function ∆ ( θ ) =˜ S ( θ ) − S . Direct calculations show that for every choiceof model parameter q the function ˜ S ( θ ) and hence ∆ ( θ )possess an important property, namely their first deriva-tives with respect to θ identically equal to zero at bothendpoints θ = 0 and θ = π/ S ′ (0) = ∆ ′ (0) ≡ , ˜ S ′ ( π/
2) = ∆ ′ ( π/ ≡ . (20)From Eqs. (17) and (18) we get the expressions for thepost-measurement entropy at the endpoint θ = 0,˜ S ( q ) = − (1 − q ) log(1 − q ) − q log( q/ , (21) . V. Moreva et al. : Quantum correlation regions with the variable optimal measurement angle 5 Fig. 1.
Dependencies ˜ S ( q ) (curve 1) and ˜ S π/ ( q ) (curve 2).Longer bars mark the position of interval [0 . , . Fig. 2.
Post-measurement entropy ˜ S vs θ by q = 0 .
05 (1),0.2 (2), 0.4 (3), 0.6155 (4), and 0.8 (5) and at the second endpoint θ = π/ S π/ ( q ) = log 2 + h ((1 + p (1 − q ) + q ) / , (22)where h ( x ) = − x log x − (1 − x ) log(1 − x ) is the Shannonbinary entropy function. The behavior of functions ˜ S ( q )and ˜ S π/ ( q ) is depicted in Fig. 1. The local maxima ofthese functions lie at q = 2 / /
2, and equal log ≈ . [10 − √ (3 + 2 √ ≈ . S π/ ( q ) is symmetric under thereplacement q → − q . The curves 1 and 2 intersect atthe point (0 . , . ∆ = ∆ (0) and ∆ π/ = ∆ ( π/ ∆ = q log 2 (= q bits).At the 0- and π/ ∆ ′′ (0) = 0 and ∆ ′′ ( π/
2) = 0 (23) ˜ S ′′ (0) = 0 and ˜ S ′′ ( π/
2) = 0 (24)will be needed below.As calculations yield,˜ S ′′ (0) = 1 − q + 2 q − q ln 2(1 − q ) q . (25)The roots of equation ˜ S ′′ (0) = 0 are 1/2 and 1.On the other hand, calculations show that the secondderivative ˜ S ′′ ( θ ) with respect to θ at θ = π/ S ′′ ( π/
2) = q r [ r − (1 − q ) ] ln 1 + r − r − (1 − q ) − r [1 − − q )(1 − − q r )] , (26)where r = p (1 − q ) + q . (27)The results of numerical solution of the equation ˜ S ′′ ( π/
2) =0 are q = 0 . q ∈ (0 . , . q in the segment[0 , θ by different values of parameter q . The curves˜ S ( θ ) for q ≤ . θ = 0 and π/ q = 0 . q > . S (0) − ˜ S ( π/ / ˜ S (0) equals 6% for q = 0 . π/ . < q < . S ( θ ) with respect to parameter q . Thecurve presents a monotonic increase when the parameter q varies from zero to q = 1 /
2. At the point q = 1 /
2, asvisible in Fig. 3(a), a bifurcation of the minimum at θ = 0occurs. In the range 0 . < q < . S ( θ )reaches, as shown in Figs. 3(b) and (c), the interior min-imum. So, the region with variable optimal angle ϑ takesup a part 0 . ≈ .
5% on the section [0 ,
1] of q axis,and the fidelity between the states at bound points q = 0 . q = 0 . F = 96 . . The position of sucha local minimum ϑ smoothly moves from zero to π/ Note for comparison that experimenters achieve now thevalues of fidelity F = 99 . et al. : Quantum correlation regions with the variable optimal measurement angle Fig. 3.
Post-measurement entropy ˜ S vs θ by q = 0 . The interior minimum of post-measurement entropy isbest observed when the values of ˜ S and ˜ S π/ equal to oneanother. This occurs (see Fig. 1) at the point q = 0 . S = ˜ S π/ = 1 . ∆ π/ = ∆ = q = 0 . θ min ≡ ϑ = 0 . ≈ .
8. Its depth is0.01397 bit what yields relative corrections to the post-measurement entropy and one-way deficit equaled δ ˜ S =0 .
9% and δ ∆ = 2 . q = 0 . ϑ in the full interval (from 0 to π/ S π/ with constant optimal measurement angleequaled π/
2. From here and up to q = 1, the curves ofpost-measured entropy exhibit monotonically decreasingbehavior as illustrated in Fig. 3(d).One should emphasize here that the behavior of theminimized one-way quantum deficit ∆ = min θ ∆ ( θ ) withrespect to the argument q is continuous and smooth. Nev-ertheless, the function ∆( q ) is a piecewise one,∆( q ) = ( ∆ , ≤ q ≤ . ∆ π/ , . ≤ q ≤ ∆ ϑ , q ∈ (0 . , . Fig. 4.
High-resolved post-measurement entropy ˜ S vs θ by q = 0 . Fig. 5.
Concentration of Bell-diagonal state, q ∈ [0 . , . ϑ . and therefore presents nonanalyticities at the border points q = 0 . ϑ with q ∈ [0 . , . q ( ϑ ) is biunique (one-to-one) and the presented curve al-lows to estimate the value of parameter q in the mixedquantum state ρ AB . Hence, this can serve as one of pos-sible applications of quantum correlation in practice. For testing the theoretical approach, described in the pre-vious sections, we propose a design of an experimentalsetup allowing to prepare a family of polarization states ρ AB (9) and estimate the post-measured entropy ˜ S throughthe tomographic reconstruction of density matrix. In par-ticular, we suggest to use the well-known method of po-larization state preparation via spontaneous parametricdown-conversion process (SPDC) [46] and quantum statetomography protocol based on tetrahedral symmetry [47],guaranteeing simple realization and high quality of re-construction. We intentionally selected commonly used . V. Moreva et al. : Quantum correlation regions with the variable optimal measurement angle 7 Fig. 6. (Color online) Experimental setup for preparing the initial quantum state and measuring its one-way deficit. The first(green) block prepares the specific quantum states ρ AB defined in (9), the second (blue) block performs local measurements whilethe third (yellow) block of tomography measures the transformed quantum state ˜ ρ AB . Ar laser: argon laser with wavelength351 nm, M: mirror, V: vertical oriented Glan-Thompson prism, BBO: nonlinear barium borate crystals of Type-I, HWP andQWP: half- and quarter-wave plates, QP: (dichroic) quartz plates, BS: beamsplitter, PBS: polarizing beamsplitter, DBS: dichroicbeamsplitter (dichroic mirror), IF: interference filter, D: single photon avalanche detectors (SPAD), CC: coincidence circuit methods to show that the nontrivial behavior of quantumdeficit can be observed without requiring specific appara-tus. The setup is schematically depicted in Fig. 6. A cw ar-gon laser beam at λ = 351 nm passes through a Glan-Thompson prism (V) with vertical orientation, half-waveplate HWP p and polarizing beamsplitter (PBS). The HWP p and PBS serve to control the q parameter in the state ρ AB (9). In the upper arm of non-balanced interferometerthe maximum entangled Bell state | Φ + i is produced. Twononlinear type-I BBO crystals, positioned with the planescontaining optical axes orthogonal to each other, gener-ate a pair of the basic states | H H i , | V V i via collinear,frequency nondegenerated regime of SPDC. The relativephase φ between basic states is controlled by two quartzplates QP , while amplitudes are controlled by the half-wave plate HWP p .Among th others, and as a practical example, let ussuppose that the wavelengths of the collinear downcon-verted photons are λ = 763 nm and λ = 650 nm. Inthe bottom arm, where the horizontal polarization of thepump is reflected, the second component of the state ρ AB is prepared. We use the technique, suggested in the papers[45,46], to perform the transformation | V V i ⇒ | H V i .This transformation can be achieved by using dichroicwave plates QP , which act separately on the photonswith different frequencies and introduce a phase shift of2 π for a vertically polarized photon at 650 nm, a phaseshift of π for the conjugated photon at 763 nm. The waveplates are oriented at 45 ◦ to the vertical direction. Using quartz plates as retardation material it is easy to calculatethat the thickness of the wave plate operating the trans-formation should be equal to 1.585 mm or 3.464 mm. Thetheoretical estimated fidelity is more than 0 . β , then the optical thickness of the effective wave plateformed by QP will change, and, at a certain value of β ,the desired transformation will be achieved. Ultimately,non-polarizing beamsplitter (BS) mixes the states fromthe upper and bottom arms of non-balanced interferome-ter and, as a result, prepares the initial state ρ AB .The local projective measurements at a variable an-gle are realized in the (blue) block in Fig. 6. Accordingto Eq. (19) they are implemented only over one of thephoton of the pair, so a dichroic beamsplitter (DBS) sep-arates photons with different frequencies into the two spa-tial modes. In the upper spatial mode a half-wave plate(HWP λ ) is oriented at the angle ˜ θ [˜ θ = θ/ θ is the polar angle in the transformation (16)] anda couple of polarizing beam splitters (PBS), forming anon-balanced interferometer, perform two orthogonal pro-jections at different angles. To obtain the statistical mix-ture (19) the length difference between the arms of non-balanced interferometer must be larger then the coherence E. V. Moreva et al. : Quantum correlation regions with the variable optimal measurement angle length of the photons with orthogonal polarization, whichfor the selected wavelength ( λ ) and a full width at halfmaximum (FWHM) δλ = 3 nm of the interference filterIF equals λ /δλ ≈ µ m.After the non-selective projective measurement the statewas sent to the reconstruction block of the setup (the third(yellow) block in Fig. 6). The post-measured quantity ofthe entropy (18) at varying angle ˜ θ was numerically calcu-lated through the density matrices of post-measurementstate ˜ ρ AB using quantum tomography protocols [46]. Theprojective measurements in each arm to perform the quan-tum state tomography can be realized by means of a polar-ization filtering system consisting of a sequence of quarter-and half-wave plates, followed by a polarization prism,which transmits vertical polarization ( V ). The detectionis operated by taking advantage of silicon single-photonavalanche detectors (SPAD). In the case of independentmeasurement of two qubits the projective measurementscan be chosen arbitrarily. We propose quantum state to-mography protocol, where projections on the states pos-sess tetrahedral symmetry [47]. There are several worksshowing that due to the high symmetry such protocol pro-vides a better quality of reconstruction [47,48,49]. As al-ternative, adaptive quantum tomography protocols can beused: in fact, despite these require more complex analysis,they guarantee at the same time the highest quality ofstate reconstruction [50,51]. According to the theory, the region with the variable op-timal measurement angle is narrow and therefore requiresprecise quantum state preparation and reconstruction. Itis worth to stress that the standard procedure of the statereconstruction from the likelihood equation associates withfinite statistics of the registered outcomes (sample size) ofan experiment and therefore takes random values [46,52].For the state ρ AB we can calculate the statistical distri-bution of fidelity F for the given sample size.Usually the quantum tomography protocol can be de-fined by a so-called instrumental matrix X that has m rows and s columns [53,54,55], where s is the Hilbert spacedimension and m the number of projections in such space.For every row, i.e. for every projection, there is a corre-sponding amplitude M j . This matrix equals: M j = X jl c l ( j = 1 , , ..., m ; l = 1 , , ..., s ) , (29)where c l are the expansion coefficients.The square of the absolute value of the amplitude de-fines the intensity of a process, which is the number ofevents in one second λ j = | M j | . (30)The number of registered events k j is a random variableexhibiting Poisson distribution, t j is the time of expositionof the selected row of the protocol and λ j t j the averagevalue, P ( k j ) = ( λ j t j ) k j k j ! exp( − λ j t j ) . (31) Fig. 7.
The universal statistical fidelity loss histogram dis-tribution for 500 numerical experiments. Sample size of eachexperiment is 10 The normalization condition for the protocol defines thetotal expected number of events n summarized by all rows: m X j =1 λ j t j = n. (32)Equation (32) substitutes the traditional normalizationcondition for the density matrices, tr( ρ ) = 1.The fidelity achieved now by experimenters has thevalues F ≥ .
8% for the given optical states (9) [38,43,44], that is why in the numerical experiments test-ing the universal statistical distribution for fidelity losseswe used statistics that guarantees the same fidelity: wenumerically generated sets of experimental data and re-constructed 500 states ρ AB ( q = 0 . n = 10 for each state, obtained by per-forming the maximum-likelihood estimation on randomvariables of the measured counts according to Poissonianstatistics. The estimated average fidelity with absoluteerror is F = 0 . ± .
02. The fluctuations of F canbe formally taken into account by introducing the so-called loss of fidelity 1 − F , which is associated only withstatistical errors and does not take into account experi-mental imperfections.We can also introduce the variable z = − log (1 − F ), which is the number of digit 9 in thedecimal representation of the parameter F (e.g., z = 3corresponds to F = 0 . ρ AB . The form of the distribution besidessample size depends on the initial state and the used pro-tocol of quantum state reconstruction. In our simulationthe tomography protocol based on the states possessingtetrahedral symmetry had been used [47].For the same sample size we estimated numerically thepost-measured entropy for the best observed interior min-imum at the point q = 0 . θ , which . V. Moreva et al. : Quantum correlation regions with the variable optimal measurement angle 9 Fig. 8.
Post-measurement entropy ˜ S vs ˜ θ by q = 0 . S , barsare standard deviations. Sample size of each experiment is 10 .It is clearly seen a minimum of ˜ S (˜ θ ) near the angle ˜ θ = 10 ◦ ,that is in good agreement with the theoretical prediction shownin Fig. 4 varies from 0 ◦ to 22 ◦ (with step 1 ◦ ), a set of 100 den-sity matrices of ˜ ρ AB was generated, then eigenvalues werecalculated and finally the mean values together with stan-dard deviations of the post-measured entropy ˜ S were es-timated. The dependence ˜ S on ˜ θ is plotted on Fig. 8. Thegraph shows that with a good experimental accuracy, itis possible to observe the region with the variable opti-mal measurement angle taking advantage of the proposedsetup, even if the corrections to the post-measurement en-tropy values are small (in the order of 0.9% or less). In this work we have found the specific quantum state (9)with the record wide region of variable optimal measure-ment angle, presented for it the detailed calculations ofentropy after measurement and hence of one-way deficit,and performed the numerical simulation of the proposedexperiment.The described experimental setup opens a possibil-ity to construct universal facilities allowing to measurethe one-way deficit of any symmetric (up to local uni-tary transformation) two-photon quantum states. More-over, we have considered in detail the preparation of two-component mixture of quantum states which, according tothe theory, contains a fraction with the variable optimalmeasurement angle.The performed numerical simulations show that theavailable experimental techniques allows one to investigatea fine structure of quantum correlation domain includingthe region with the variable optimal measurement angleand opens a way to implement the real physical experi-ment to study the one-way deficit behavior.
Acknowledgments
One of us (M. Yu.) is grateful to participants of the Semi-nar of the Quantum Technologies Center of the LomonosovMoscow State University for fruitful discussion.
M.Yu. contributed with theoretical calculations of one-way quantum deficit and wrote the theoretical part of themanuscript, E.M. contributed to the experimental setupdesign, numerical simulation and wrote the experimentalpart of the manuscript, M.G. helped with experimentalsetup design and critical revision of the manuscript.
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