Linear optics and quantum maps

Linear optics and quantum maps

Aiello, A.; Puentes, G.; Woerdman, J.P.

Citation

Aiello, A., Puentes, G., & Woerdman, J. P. (2007). Linear optics and quantum maps. Physical

Review A, 76, 032323. doi:10.1103/PhysRevA.76.032323

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Linear optics and quantum maps

A. Aiello, G. Puentes, and J. P. Woerdman

Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands 共Received 17 November 2006; published 21 September 2007兲

We present a theoretical analysis of the connection between classical polarization optics and quantum mechanics of two-level systems. First, we review the matrix formalism of classical polarization optics from a quantum information perspective. In this manner the passage from the Stokes-Jones-Mueller description of classical optical processes to the representation of one- and two-qubit quantum operations, becomes straight- forward. Second, as a practical application of our classical-vs-quantum formalism, we show how two-qubit maximally entangled mixed states can be generated by using polarization and spatial modes of photons generated via spontaneous parametric down conversion.

DOI:10.1103/PhysRevA.76.032323 PACS number共s兲: 03.67.Mn, 03.65.Ud, 42.25.Ja

I. INTRODUCTION

Quantum computation and quantum information have been among the most popular branches of physics in the last decade 关1兴. One of the reasons for this interest is that the smallest unit of quantum information, the qubit, could be reliably encoded in photons that are easy to manipulate and virtually free from decoherence at optical frequencies关2,3兴.

Thus, recently, there has been a growing interest in quantum information processing with linear optics关4–7兴 and several techniques to generate and manipulate optical qubits have been developed for different purposes ranging from, e.g., teleportation关8,9兴, to quantum cryptography 关3兴, to quantum measurements of qubits states关10兴 and processes 关11兴, etc. In particular, Kwiat and co-workers关12,13兴 were able to create and characterize arbitrary one- and two-qubit states, using polarization and frequency modes of photons generated via spontaneous parametric down conversion共SPDC兲 关14兴.

Manipulation of optical qubits is performed by means of linear optical instruments such as half- and quarter-wave plates, beam splitters, polarizers, mirrors, etc., and networks of these elements. Each of these devices can be thought as an object where incoming modes of the electromagnetic fields are turned into outgoing modes by a linear transformation.

From a quantum information perspective, this transforms the state of qubits encoded in some degrees of freedom of the incoming photons, according to a completely positive mapE describing the action of the device. Thus, an optical instru- ment may be put in correspondence with a quantum map and vice versa. Such correspondence has been largely exploited 关7,12,13,15兴 and stressed 关16,17兴 by several authors. More- over, classical physics of linear optical devices is a textbook matter 关18,19兴, and quantum physics of elementary optical instruments has been studied extensively关20兴, as well. How- ever, surprisingly enough, a systematic exposition of the con- nection between classical linear optics and quantum maps is still lacking.

In this paper we aim to fill this gap by presenting a de- tailed theory of linear optical instruments from a quantum information point of view. Specifically, we establish a rigor- ous basis for the connection between quantum maps describ- ing one- and two-qubit physical processes operated by polarization-affecting optical instruments and the classical

matrix formalism of polarization optics. Moreover, we will use this connection to interpret some recent experiments in our group关21兴.

We begin in Sec. II by reviewing the classical theory of polarization-affecting linear optical devices. Then, in Sec. III we show how to pass, in a natural manner, from classical polarization-affecting optical operations to one-qubit quan- tum processes. Such passage is extended to two-qubit quan- tum maps in Sec. IV. In Sec. V we furnish two explicit ap- plications of our classical-vs-quantum formalism that illustrate its utility. Finally, in Sec. V we summarize our re- sults and draw the conclusions.

II. CLASSICAL POLARIZATION OPTICS

Many textbooks on classical optics introduce the concept of polarized and unpolarized light with the help of the Jones and Stokes-Mueller calculi, respectively 关19兴. In these cal- culi, the description of classical polarization of light is for- mally identical to the quantum description of pure and mixed states of two-level systems, respectively 关22兴. In the Jones calculus, the electric field of a quasimonochromatic polar- ized beam of light which propagates close to the z direction, is represented by a complex-valued two-dimensional vector, the so-called Jones vector E僆C²: E = E₀x + E₁y, where the three real-valued unit vectors 兵x,y,z其 define an orthogonal Cartesian frame. The same amount of information about the state of the field is also contained in the 2⫻2 matrix J of components Jij= EiE^ⴱ_j,共i, j=0,1兲, which is known as the co- herency matrix of the beam关18兴. By definition, the matrix J is Hermitean and positive semidefinite. Further, J has the projection property J²= J Tr J, and its trace equals the total intensity of the beam Tr J =兩E0兩²+兩E1兩². If we choose the electric field units in such a way that Tr J = 1, then J has the same properties of a density matrix representing a two-level quantum system in a pure state. In classical polarization op- tics the coherency matrix description of a light beam has the advantage, with respect to the Jones vector representation, of generalizing to the concept of partially polarized light. In this case the projection property is lost and J has the same prop- erties of a density matrix representing a two-level quantum system in a mixed state. Coherency matrices of partially po- larized beams of light may be obtained by tacking linear

1050-2947/2007/76共3兲/032323共12兲 032323-1 ©2007 The American Physical Society

combinations兺NwNJNof coherency matrices JNof polarized beams共all traveling along the same optical axis z兲, where the index N runs over an ensemble of field configurations and wNⱖ0. It should be noted that the off-diagonal elements of the coherency matrix are complex valued and, therefore, not directly observables. However, as with any 2⫻2 matrix, J can be written either in the Pauli basis ␴_␣ 关23兴 or in the standard basis Y_␣:共Y_␣兲ij=␦2i+j,␣ 共i, j=0,1兲, as

J =1

2_␣=0

兺

兺

where s_␣= Tr共␴␣J兲僆R, y_␤= Tr共Y_␤^†J兲僆C, and, from now on, all Greek indices␣^,␤^,␮^,␯, . . ., take the values 0,1,2,3. The four real coefficients s_␣, called the Stokes parameters关24兴 of the beam, can be actually measured thus relating J with ob- servables of the optical field. The real and complex represen- tations s_␣ and y_␤, respectively, are related via the matrix V : V_␣␤= Tr共␴_␣^Y_␤兲, such that s_␣=兺_␤V_␣␤y_␤, where V^†V / 2 = I₄ and I₄ is the 4⫻4 identity matrix.

When a beam of light passes through an optical system its state of polarization may change. Within the context of po- larization optics, a polarization-affecting linear optical in- strument is any device that performs a linear transformation upon the electric field components of an incoming light beam without affecting the spatial modes of the field. Half- and quarter-wave plates, phase shifters, polarizers, are all ex- amples of such devices. The class of polarization-affecting linear optical elements comprises both nondepolarizing and depolarizing devices. Roughly speaking, a nondepolarizing linear optical element transforms a polarized input beam into a polarized output beam. On the contrary, a depolarizing lin- ear optical element transforms a polarized input beam into a partially polarized output beam关25兴. A nondepolarizing de- vice may be represented by a classical map via a single 2

⫻2 complex-valued matrix T, the Jones matrix 关19兴, such that

J_in→ Jout= TJ_inT^†. 共2兲 There exist two fundamental kinds of nondepolarizing opti- cal elements, namely, retarders and diattenuators; any other nondepolarizing element can be modeled as a retarder fol- lowed by a diattenuator 关26兴. A retarder 共also known as a birefringent element兲 changes the phases of the two compo- nents of the electric-field vector of a beam, and may be rep- resented by a unitary Jones matrix TU. A diattenuator 共also known as a dichroic element兲 instead changes the amplitudes of components of the electric-field vector 共polarization- dependent losses兲, and may be represented by a Hermitean Jones matrix TH.

Since TikT^ⴱ_jl=共T丢T^ⴱ兲ij,kl⬅Mij,kl we can rewrite Eq. 共2兲 as关27兴

共Jout兲ij=Mij,kl共Jin兲kl, 共3兲 where, from now on, summation over repeated indices is understood and all Latin indices i , j , k , l , m , n , . . ., take the values 0 and 1.M is also known as the Mueller matrix in the standard matrix basis关28兴 and it is simply related to the more commonly used real-valued Mueller matrix M 关19兴 via the

change of basis matrix V : M = VMV^†/ 2. For the present case of a nondepolarizing device, M is called the Mueller-Jones matrix.

With respect to the Jones matrix T, the Mueller matrices M and M have the advantage of generalizing to the repre- sentation of depolarizing optical elements. Mueller matrices of depolarizing devices may be obtained by taking linear combinations of Mueller-Jones matrices of nondepolarizing elements as

M =

兺

兺

where peⱖ0. Index e runs over an ensemble 共either deter- ministic 关29兴 or stochastic 关30兴兲 of Mueller-Jones matrices Me= Te丢T_e^ⴱ, each representing a nondepolarizing device. In the current literature M is often written as关26兴

M =

冋

册

where p僆R³, d僆R³ are known as the polarizance vector and the diattenuation vector共superscript T indicates transpo- sition兲, respectively, and W is a 3⫻3 real-valued matrix.

Note that p is zero for pure depolarizers and pure retarders, while d is nonzero only for dichroic optical elements关26兴.

Moreover, W reduces to a three-dimensional orthogonal ro- tation for pure retarders. It the next section, we shall show that if we choose M₀₀= 1共this can always be done since it amounts to a trivial polarization-independent renormaliza- tion兲, the Mueller matrix of a nondichroic optical element 共d=0兲, is formally identical to a nonunital, trace-preserving, one-qubit quantum map 共also called channel兲 关31兴. If also p = 0共pure depolarizers and pure retarders兲, then M is iden- tical to a unital one-qubit channel 共as defined, e.g., in Ref.

关1兴兲.

III. FROM CLASSICAL TO QUANTUM MAPS:

THE SPECTRAL DECOMPOSITION

An important theorem in classical polarization optics states that any linear optical element共either deterministic or stochastic兲 is equivalent to a composite device made of at most four nondepolarizing elements in parallel 关32兴. This theorem follows from the spectral decomposition of the Her- mitean positive semidefinite matrix H关33兴 associated to M, and defined as H_ij,kl⬅Mik,jl关27,34兴. In view of the claimed connection between classical polarization optics and one- qubit quantum mechanics, it is worth noting that H is for- mally identical to the dynamical共or Choi兲 matrix, describing a one-qubit quantum process 关35兴. After a straightforward calculation, it can be shown that关27,34兴

M =_␮=0

兺

where共T_␮兲ij⬅共u_␮兲_␣=2i+j.␭_␮ⱖ0 are the non-negative eigen- values of H, and兵u_␮其=兵u0, u₁, u₂, u₃其 is the orthonormal ba- sis of eigenvectors of H: Hu_␮=␭_␮u_␮. Equation共6兲 represents the content of the decomposition theorem in classical polar-

ization optics, as given by Cloude关32,36兴. It implies, via Eq.

共3兲, that the most general operation that a linear optical de- vice can perform upon a beam of light can be written as

J_in→ Jout=_␮=0

兺

where the four Jones matrices T_␮ represent four different nondepolarizing optical elements. Since ␭_␮ⱖ0, Eq. 共7兲 is formally identical to the Kraus form 关1兴 of a completely positive共CP兲 one-qubit quantum map E. It worth to note 关37兴 that a classical Mueller matrix is always equivalent to a quantum CP map and not, for example, to a positive map.

The reason for this fact is twofold: First, it can be easily shown关28兴 that the matrix H associated to a Mueller-Jones matrix representing a nondepolarizing optical device, is nec- essarily positive semidefinite. Second, in Ref. 关38兴, starting from the exact equations for the propagation of a paraxial electromagnetic field through an arbitrary linear medium, we have calculated the form of the corresponding phenomeno- logical Mueller matrix M that could be measured with a standard polarization tomography setup. From such calcula- tion, it straightforwardly follows thatM has necessarily the form given in Eq.共4兲 and, therefore, is equivalent to a CP map.

Because of the isomorphism between J and␳ 关22兴, from Eq. 共7兲, it follows that when a single photon encoding a polarization qubit 共represented by the 2⫻2 density matrix

␳in兲, passes through an optical device classically described by the Mueller matrixM=兺_␮␭_␮T_␮丢T_␮^ⴱ, its state will be trans- formed according to

␳in→␳out⬀_␮=0

兺

where the proportionality symbol “⬀” accounts for a possible renormalization to ensure Tr␳out= 1. Such renormalization is not necessary in the corresponding classical Eq. 共7兲 since Tr J_outis equal to the total intensity of the output light beam that does not need to be conserved. Note that by using the definition共6兲 we can rewrite explicitly Eq. 共8兲 as

␳out,ij⬀^˜␳out,ij=Mij,kl␳in,kl, 共9兲 where共␳兲ij=具i兩␳兩j典 are density matrix elements in the single- qubit standard basis 兵兩i典其, i僆兵0,1其, and ^˜␳out is the un- normalized single-qubit density matrix such that ␳out

=^˜␳out/ Tr^˜␳out. From Eq.共9兲, it readily follows that Tr^˜␳out= M₀₀+ M₀₁共␳in,01+␳in,10兲 + iM02共␳in,01−␳in,10兲

+ M₀₃共␳in,00−␳in,11兲, 共10兲

where we have assumed Tr␳in= 1. The equation above shows that M represents a trace-preserving map only if M₀₀= 1 and d^T=共M01, M₀₂, M₀₃兲=共0,0,0兲, namely, only if M describes the action of a nondichroic optical instrument.

In addition, if␳inrepresents a completely mixed state, that is, if␳in= X₀/ 2, then from Eq.共9兲 it follows that

␳

˜_out=1

2_␮=0

兺

where we have defined p₀⬅M₀₀ and 共p₁, p₂, p₃兲=p is the polarizance vector. Equation 共11兲 shows that in this case Tr^˜␳out= M₀₀, and ␳out=␳^˜out/ M₀₀⫽X0/ 2 if p⫽0, that is, the map represented byM 共or M兲 is unital only if p=0.

By writing Eqs.共7兲–共11兲 we have thus completed the re- view of the analogies between linear optics and one-qubit quantum maps. In the next section we shall study the con- nection between classical polarization optics and two-qubit quantum maps.

IV. POLARIZATION OPTICS AND TWO-QUBIT QUANTUM MAPS

Let us consider a typical SPDC setup where pairs of pho- tons are created in the quantum state␳ along two well de- fined spatial modes共say, path A and path B兲 of the electro- magnetic field, as shown in Fig.1. Each photon of the pair encodes a polarization qubit and ␳ can be represented by a 4⫻4 Hermitean matrix. Let TAandTBbe two distinct optical devices put across path A and path B, respectively. Their action upon the two-qubit state␳can be described by a bilo- cal quantum map␳→EA丢EB关␳兴 关39兴. A subclass of bilocal quantum maps occurs when eitherTA orTB is not present in the setup, then eitherEA=I or EB=I, respectively, and the corresponding map is said to be local. In the above expres- sionsI represents the identity map: It does not change any input state. When a map is local, that is when it acts on a single qubit, it is subjected to some restrictions. This can be easily understood in the following way: For definiteness, let assume EB=I so that the local map E can be written as E关␳兴=EA丢I关␳兴. Let Alice and Bob be two spatially sepa- rated observer who can detect qubits in modes A and B, respectively, and let␳ ^and␳_Edenote the two-qubit quantum state before and after TA, respectively. In absence of any causal connection between photons in path A with photons in path B, special relativity demands that Bob cannot detect via any type of local measurement the presence of the deviceTA

located in path A. Since the state of each qubit received by Bob is represented by the reduced density matrix ␳_E^B

TB Path B

Path A

DB Nonlinear

crystal

DA

&

TA Pump

FIG. 1. 共Color online兲 Layout of a typical SPDC experimental setup. An optically pumped nonlinear crystal emits photon pairs that propagate along path A and B through the scattering devicesTAand TB, respectively. Scattered photons are detected in coincidence by detectors D_Aand D_Bthat permit a tomographically complete two- photon polarization state reconstruction.

=兩Tr兩A共␳_E兲, the locality constraint can be written as

␳_E^B=␳^B. 共12兲 We can write explicitly the mapEA丢I as a Kraus operator- sum decomposition关1兴

␳哫␳E⬀_␮=0

兺

where, from now on, the symbol I denotes the 2⫻2 identity matrix and兵A_␮其 is a set of four 2⫻2 Jones matrices describ- ing the action ofTA. Then, Eq.共12兲 becomes

兺

␳li,kj

冉

^兺

冊

⬀

兺

which implies the trace-preserving condition on the local mapEA丢I:

␮=0

兺

3

␭_␮A_␮^†A_␮⬀ I. 共15兲

Local maps that do not satisfy Eq.共15兲 are classified as non- physical. In this section we show how to associate a general two-qubit quantum map E关␳兴=EA丢EB关␳兴 to the classical Mueller matricesM^AandM^Bdescribing the optical devices TA and TB, respectively. Surprisingly, we shall find that physical linear optical devices exist共dichroic elements兲 that may generate nonphysical two-qubit quantum maps关40兴.

Let denotes with 兩ij典⬅兩i典丢兩j典, i, j僆兵0,1其 the two-qubit standard basis. A pair of qubits is initially prepared in the generic state ␳⁼␳ij,kl兩ij典具kl兩=␳ik,jl

R 兩i典具k兩丢兩j典具l兩, where super- script R indicates reshuffling关34兴 of the indices, the same operation we used to pass from M to H: ␳ik,jl

R ⬅␳ij,kl

=具ij兩␳兩kl典. ␳ is transformed under the action of the bilocal linear mapE关␳兴=EA丢EB关␳兴 into the state

␳E=EA丢EB关␳兴 ⬀

兺

where 兵A_␮其 and 兵B_␯其 are two sets of 2⫻2 Jones matrices describing the action of TA and TB, respectively. From Eq.

共16兲 we can calculate explicitly the matrix elements 具ij兩␳_E兩kl典=共␳_E兲ij,klin the two-qubit standard basis

共␳E兲ij,kl⬀ ␭_␮^A共A_␮兲im共A_␮^ⴱ兲kp␳mp,nq

R ␭_␯^B共B_␯兲jn共B_␯^ⴱ兲lq

=M_ik,mp^A M^B_jl,nq␳mp,nq

R , 共17兲

where summation over repeated Latin and Greek indices is understood. Since by definition共␳E兲ij,kl=共␳_E^R兲ik,jl we can re- write Eq.共17兲 using only Greek indices as

共␳_E^R兲_␣␤⬀ M_␣␮^A M_␤␯^B ␳^R_␮␯⁼共M^A丢M^B兲_{␣␤,␮␯}␳_␮␯^{R ,} 共18兲 where summation over repeated Greek indices is again un- derstood. Equation共18兲 relates classical quantities 共the two Mueller matricesM^A andM^B兲 with quantum ones 共the in- put and output density matrices␳^Rand␳_E^R, respectively兲, thus realizing the sought connection between classical polariza- tion optics and two-qubit quantum maps.

An important case occurs whenEB=I⇒M^B= I₄ and Eq.

共18兲 reduces to

␳_E^R⬀ M^A␳^R. 共19兲 Equation共19兲 illustrates once more the simple relation exist- ing between the classical Mueller matrixM^Aand the quan- tum state␳_E^.

With a typical SPDC setup it is not difficult to prepare pairs of entangled photons in the singlet polarization state.

Via a direct calculation, it is simple to show that when ␳ represents two qubits in the singlet state ␳s=¹₄共␴0丢␴0−␴1 丢␴1−␴2丢␴2−␴3丢␴3兲 and M^A is normalized in such a way that M₀₀^A = 1, then the proportionably symbol in the last equation above can be substituted with the equality symbol:

␳_E^R=M␳s

R⇒␳_E⁼共M␳s

R兲^R, 共20兲

where, from now on, we write M for M^A to simplify the notation. Note that this pleasant property is true not only or the singlet but for all four Bell states关1兴, as well. Equation 共20兲 has several remarkable consequences: Let M denotes the real-valued Mueller matrix associated toM and assume M₀₀= 1. Then, the following results hold:

Tr共␳_E²兲 = Tr共MM^T兲/4, 共21兲 兩Tr兩A共␳^˜E兲 = 共A + D兲 + M01共B + C兲 + iM02共B − C兲

+ M₀₃共A − D兲, 共22兲

where^˜␳E⬅共M␳^R兲^Ris the un-normalized output density ma- trix. Equation 共22兲 is more general than Eq. 共21兲, since it holds for any input density matrix ␳ and not only for the singlet one ␳s. In addition, in Eq. 共22兲 we wrote the input density matrix␳ in a block-matrix form as

␳⁼

冋

册

where A, B, C = B^†, and D are 2⫻2 submatrices and A+D

=兩Tr兩A共␳兲. Equation 共21兲 shows that the degree of mixedness of the quantum state ␳_E is in a one-to-one correspondence with the classical depolarizing power关25兴 of the device rep- resented by M. Finally, Eq.共22兲, together with Eqs. 共5兲 and 共12兲, tells us that the two-qubit quantum map Eq. 共20兲 is trace preserving only if the device is not dichroic, namely, only if d^T=共M01, M₀₂, M₀₃兲=共0,0,0兲. This last result shows that despite of their physical nature 共think of, e.g., a polar- izer兲, dichroic optical elements must be handled with care when used to build two-qubit quantum maps. We shall dis- cuss further this point in the next section.

Before concluding this section, we want to point out the analogy between the 16⫻16 Mueller matrix M=M^A

丢M^B associated to a bilocal two-qubit quantum map, and the 4⫻4 Mueller-Jones matrix M=T丢T^ⴱ representing a nondepolarizing device in a one-qubit quantum map. In both cases the Mueller matrix is said to be separable. Then, in Eq.

共4兲 we learned how to build nonseparable Mueller matrices representing depolarizing optical elements. By analogy, we can now build nonseparable two-qubit Mueller matrices rep- resenting nonlocal quantum maps, as

M =

兺

A,B

wABM^A丢M^B, 共24兲 where wABⱖ0, wAB⫽wA⫻wB, and indices A, B run over two ensembles of arbitrary Mueller matrices M^A and M^B representing optical devices located in path A and path B, respectively.

V. APPLICATIONS

In this section we exploit our formalism, by applying it to two different cases. As a first application, we build a simple phenomenological model capable to explain certain of our recent experimental results关21兴 about scattering of entangled photons. The second application consists in the explicit con- struction of a bilocal quantum map generating two-qubit MEMS states. A realistic physical implementation of such map is also given.

A. Example 1: Simple phenomenological model In Ref.关21兴, by using a setup similar to the one shown in Fig. 1, we have experimentally generated entangled two- qubit mixed states that lie upon and below the Werner curve in the linear entropy-tangle plane关41兴. In particular, we have found the following.共a兲 Birefringent scatterers always pro- duce generalized Werner states of the form␳GW= V丢I␳WV^†

丢I, where ␳W denotes ordinary Werner states 关42兴 and V represents an arbitrary unitary operation.共b兲 Dichroic scat- terers generate sub-Werner states, that is states that lie below the Werner curve in the linear entropy-tangle plane. In both cases, the input photon pairs were experimentally prepared in the polarization singlet state␳s. In this subsection we build, with the aid of Eq.共20兲, a phenomenological model explain- ing both results共a兲 and 共b兲.

To this end let us consider the experimental setup repre- sented in Fig.1. According to the actual scheme used in Ref.

关21兴, where a single scattering device was present, in this subsection we assume TB=I, so that the resulting quantum map is local. The scattering elementTA inserted across path A can be classically described by some Mueller matrixM.

In Ref. 关26兴, Lu and Chipman have shown that any given Mueller matrixM can be decomposed in the product

M = MDMBM_⌬, 共25兲 whereM_⌬,MB, and MDare complex-valued Mueller ma- trices representing a pure depolarizer, a retarder, and a diat- tenuator, respectively. Such decomposition is not unique, for example, M=M_⌬MDMB is another valid decomposition 关43兴. Of course, the actual values of M_⌬, MB, and MD

depend on the specific order one chooses. However, in any case they have the general forms given below:

M_⌬=

冤

冥

MB= TU丢T_U^ⴱ, 共27兲

MD= T_H丢T_H^ⴱ, 共28兲 where a , b , c僆R and TU, T_Hare the unitary and Hermitean Jones matrices representing a retarder and a diattenuator, re- spectively. Actually, the expression ofM_⌬given in Eq.共26兲 is not the most general possible关26兴, but it is the correct one for the representation of pure depolarizers with zero polari- zance, such as the ones used in Ref.关21兴. Note that although MBandMDare Mueller-Jones matrices,M_⌬is not. When a = b = c⬅p:p僆关0,1兴 the depolarizer is said to be isotropic 共or, better, polarization isotropic兲. This case is particularly relevant when birefringence and dichroism are absent. In this case MB= I₄=MD, and Eq. 共25兲 gives M=M_⌬. Thus, by using Eq.共26兲 we can calculate M_⌬共p兲 and use it in Eq. 共20兲 to obtain

␳E= p␳s+ 1 − p

4 I₄⬅␳W, 共29兲

that is, we have just obtained a Werner state:␳_E⁼␳W. Thus, we have found that a local polarization-isotropic scatterer acting upon the two-qubit singlet state, generates Werner states.

Next, let us consider the cases of birefringent共retarders兲 and dichroic 共diattenuators兲 scattering devices that we used in our experiments. In these cases the total Mueller matrices M of the devices under consideration can be written as M

=MZM_⌬, where either Z = B or Z = D, andM_⌬=M_⌬共p兲 rep- resents a polarization-isotropic depolarizer. For definiteness, let consider in detail only the case of a birefringent scatterer, since the case of a dichroic one can be treated in the same way. In this case

MBM_⌬共p兲 =

兺

␮=0 3

␭_␮共p兲TUT_␮丢T_U^ⴱT_␮^ⴱ 共30兲

and, as result of a straightforward calculation, ␭0=共1 + 3p兲/2, ␭1=␭2=␭3=共1−p兲/2, T_␮= X_␮/

冑

共20兲, we prefer to rewrite Eq. 共16兲 adapted to this case as

␳E=_␮=0

兺

= TU丢I

冋

^兺

册

= TU丢I␳WT_U^† 丢I =␳GW, 共31兲 where Eq. 共29兲 has been used. Equation 共31兲 clearly shows that the effect of a birefringent scatterer is to generate what we called generalized Werner states, in full agreement with our experimental results关21兴.

The analysis for the case of a dichroic scatterer can be done in the same manner leading to the result

␳E⬀^˜␳E= TH丢I␳WT_H^† 丢I, 共32兲 where THis a 2⫻2 Hermitean matrix representing a generic diattenuator关19兴

TH=

冋

册

共33兲 where di僆关0,1兴 are the diattenuation factors, while

␪僆共0,2␲兴 gives the direction of the transmission axis of the linear polarizer to which TH reduces when either d₀= 0 or d₁= 0. Figure2reports, in the linear entropy-tangle plane, the results of a numerical simulation were we generated 10⁴ states ␳E from Eq. 共32兲, by randomly generating 共with uni- form distributions兲 the four parameters p, d0, d₁, and␪in the ranges p , d₀, d₁僆关0,1兴, ␪僆共0,2␲兴. The numerical simula- tion shows that a local dichroic scatterer may generate sub- Werner two-qubit states, that is, states located below the Werner curve in the linear entropy-tangle plane. The qualita- tive agreement between the result of this simulation and the experimental findings shown in Fig. 3 of Ref.关21兴 is evident.

Discussion. It should be noticed that while we used the equality symbol in writing Eq.共31兲, we had to use the pro- portionality symbol in writing Eq. 共32兲. This is a conse- quence of the Hermitean character of the Jones matrix TH

that generates a non-trace-preserving map. In fact, in this case from M = M_DM_⌬共p兲, where MD=共VTH丢T_H^ⴱV^†兲/2 and M_⌬共p兲=关VM_⌬共p兲V^†兴/2, we obtain Tr共^˜␳E兲=共d₀²+ d₁²兲/2⫽1.

Moreover, Eq.共22兲 gives

␳EB

=兩Tr兩A共␳E兲 =X₀

2 − p

冉

冊

共34兲 where␳_E^=˜␳_E/ Tr共^˜␳_E兲. This result is in contradiction, for d0

⫽d1, with the locality constraint expressed by Eq. 共12兲 which requires

␳_E^B=X0

2 . 共35兲

As we already discussed in the previous section, only the latter result seems to be physically meaningful since photons in path B, described by ␳_E^B, cannot carry information about device TA which is located across path A. On the contrary, Eq. 共34兲 shows that ␳_E^B is expressed in terms of the four physical parameters p, d₀, d₁, and␪ that characterizeTA. Is there a contradiction here?

In fact, there is not. One should keep in mind that Eq.共34兲 expresses the one-qubit reduced density matrix ␳_E^{B that is} extracted from the two-qubit density matrix␳_Eafter the latter has been reconstructed by the two observers Alice and Bob by means of nonlocal coincidence measurements. Such ma- trix contains information about both qubits and, therefore, contains also information aboutTA. Conversely,␳_E^{B= X}0/ 2 in Eq.共35兲, is the reduced density matrix that could be recon- structed by Bob alone via local measurements before he and Alice had compared their own experimental results and had selected from the raw data the coincidence counts.

From a physical point of view, the discrepancy between Eqs.共34兲 and 共35兲 is due to the polarization-dependent losses 共that is, d0⫽d1兲 that characterize dichroic optical devices and it is unavoidable when such elements are present in an experimental setup. Actually, it has been already noticed that a dichroic optical element necessarily performs a kind of post-selective measurement 关16兴. In our case coincidence measurements post-select only those photons that have not been absorbed by the dichroic elements present in the setup.

However, since in any SPDC setup even the initial singlet state is actually a post-selected state共in order to cut off the otherwise overwhelming vacuum contribution兲, the practical use of dichroic devices does not represent a severe limitation for such setups.

B. Example 2: Generation of two-qubit MEMS states In the previous subsection we have shown that it is pos- sible to generate two-qubit states represented by points upon and below the Werner curve in the linear entropy-tangle plane, by operating on a single qubit 共local operations兲 be- longing to a pair initially prepared in the entangled singlet state. In another paper 关40兴 we have shown that it is also possible to generate MEMS states共see, e.g., Refs. 关41,44兴, and references therein兲, via local operations. However, the price to pay in that case was the necessity to use a dichroic device that could not be represented by a “physical,” namely, a trace-preserving, quantum map. In the present subsection, as an example illustrating the usefulness of our conceptual scheme, we show that by allowing bilocal operations per- formed by two separate optical devicesTAandTBlocated as in Fig.1, it is possible to achieve MEMS states without using dichroic devices.

To this end, let us start by rewriting explicitly Eq.共16兲, where the most general bilocal quantum map E关␳兴=EA 丢EB关␳兴 operating upon the generic input two-qubit state␳^{, is} represented by a Kraus decomposition

0 0.2 0.4 0.6 0.8 1

Linear entropy, SL 0

0.2 0.4 0.6 0.8 1

elgnaT,T

0 0.2 0.4 0.6 0.8 1

Linear entropy, SL 0

0.2 0.4 0.6 0.8 1

elgnaT,T

FIG. 2. Numerical simulation from our phenomenological model qualitatively reproducing the behavior of a dichroic scatter- ing system. The gray region represents unphysical states and it is bounded from below by MEMS共dashed curve兲. The lower continu- ous thick curve represents Werner states.

␳E=EA丢EB关␳兴 =

兺

where now the equality symbol can be used since we assume that both single-qubit mapsEAandEB are trace preserving

␮=0

兺

3

␭_␮^AA_␮^†A_␮= I =

兺

␯=0 3

␭_␯^BB_␯^†B_␯ 共37兲

but not necessarily unital: EF关I兴⫽I, F僆兵A,B其 关39兴. Under the action ofE, the initial state of each qubit traveling in path A or path B is transformed into either the output state

␳_E^A=兩Tr兩B共␳_E兲 =_␮=0

兺

or

␳_E^B=兩Tr兩A共␳_E兲 =

兺

respectively, where ␳^A=兩Tr兩B共␳兲 and ␳^B=兩Tr兩A共␳兲. Without loss of generality, we assume that the two qubits are initially prepared in the singlet state␳⁼␳s. Then Eqs.共38兲 and 共39兲 reduce to ␳_E^F=兺_␣␭_␣^FF_␣F_␣^†/ 2, F僆兵A,B其. From the previous analysis 关see Eqs. 共16兲–共18兲兴 we know that to each bilocal quantum map EA丢EB can be associated a pair of classical Mueller matricesM^A andM^B such that

共␳_E^R兲_␣␤=

兺

The real-valued Mueller matrices M^A and M^B associated to M^AandM^B, respectively, can be written as

M^A=

冋

册

冋

册

where Eq.共5兲 with dA= 0 = dBand M₀₀= 1 has been used, and

pA⬅ a =

冤

冥

冤

冥

are the polarizance vectors of M^A and M^B, respectively. We remember that the condition dA= dB= 0 is a consequence of the fact that both mapsEAandEBare trace preserving, while the conditions a⫽0 and b⫽0 reflect the nonunital nature of EA andEB. With this notation we can rewrite Eqs.共38兲 and 共39兲 as

␳_E^A=1

2_␮=0

兺

␳_E^B=1 2

兺

␯=0 3

b_␯X_␯, 共44兲

where we have defined a₀= 1 = b₀. Moreover, the output two- qubit density matrix␳E=E关␳s兴 can be decomposed into a real and an imaginary part as␳E=␳_E^{Re+ i}␳_E^{Im, where}

␳_E^Re=1

4

冤

冥

and

␳_E^Im=1

4

冤

冥

with

␣±+⬅ 共1 + a3兲 ± 关b3共1 + a3兲 − C33兴,

␣±−⬅ 共1 − a3兲 ± 关b3共1 − a3兲 + C33兴, 共47兲 and

␤±⬅ b1±共a3b₁− C₃₁兲, ␥±⬅ a1±共a1b₃− C₁₃兲,

␦±⬅ a1b₁− C₁₁⫿ 共a2b₂− C₂₂兲, 共48兲 and

␰±⬅ b2±共a3b₂− C₃₂兲, ␩±⬅ a2±共a2b₃− C₂₃兲,

␶±⬅ a2b₁− C₂₁±共a1b₂− C₁₂兲, 共49兲 where Cij⬅共AB^T兲ij, i , j僆兵1,2,3其.

At this point, our goal is to determine the two vectors a, b and the two 3⫻3 matrices A, B such that␳_E^{Im= 0 and}

␳_E^Re=␳MEMS=

冤

冥

where

g共p兲 =

再

冎

To this end, first we calculate a and b by imposing

␳_E^A=␳MEMS

A =

冋

册

␳_E^B=␳MEMS

B =

冋

册

respectively. Note that only fulfilling Eqs.共52兲 and 共53兲, to- gether with␳_E^Re=␳MEMSand␳_E^Im= 0, will ensure the achieve- ment of true MEMS states. It is surprising that in the current literature the importance of this point is neglected. Thus, by solving Eqs.共52兲 and 共53兲 we obtain a1= a₂= 0, a₃= 1 − g共p兲, and b = −a, where Eqs.共43兲 and 共44兲 have been used. Then, after a little of algebra, it is not difficult to find that a pos- sible bi-local mapE=EA丢EBthat generates a solution␳_E^for the equation␳_E⁼␳MEMS, can be expressed as in Eqs.共40兲 and

共41兲 in terms of the two real-valued Mueller matrices

M^A=

冤

^冑

^冑

冥

M^B=

冤

^冑

^冑

冥

It is easy to check that both M^A and M^B are physically ad- missible Mueller matrices since the associated matrices H^A and H^Bhave the same spectrum made of non-negative eigen- values兵␭_␮^A其=兵␭_␮^B其⬅兵␭_␮其=兵␭0,␭1,␭2,␭3其. In particular:

兵␭_␮其 = 兵0, 1 − p, 0, 1 + p其, for 2/3 ⬍ p ⱕ 1 共55兲 and

兵␭_␮其 =

再

^冑

^冑

冎

for 0ⱕpⱕ2/3. It is also easy to see that the map E can be decomposed as in Eq.共36兲 in a Kraus sum with A0= A₂= 0

A₁

冑

冋

^冑

册

^冑

冋

^冑

册

and B₀= B₂= 0

B₁

冑

冋

^冑

册

^冑

冋

^冑

册

for 2 / 3⬍pⱕ1. Analogously, for 0ⱕpⱕ2/3 we have A0

= 0

A₁

冑

冋

^冑

册

A₂

冑

冋

册

^冑

冋

册

and B₀= 0

B₁

冑

冋

^冑

册

B₂

冑

冋

册

^冑

冋

册

where

␾±⬅

冑

冉

^冑

冊

␺±⬅

冑

冉

冑

冊

Note that these coefficients satisfy the following relations:

␾₊²+␾₋²= 1, 共65兲

1

3+␺+2+␺−2= 1. 共66兲

A straightforward calculation shows that the single-qubit mapsEA andEBare trace-preserving but not unital, since

␮=0

兺

3

␭_␮A_␮A_␮^†=

冋

册

and

兺

␭_␯B_␯B_␯^†=

冋

册

At this point our task has been fully accomplished. How- ever, before concluding this subsection, we want to point out that both mapsEA andEB must depend on the same param- eter p in order to generate proper MEMS states. This means that either a classical communication must be established betweenTA and TB in order to fix the same value of p for both devices or a classical signal encoding the information about the value of p must be sent toward bothTA andTB.

Physical implementation. Now we furnish a straightfor- ward physical implementation for the quantum maps pre- sented above. Up to now, several linear optical schemes gen- erating MEMS states were proposed and experimentally tested. Kwiat and co-workers 关41兴 were the first to achieve MEMS using photon pairs from spontaneous parametric down conversion. Basically, they induced decoherence in SPDC pairs initially prepared in a pure entangled state by coupling polarization and frequency degrees of freedom of the photons. At the same time, a somewhat different scheme was used by De Martini and co-workers 关44兴 who instead used the spatial degrees of freedom of SPDC photons to induce decoherence. In such a scheme the use of spatial de- grees of freedom of photons required the manipulation of not only the emitted SPDC photons, but also of the pump beam.

In this subsection, we show that both single-qubit maps EA andEB can be physically implemented as linear optical networks 关6兴 where polarization and spatial modes of pho- tons are suitably coupled, without acting upon the pump beam. The basic building blocks of such networks are polar- izing beam splitters 共PBSs兲, half-waveplates 共HWPs兲, and mirrors. Let兩i,N典 be a single-photon basis, where the indices i and N label polarization and spatial modes of the electro- magnetic field, respectively. We can also write兩i,N典=aˆiN

†兩0典 in terms of the annihilation operators aˆiN and the vacuum state 兩0典. A polarizing beam splitter distributes horizontal 共i

= H兲 and vertical 共i=V兲 polarization modes over two distinct spatial modes, say N = n and N = m, as follows:

兩H,n典in→ 兩H,n典out and 兩V,n典in→ 兩V,m典out,

兩H,m典in→ 兩H,m典out and 兩V,m典in→ 兩V,n典out, 共69兲 as illustrated in Fig. 3. A half-waveplate does not couple polarization and spatial modes of the electromagnetic field and can be represented by a 2⫻2 Jones matrix THWP共␪兲 as