New J. Phys. 18 (2016) 073010 doi:10.1088/1367-2630/18/7/073010

PAPER

Generating and protecting correlated quantum states undercollective dephasing

EdoardoGCarnio1,2,6, Andreas Buchleitner1,3 andManuelGessner1,4,5

1 Physikalisches Institut, Albert-Ludwigs-Universität Freiburg,Hermann-Herder-Straße 3,D-79104 Freiburg, Germany2 Department of Physics, University ofWarwick, Coventry, CV4 7AL,UK3 Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universität Freiburg, Albertstraße 19, D-79104 Freiburg, Germany4 QSTAR, CNR-INO, and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy5 IstitutoNazionale di RicercaMetrologica (INRiM), I-10135Torino, Italy6 Author towhomany correspondence should be addressed.

E-mail: e.carnio@warwick.ac.uk

Keywords: entanglement, quantum correlations, dephasing, decoherence

AbstractWe study the collective dephasing process of a systemof non-interacting atomic qubits, immersed in aspatially uniformmagnetic field offluctuating intensity. The correlation properties of bipartite statesare analysed based on a geometric representation of the state space. Particular emphasis is put on thedephasing-assisted generation of states with a high correlation rank, which can be related to discord-type correlations and allow for direct applications in quantum information theory. Finally we studythe conditions that ensure the robustness of initial entanglement and discuss the phenomenon oftime-invariant entanglement.

1. Introduction

Ensembles of trapped, laser-cooled atomic particles provide some of the best-controlled experimental platformsto study quantumdynamics, to engineer effective interactions, or to generate specific quantum states [1–8]. Anessential requirement for such levels of control is the efficient isolation of the system from its environment,whose detrimental influence leads to the decay of coherent superpositions [9]. The loss of coherence often alsoimplies the loss of quantum correlations, such as entanglement, which are required, e.g., to process quantuminformation [10, 11].

One persistently dominant source of error, common tomost experiments on trapped atomic particles, iscaused by intensity fluctuations of external electromagnetic fields, which are needed to lift degeneracies, tocompensate background fields, or tomanipulate the quantum state of the system [5, 7]. Since suchfields aretypically generated by large coils outside the vacuumchamber, the resultingfield is spatially homogeneous alongthe positions of the trapped particles. The unavoidable fluctuations of thefield strength therefore lead to acollective dephasing process, which is formally described by an ensemble average over thefluctuating parameter[6, 12–16]. The collective nature of this noise provides new possibilities to protect coherent superpositions [15],and, as demonstrated in a recent experiment [6], to generate robust, strongly correlated, albeit separablequantum states.

In this article we study the impact of the collective dephasing process on different types of correlationsinscribed into quantum states, using the analytical description of the collective dephasing dynamics outlined in[15]. Our analysis focuses on the correlation rank, which in turn entails direct consequences for the discord-typecorrelations [17]. Specifically, in section 4, we study how strongly correlated two-qubit states can be generatedvia collective dephasing, as a function of tunable external control parameters, e.g., themagnetic field direction.We discuss specific applications of the produced quantum states in the context of quantum informationprotocols. In section 5we follow a complementary approach and analyse the protection of existing correlationsduring the collective dephasing process.We further discuss robust conditions that lead to the striking

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11May 2016

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10 June 2016

PUBLISHED

6 July 2016

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phenomenon of time-invariant entanglement: the perfect conservation of the initial entanglement, even forstates whose purity is reduced due to the dephasing process.

2. Correlations in quantum states

2.1. Entanglement, discord and correlation rankWebegin by reviewing different notions of correlations in quantum states that will become relevant in the courseof this article.Quantum entanglement captures the non-classical correlations of a quantum state in a compositeHilbert space ÄA B, i.e., it expresses the inability to characterize the full quantum state ρ via a classicalprobability distribution pi and local density operators { }ri

Ai and { }ri

Bi on the localHilbert spacesA andB,

respectively. In particular, a quantum state is separable (i.e., not entangled) if it can be expressed as a convexlinear combination of product states [10, 18, 19]

( )år r r= Äp . 1i

i iA

iB

s

Determining whether a givenmixed quantum state is entangled, or evaluating an appropriatemeasure toquantify entanglement, is generally a very hard task [10, 19]. Only for the special case of two-qubit systems( = =A B 2), an algebraic expression that exactly quantifies the entanglement of arbitrarymixed states isavailable [20]. Thismeasure, the concurrence, is determined on the basis of the eigenvalues li of ˜rr r ,

labelled in decreasing order, where the spin-flipped state ˜ ( ) ( )*r s s r s s= Ä Äy y y y is obtained by collectiveapplication of the Paulimatrix sy to *r , and complex-conjugation is performed in the computational basis. Theconcurrence of the state ρ is then given by ( ) { }r l l l l= - - -C max 0, 1 2 3 4 . For higher dimensionalproblems, only algebraic lower bounds are available [19].

The quantumdiscord describes the disturbance of localmeasurements on quantum states of compositesystems [21]. Aswewill see in the following, these features related to the quantum-mechanicalmeasurementprocess can be expressed via non-vanishing commutators, and can only occur in correlated quantum states.They can, however, alsomanifest in separable states that exhibit only classical correlations, and thereforedescribe amore general type of quantumproperties than entanglement. Formally, a state ρ has zero discord if itcan bewritten as [21]

∣ ∣ ( )år y y r= ñá Äp , 2ci

i iA

iA

iB

where { }yiA

i is an orthonormal basis ofA. This definition is equivalent to the following: a state ρ has zerodiscord if and only if there exists a non-selective local projectivemeasurement onA that leaves the stateinvariant, i.e., ( ) ( ) r r= å P Ä P Äi i

A BiA B , where ∣ ∣y yP = ñái

AiA

iA . The definition presented here considers

measurements onA and is straight-forwardly extended tomeasurements onB. Due to the asymmetry of thedefinition, one should always specify in which subsystemmeasurements are performed, when discussingquantumdiscord.

Notice that every zero-discord state is separable but the converse is not true. The two concepts coincide onlyin the case of pure states. In contrast to entanglement, local operations on one of the subsystems can generatediscord [6, 22], which confirms that discord is not a propermeasure for correlations. To quantify thecorrelations of a bipartite quantum state, we employ the rank of an appropriately constructed correlationmatrix(the correlation rank), which is theminimal number of bipartite operator products needed to describe the densityoperator [16, 17]. For the formal definition, wewrite the density operator ρ in an arbitrary basis of localHermitian operators { }Ai i and { }Bj j [22]:

( )åår = Ä= =

r A B , 3i

d

j

d

ij i j1 1

A B2 2

where =d dimA BA B

,, . Here ( )=R rij is the correlationmatrix, a real-valued ´d dA B

2 2-dimensionalmatrixwhose rank L is then called the correlation rank [17]. Employing a singular value decomposition, wefind non-zero singular values { }¼c c, , L1 and orthogonalmatrices ( )=U uij and ( )=V vij such that

( )= ¼R U c c Vdiag , , LT

1 .We obtain

( )å åå år = Ä = Ä = Ä= =

r A B u c v A B c S F , 4i j

ij i ji j k

L

ik k jk i jk

L

k k k, , 1 1

where = åS u Ak i ik i and = åF v Bk j jk j. The above decomposition can be regarded as a Schmidt decompositionof a densitymatrix in terms of local operator bases [16]. Amore familiar application of the Schmidtdecomposition is known for pure bipartite quantum states, which are decomposed in terms of local vectors [10].The associated singular value decomposition yields the Schmidt rank, which quantifies how entangled a pure

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New J. Phys. 18 (2016) 073010 EGCarnio et al

state is [10]. Consequently, the correlation rank stands in close analogy to the Schmidt rank. In general, thecorrelation rank does not quantify entanglement but rather total correlations, i.e., any incompatibility with anuncorrelated product state, without an explicit distinction between classical and quantumnature of thecorrelations. In the special case of a pure state one obtains =L 2, where denotes that stateʼs Schmidtrank [16].

Conclusions about the local quantumnature of the state can be drawn from the correlation rank by realizingthat itsmaximal value is bounded from above for all zero-discord states. Generally, the correlation rank cannotexceed dmin

2 , where { }=d d dmin ,A Bmin . However, as can be seen from the definition(2), states of zero discordwith respect tomeasurements onA B, are represented in terms of local projectors ∣ ∣y yñái

A BiA B, , onto

orthogonal subspaces. Since there exist nomore than dA B, orthogonal subspaces inA B, , the correlation rank ofzero-discord states is bounded by L dmin .

This is directly related to the observation that the operators Si, which appear in equation (4), can be used toassess the discord of ρ: the state ρhas zero discord (with respect tomeasurements onA) if and only if all of theSi commute [22].While local operations cannot increase the correlation rank L [17], they can change thecommutativity of the local operators Si, and thereby generate discord [22]without actually generatingcorrelations [17].

The correlation rank allows to distinguish separable states with high correlation rank ( <d L dmin min2 )

from those that can be generated from states of zero discord via local operations, as characterized by a lowcorrelation rank ( L dmin ). Separable operations of the form å F Ä Fpi i i

AiB, with local operations FA and FB,

can generate classical correlations among the particles and, consequently, are able to increase the correlationrank [6]. The collective dephasing operation to be discussed in this article represents such a separable operation.

2.2. Representation of bipartite statesThe density operator of any bipartite system can be represented in terms of the operator bases{ } { } s sÄ, ,d A d BA B

, where sA B, denote vectors whose entries are the generators of SU (dA B, ), and dA B,denote

the respective identity operators.We obtain the stateʼs Fano form as [23, 24]:

· · ( ) ( ) ( ) å ås s s sr b= + Ä + Ä + Ä=

-

=

-⎛⎝⎜⎜

⎞⎠⎟⎟r r

d d

1, 5

A Bd d A A d d B B

i

d

j

d

ij A i B j1

1

1

1

A B B A

A B2 2

where rA and rB are the (generalized)Bloch vectors of the reduced subsystems, andβ is a real( ) ( )- ´ -d d1 1A B

2 2 matrix that describes the correlations between the subsystems. Because the state iscompletely characterized by rA, rB and theβmatrix, throughout this article wewill use the compact notation

( )r br r, ,A B [25]. From the representation (5) it immediately follows that the correlationmatrix of ρ is givenby

( )b

=⎛⎝⎜

⎞⎠⎟

r

rR

d d

1 1, 6

A B

BT

A

whose rank rk is [26]

( ) ( ) ( )b= = + - Är rL Rrk 1 rk . 7A B

While for a rigorous proof of the above identity we refer to [26], we remark that the result can be obtained viablock-diagonalization of thematrix ( )=R P M Qdiag 1, , where P andQ are rank-4matrices and, in this case,

b= - Är rM A B is a 3×3matrix called the Schur complement (of the submatrix 1).In the remainder of this article, wemostly employ the reduced Bloch vectors and theβ-matrix to investigate

the impact of the collective dephasing process, which allows for an intuitive geometric description. A tool that weoften employ to simplify our analysis are local unitary transformations, since these affect neither the stateʼsentanglement properties (by definition [10, 19])nor its correlation rank (as we show explicitly later). Following[25], we consider transformations = ÄU U UA B such that †r r¢ = U U . For every unitary transformationUA,there exists a rotationOA on the respective (generalized)Bloch sphere such that ( · ) ( ) ·†s s=v vU U OA A A A A.We then obtain the following transformation rules:

( )b b¢ = ¢ = ¢ =r r r rO O O O, , . 8A A A B B B A BT

If twomatricesβ and b¢ can be transformed into each other by such an operation, wewrite b b¢ ~ . Inparticular, we can always diagonalize theβmatrix by applying unitary transformations to the underlyingquantum state.

We now show that such transformations do not change the rank of the correlationmatrix. Ifb= - Är rM A B, we have

( )b¢ = ¢ - ¢ Ä ¢r rM 9A B

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New J. Phys. 18 (2016) 073010 EGCarnio et al

( )b= - Är rO O O O 10A BT

A A B B

( ) ( )b= - Ä =r rO O O MO . 11A A B BT

A BT

The rank ofM is defined by the image of the linearmap x xf M: :

( ) ( ) { } ( ) = = Î = Îy y x xM f Mrk dimImage dim : , . 12d dA B

SinceOA andOB are orthogonalmatrices, we have

{ } { } ( ) Î = ¢ Î = Î = Îy y x x y y x xM O MO: , : , 13d d dA B

T dA B A B

{ ( ) ( ) } ( ) = Î = Îy y x xO M O: , . 14dAT

BT dB B

Since themaps =y z yOAT and =x w xOB

T are bijective, wefinally obtain

( ) { } ( ) ( ) ¢ = Î = Î =z z w wM M Mrk dim : , rk . 15d dA B

In the following, we analyse the impact of the collective dephasing process, to be introduced in the nextsection, on the different concepts that were introduced in this section, all of which are intimately related tocorrelations in quantum states.

3. Collective dephasing: Krausmap representation

3.1. Ensemble average dynamicsCollective dephasing describes the dynamics ofN particles that share the same environment, but experience nodirect particle–particle interactions among each other. The environment does not induce exchange of energywith the particles, and hence does not lead to dissipation, but rather to pure dephasing, i.e., loss of phaserelationswithout loss of populations. A physically relevant example of such a scenariowas alreadymentioned inthe introduction: when an ensemble of atomic dipoles is spatially confined in a regionwhere the electromagneticfield is homogeneous, all dipoles share the same transition frequency. Thefluctuations of the field then lead to acollective dephasing process [5, 7]. The quantum state that predicts themeasurement results aftermanyexperimental repetitions is described by the ensemble average over the actual realizations of thesefluctuations [6, 15].

Let us considermagnetic dipoles, described by two-level systems, in a constantmagnetic field B. TheHamiltonianH of theN-particle system is given by ·g= B SH , with ( )s= å =S i

N i1 , where here ( )s i is a vector

of Paulimatrices on theHilbert space of the ith particle, and further constants are absorbed into γ. Identifyingw g= B2 , with =B nB , wewrite

· ( )( ) å sw=

=

nH2

. 16i

Ni

1

For afixedmagnetic field strengthB, the fully coherent dynamics of the total system is given by

( ) ( ) ( )· · · · r r= Ä Ä Ä Äs s s sw w w w- -t e e 0 e e , 17n n n nt t t ti 2 i 2 i 2 i 2

where ( )r t is theN-particle density operator at time t. The dephasing process is nowdescribed by an ensembleaverage over thefluctuations of B. The physically intuitive reason for such a description is the necessity to repeatexperimentsmany times to produce significant statistics for the efficient estimation of the populations.Wemakethe following assumptions on the fluctuations of B:

• the direction n of themagnetic field is constant, and thefluctuations only affect the field strengthB;

• themagnetic fieldmay change from experimental run to the next, butwithin each runwe assume themagneticfield to be constant.

Both of these assumptions can bemotivated at the hand of state-of-the-art experiments on cold atoms ortrapped ions: the externalfield influences the energy splitting of the atomic two-level systems through a Zeemaneffect, as described above, and, thus, the field is chosen strong enough to dominate over the effect of possiblestrayfields. Thefield therefore has a fixed direction (satisfying thefirst of the two above assumptions), but smallfluctuations of the supplying currents will produceweak intensityfluctuations ofB, on top of a relatively largemean value. Themean value determines the time scale of the atomic evolution, which is thereforemuch fasterthan the time scale onwhich fluctuations occur (satisfying the second assumption).

Characterizing the intensity fluctuationswith the probability distribution ( )wp , the collective dephasingdynamics is described by

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New J. Phys. 18 (2016) 073010 EGCarnio et al

( ) ( ) ( ) ( )· · · · òr w r w= Ä Ä Ä Äs s s sw w w w- -t p e e 0 e e d . 18n n n nt t t ti 2 i 2 i 2 i 2

In [15], the above integral was solved analytically without further assumptions, and in the followingwewill recallthe resulting solution and some of its properties.

3.2. General propertiesThe transient time evolution of a quantum state under collective dephasing (18) is determined by thecharacteristic function

( ) ( ) ( )òj w w= wt pd e 19ti

of the probability distribution ( )wp . Introducing thematrix elements

( ) [( ) ] ( )j= -M t i j t 20ij

and theHermitian operators

!( )![ ] ( )†åQ =

-L Ä L

ÎS-Ä

+Ä -

j N jV V

1, 21j

s

j N js s

N

where ( · ) sL = n1

2 2 , and ∣ ∣( ) ( )= å ñáV i i i i... ...i i s s N Ns ... 1 1N1represents the permutation s in theHilbert

space ofN qubits, we can express the collective dephasing dynamics with the followingmap [15]:

( ) ( ) ( ) ( ) ( ) år r r = Q Q=

t M t: 0 0 . 22nt

i j

N

ij i j,0, 0

Thematrix of elementsMij is positive semi-definite, and can be diagonalized to obtain the canonical Kraus formof the abovemap [15]. One can further show [15] that themap(22) is always completely positive and tracepreserving [11, 24]. The obtained dynamics therefore exhibits the properties of dynamicalmaps associatedwiththe dynamics of open quantum systems, indicating the equivalence of ensemble average approaches with open-system treatments based on amicroscopicmodel for the environment and its coupling to the system [9, 13].

In the context of the present article, we are only interested in the asymptotic limit, which is described,independently of ( )wp (assuming that ( )wp is absolutely integrable), by [15]

( ) ( ) ( ) ( ) år r r r = = Q Q¥ =

t: 0 lim 0 . 23n

t j

N

j js0

In performing this limit, we assume that the time evolution of the atomic ensemble is recorded for an intervallong enough that the atomic evolution has reached its stationary state, but not too long to compromise with theassumption that the field strength can be considered constant during the evolution.

3.3. Integral ofmotionIn [15] themap (22)was shown to conserve the trace of theβmatrix for bipartite systems, as defined in (5). Thisintegral ofmotion can, in fact, be understood as themanifestation of themore general conservation of angularmomentum in the special case ofN=2. To see this, recall that the total spin S commutes with theHamiltonianfor every choice of themagneticfield B, hence the expectation value of ·=S S S2 is conserved, even in thepresence of the ensemble average over the fluctuations of themagnetic field.

We express the squared total spin as

· · ( )( ) ( ) ( ) ( ) å ås s s s= += =

>

S4 2

, 24i

Ni i

i ji j

Ni j2

2

1

2

, 1

with ·( ) ( ) ( ) ( )s s s s= å =i j

k ki

kj

13 , and the index k labels the spatial directions.We generalize the definition of theβ

matrix to

( ) ( ) · ( )( ) ( )åb r s s= Ä> =

⎪

⎪

⎪

⎪

⎧⎨⎩

⎫⎬⎭

t ttr . 25abi j

N

ai

bj

1

Note that this definition reduces to the bipartiteβmatrix, as introduced in (5), in the special case ofN=2. Thetotal angularmomentum is expressed via the quantummechanical expectation value, using equations (24) and(25)

{ ( ) } ( ) ( ) r bá ñ = = +S St N ttr

3

4 2tr , 262 2

2 2

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New J. Phys. 18 (2016) 073010 EGCarnio et al

and from the time-independence of á ñS2 , we obtain the conservation of the trace of the generalizedβmatrix:

( ) ( )b =t

td

dtr 0. 27

3.4. Asymptotic collective dephasing of two qubitsLet us discuss the description of the collective dephasing of an initial two-qubit state ρ into the stationary state rs,using themap(23) forN=2:

[ ] ( ) år r r= = Q Q=

. 28n

ii is

0

2

Based on (21), theKraus operatorsQi can be explicitly given as [6]

( · · · · ) ( ) s s s sQ = L Ä L = Ä - Ä - Ä + Ä- - n n n n1

4, 290 2 2 2 2

( · · ) ( ) s sQ = L Ä L + L Ä L = Ä - Ä+ - - + n n1

2, 301 2 2

( · · · · ) ( ) s s s sQ = L Ä L = Ä + Ä + Ä + Ä+ + n n n n1

4. 312 2 2 2 2

To efficiently describe the impact of the collective dephasing on an arbitrary initial state ( )r br r, ,A B , wenowderive a description of itsmap on the level of the reduced Bloch vectors rA and rB, togetherwith theβmatrix.Wefirst express theβ-matrix of the initial state in terms of a diagonal, singular value decomposition[recall (8)], as

( )åb = Ä=

v wd , 32i

i i i1

3

where vi and wi are normalized vectors, di are non-negative real numbers, and the tensor product is definedelement-wise as ( )Ä =a b a bkl k l.

By direct application of the operatorsQi and of the properties of the scalar product, (28) leads to [6]

( ) ( ) Ä = Ä , 33n2 2 2 2

( · ) ( · ) · ( ) s sÄ = Är r n n , 34n2 2

( · ) ( · ) · ( ) s sÄ = Är r n n , 35n2 2

( · · ) { ( · ) · ( · ) · ( ) · ( ) ·

[ ( · ) ] · [ ( · ) ] · } ( )

s s s s s s

s s

Ä = Ä + ´ Ä ´

+ - Ä -

v w n v n n w n v n w n

v v n n w w n n

1

22

. 36

n

Wecan thus formulate transformation rules for vectors, and tensor products thereof, to express how they arealtered by the collective dephasing, as a function of the direction n of thefluctuating, externalfield:

⟶ ( · ) ( )

r r n n 37n

⟶ { ( · ) ( · ) ( ) ( )

[ ( · ) ] [ ( · ) ]} ( )

Ä Ä + ´ Ä ´

+ - Ä -

v w n v n n w n v n w n

v v n n w w n n

1

22

. 38

n

The local vectors that determine the decomposition on the right-hand side of (38) form an orthogonal basisconsisting (in the first subsystem) of the direction n of themagnetic field, the vector orthogonal to the planespanned by n and v , and the vector orthogonal to these; the same holds for the second subsystemwhen v isreplaced by w. The decisive parameters are the angles · q = n v vcos v and · q = n w wcos w .

Notice that if either one of v and w is parallel or orthogonal to n, some of the terms in (38) disappear. Fornow,we assume that ∣ ∣q< <0 cos 1v and ∣ ∣q< <0 cos 1w . In this case, we can introduce two orthonormalbases of 3 as

{ } ( · ) ( ) ( )q q

=- ´⎧⎨⎩

⎫⎬⎭s s sv v n n

nv n

, ,sin

, ,sin

, 39v v

1 2 3

{ } ( · ) ( ) ( )q q

=- ´⎧⎨⎩

⎫⎬⎭t t tw w n n

nw n

, ,sin

, ,sin

, 40w w

1 2 3

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New J. Phys. 18 (2016) 073010 EGCarnio et al

allowing us to re-express (38) as

⟶ ( )

åaÄ Ä

=

v w s t2

, 41i

ii i

1

3n

with the coefficients

{ } { } ( )a a a q q q q q q=, , sin sin , 2 cos cos , sin sin . 42v w v w v w1 2 3

4.Generating correlations by collective dephasing

In this sectionwe investigate towhat extent the collective dephasingmap (23) can generate or increase thecorrelations between the subsystems. The present section extends the analysis of few special cases provided in thetheoretical treatment of the experiment reported in [6] to a complete picture.

We know themap is separable, hence it cannot create entanglement; however, it contains stochasticity, andtherefore can create classical correlations between subsystems, thereby increasing the correlation rank L of theinitial state. For this reason, we focus on the analysis of the correlations in the asymptotic state, based on thecorrelation rank. As discussed in section 2.1, a correlation rank of >L dmin can be interpreted as awitness fornon-zero discord. Furthermore, we can exclude that the thereby detected discordant states can be generated byapplying a local operation to a zero-discord state [17]. For two-level systemswe have = =d d 21 2 , and, thus, themaximal correlation rank for zero-discord states is = =L d 2min . States with L=3 or L=4 are consideredstrongly correlated, since their correlations are beyond the reach of any zero-discord states, and neither can beattained by states whose discordwas generated by a local operation.

4.1. Initially uncorrelated statesWebegin by considering an initially completely uncorrelated state, i.e., a product state r r r= ÄA B0 . In theFano form (5) this reads

( · ) ( · ) ( ) ( ) s sr = + Ä + Är r r r r r1

2

1

2, , . 43A B A B A B0 2 2

Thismeans that the initialβmatrix is b = Är rA B0 and the initial rank (7) is =L 10 , which is consistent withthe state having no correlations [6].

Application of the collective dephasingmap, assuming that n does not coincide with the direction of theBloch vectors rA and rB of the respective reduced systems, yields

( ) ( · ) ( · ) ( ) år r= Ä=

⎛⎝⎜

⎞⎠⎟r n n r n n v w

d, ,

2, 44n

A Bi

ii i1 0

1

3

with the set of orthonormal vectors (40)

{ } ( · ) ( )q q

=- ´⎧⎨⎩

⎫⎬⎭v v vr r n n

nr n

r r, ,

sin, ,

sin, 45A A

A A

A

A A1 2 3

{ } ( · ) ( )q q

=- ´⎧⎨⎩

⎫⎬⎭w w wr r n n

nr n

r r, ,

sin, ,

sin, 46B B

B B

B

B B1 2 3

and coefficients (42)

{ } ( )q q q q q qÎd r r r r r rsin sin , 2 cos cos , sin sin . 47i A B A B A B A B A B A B

The set of accessible final states is described by four real parameters, namely the norms = rrA B A B, , of thereduced Bloch vectors and their angles qA B, with themagnetic field direction.Within the set of densitymatrices,which is afifteen-dimensional real space, this represents ameasure-zero set, therefore we cannot synthesizearbitrary states by adjusting the parameters of the collective dephasingmap or of the initial state. In thefollowing, however, we specify conditions that lead to a given value of the correlation rank L after collectivedephasing.

As discussed before, the vectors { }v v v, ,1 2 3 , defined in (45), span an orthonormal basis of 3. Followingsection 2.2, we can apply a local unitary operatorUB (associatedwith a rotationmatrixOB, such that =w vOB i i)which does not change the properties of interest.We apply this rotation only to the second subsystem, asdescribed by the unitary operator = ÄU UA B. This transforms the correlationmatrix b1of the asymptoticstate(44) to thematrix b b¢ ~1 1, where b¢1, expressed in the basis { }v v v, ,1 2 3 , reads

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New J. Phys. 18 (2016) 073010 EGCarnio et al

( )bq q

q qq q

¢ =⎛

⎝⎜⎜

⎞

⎠⎟⎟

r r

2

sin sin 0 00 2 cos cos 00 0 sin sin

. 48A BA B

A B

A B

1

When expressed in the same basis, the Bloch vectors after dephasing are given by

⟶ ( · ) ( ) ( )

q=r r n n r0, cos , 0 . 49A B A B A B A BT

, , , ,

n

Thefinal correlation rank reads, using (7),

( ) ( )b q q= + ¢ - Än nL r r1 rk cos cos 50A A B B1 1

( )q q

q q= +

⎛

⎝⎜⎜

⎞

⎠⎟⎟1 rk

sin sin 0 00 0 00 0 sin sin

. 51A B

A B

The correlation rank of the final state can only have two values, determined by the relative orientations of thereduced Bloch vectors and the direction of themagnetic field [6]:

• =L 11 , if themagnetic field is parallel to one of the reduced Bloch vectors: r n;A B,

• =L 31 , if themagnetic field has a different direction than both reduced Bloch vectors: r nA B, .

The physical interpretation is immediate if we realize that, when the Bloch vector of a subsystem coincideswith themagnetic field direction, that subsystem is in an eigenstate of the localHamiltonian · swn 2, and isconsequently invariant under the action of themap. In this case, this atom can be treated separately, and thecollective dephasing acts only on the remaining atoms, whose Bloch vector differs from n. Let us suppose that=n r rA A: the atomdescribed byA is no longer affected by the collective dephasing process and the final state

can bewritten as

( ) ( ) ( ) r r r r r= Ä = Ä , 52n nA B A B1

which is an uncorrelated product state with correlation rank L=1. As is best seen from the integralrepresentation(18), this observation is easily generalized for systems of >N 2 atoms. In general, for productstates involving arbitrary local states ( )r i of qubits i that satisfy

[ · ] ( )( ) ( )s r =n , 0, 53i i

the dephasing operation factorizes:

( ) ( ) ( )( ) ( ) r r r rÄ = Ä , 54n ni i

where ρ is an arbitrary quantum state of the remaining qubits. The states that satisfy the factorizationcondition(53) encompass all incoherentmixtures of eigenstates of · ( )sn i and the identity operator.

In short, application of (23) to an uncorrelated state produces a state with high correlation rank, as long asr nA B, , but cannot reach themaximal value of L=4. Yet, is it possible to transform the resulting L=3 state

into an L=4 state by a consecutive, second application of the collective dephasingmap? Sincewe considercollective dephasing to the asymptotic state, the second applicationwould not have any effect unless we changethe direction of the externalfield. This can be shown as follows: the application of the collective dephasingmapalong the direction ¹ ºm n e2 yields a state with aβmatrix

( ) [ ( ) ( )] b q qq q

~ Ä + Ä + Äe e e e e ecos cossin sin

2.m m m

A BA B

2 2 2 1 1 3 3

If we define q q=q cot cotA B, the correlation rank of the state is given by = +L M1 rk2 , where

( · ) ( )bq q

= - Äe m m mM qsin sin

55A B

22

2

[ ( ) ( · ) )] ( ) ( ) ( ) = Ä - Ä + Ä + Äe e e m m m e e e eq 56m m m2 2 2

21 1 3 3

where q qsin sinA B is non-zero if r nA B, .We can nowobserve that, when =m e2, the termproportional to qvanishes and = Ä + Äe e e eM 1 1 3 3, yielding =L 3;2 alternatively, we can solve the equation

( ) ( ) Ä + Ä =e e e e 0m m1 1 3 3 with the constraint =m 1 to obtain( )[( ( · ) ) ( ( · ) ) ( ) ( )]= - Ä - + ´ Ä ´e e m m e e m m e m e mM q 2 2 2 2 2 2 2 , which yields again =L 32 .

Finally, one can show that thematrixM is not of full rank, e.g. by solving =Mdet 0 as a function of m given theparameter q. In any case, using linear algebra or by direct computation, one can verify that there exists atmost ameasure-zero set of directions, other than the one of the first dephasing process, alongwhich the correlationrankwill not increase. Therefore, an L=1 state can be converted into an L=4 state by a twofold application ofthe collective dephasingmap, if the direction of themagnetic field is different for the second dephasing.

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New J. Phys. 18 (2016) 073010 EGCarnio et al

4.2. Initial states of correlation rank L=2After studying the influence of the collective dephasing on an initially uncorrelated state =L 10 , we now turn tothe discussion of an initial state with low correlation rank =L 20 . This state is consideredweakly correlatedsince it contains correlations that are compatible with either a state of zero discord, or a non-zero discord statethat can be created froma state of zero discordwith a local operation—recall the discussion at the end ofsection 2.1.We consider states withmaximallymixed reduced subsystems, i.e., systemswith vanishing reducedBloch vectors, which can bewritten as ( )r b0, 0,0 0 . Since these states are diagonal in the basis of Bell states,they are also calledBell-diagonal states. In the Fano form (5) they arewritten as [25]

( · · ) ( ) s sr = + Äv wd1

4, 570 4

where ¹d 0 and the initialβmatrix was expressed in terms of its singular value decomposition (32), asb = Äv wd0 . Positivity of the state r0 requires that ∣ ∣ d 1. States of this formhave always zero discord [6], ascan be easily seen based on the cummutativity of the local operators [22], as discussed in section 2.1.

Application of the collective dephasingmap (38) produces another Bell-diagonal state, ( )r b0, 0,1 1 ,where the correlationmatrix b1 can bewritten (analogously to the case discussed before (48), and possibly aftersuitable, local orthogonal transformations that do not alter the correlation properties) as

( )bq q

q qq q

~⎛

⎝⎜⎜

⎞

⎠⎟⎟

d

2

sin sin 0 00 2 cos cos 00 0 sin sin

. 58v w

v w

v w

1

In order tofind the correlation rank of the final state, by virtue of (7), we need to determine ( )b= +L 1 rk1 1 .Except for the trivial case d=0, we notice that also in this case the rank depends on the geometric features of thestate, namely on the angle between themagnetic field direction and the left- and right-singular vectors of b0: thestate has correlation rank [6]

• =L 11 if either ( v n and ^w n), or ( w n and ^v n);

• =L 21 if v n or w n (but neither ^v n nor ^w n);

• =L 31 if ^v n or ^w n (but neither v n nor w n);

• =L 41 otherwise.

Aweakly correlated ( =L 20 )Bell-diagonal state can therefore be transformed into a state withmaximalcorrelation rank (L=4), provided that themagnetic field direction does not coincide with some very specificchoices, determined by the geometric characterization of the initial state.

4.3. ApplicationsGeneration ofWerner states—an important class of states formany applications of quantum information theoryis given by the family ofWerner states

∣ ∣ ( ) ( )r = Y ñáY + -- -s s1

4, 59W

4

with the singlet state ∣ (∣ ∣ ∣ ∣ )Y ñ = ñ ñ - ñ ñ- 0 1 1 0 2 , and - < s 11

3[18].Written in the Fano form (5), this

state has ( )b = - s s sdiag , , .Collective dephasing can generate such a state from an L=2Bell-diagonal state of the form(57)whenwe

choose the values for d, qv and qw that solve the following systemof equations:

( )q qq q

==

⎧⎨⎩k

ksin sin 2cos cos ,

60v w

v w

where = -k s d and ¹d 0. Summation of the two equations yields

( ) ( )q q q q q q+ = - = kcos cos sin sin cos 3 , 61v w v w v w

which proves that the systemof equation (60) admits solutions in the variables { }q q,v w only if ∣ ∣ k 1 3. Inparticular, when =k 1 3, the solutions lie on the lines q q=w v (see figure 1, left). The solutions are then foundas:

( ) ( )q q= = » arcsin 2 3 0.955, 62v w

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New J. Phys. 18 (2016) 073010 EGCarnio et al

or

( ) ( )q q p= = arcsin 2 3 . 63v w

We remark here thatwe have to simultaneously respect the conditions ∣ ∣ k 1 3 and ∣ ∣ d 1 (for positivityof the state), which leads to the observation that, based on the present approach, it is impossible to generateWerner states with >s 1 3. In fact, theWerner states are separable precisely when s 1 3 and entangledwhen >s 1 3. This is consistent with the fact that the initial state was separable and that themap cannot createentanglement.

The same procedure can be followed to produceWerner-like states

∣ ∣ ( ) ( )r a a= ñá + -a s s1

4, 644

where ∣añ is one of the other Bell states:

∣ (∣ ∣ ∣ ∣ )

∣ (∣ ∣ ∣ ∣ )

∣ (∣ ∣ ∣ ∣ ) ( )

Y ñ= ñ ñ + ñ ñ

F ñ= ñ ñ + ñ ñ

F ñ= ñ ñ - ñ ñ

+

+

-

1

20 1 1 0

1

20 0 1 1

1

20 0 1 1 . 65

These states haveβmatrices with the eigenvalues [25]:

∣ ∣ ( ( ))∣ ∣ ( ( ))∣ ∣ ( ( )) ( )

Y ñáY -F ñáF -F ñáF -

+ +

+ +

- -

s s s

s s s

s s s

0, 0, diag , , ,

0, 0, diag , , ,

0, 0, diag , , . 66

The values of qv and qw that yield those states are found by solving the system

( )q qq q

== -

⎧⎨⎩k

ksin sin 2cos cos .

67v w

v w

In fact, wewill show in the next section that theβmatrix of the asymptotic state always has one non-degenerate eigenvalue, with the associated eigenspace spanned by n, and one doubly degenerate eigenvalue, withthe plane orthogonal to n as eigenspace. Thus, the threeWerner-like states are distinguished by the choice of thedirection n of themagnetic field. In all previous expressions we had arbitrarily chosen to identify n with the y-direction, =n e2, thus the non-degenerate eigenvalue always appeared as second diagonal element,corresponding to ∣F ñ+ . In general, we identify

Figure 1.Pairs of angles that solve the system (60) for theWerner state (59) (left), and the system (67) for theWerner-like state (64)(right), in the case =k 1 3. The blue (darker) circles are the contour lines of q q =sin sin 2 3v w (first equation in (60) and (67)), theorange (lighter) ones, those of q q = cos cos 1 3v w (second equation). The solutions of the two systems (60) and (67) are then theintersections of the curves.

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New J. Phys. 18 (2016) 073010 EGCarnio et al

∣∣ ⟺∣ ( )

F ñ ºF ñ ºY ñ º

-

+

+

n e

n e

n e . 68

1

2

3

In order to generate theseWerner-like states we then solve the systemof equation (67) tofind the relative anglesbetween n, v and w , and rotate the basis accordingly, to obtain the target state. In order to solve (67), the sameconditions on k hold, but the solutions for =k 1 3 now lie on the lines q p q= -w v (seefigure 1, right), with qv

as before: ( )q = arcsin 2 3v or ( )q p= arcsin 2 3v . For the cases with ¹k 1 3 therewill be twice asmany solutions, which can be easily found numerically.

For the genuine (singlet-based)Werner state it does notmatter howwe choose to order the eigenvalues oftheβmatrix (by appropriate choice of the co-ordinate system relative to themagnetic field orientation), sincethey are all negative. This expresses these states’ invariance under arbitrary collective unitary rotations of theform ÄU U , which effectively rotate the co-ordinate system. In general, the set of Ä ÄU U -invariant statescan be identified as a completely time-invariant set of states under the collective,multipartite dephasingoperation, for arbitrary directions of themagnetic field [15].

Generating resource states for entanglement distribution—separable states canfind applications in quantuminformation protocols such as entanglement distribution [27–29]; for experimental realizations based onseparable states see [30–32]. This protocol allows to increase the entanglement between two parties byexchanging a carrier particle, which is not necessarily entangledwith the two parties. It is, however, necessarythat the combined state contains non-zero discord between the two parties and the carrier. In fact it was recentlyfurther shown that discord alone is not sufficient, and that discordantmixtures of two pure product states arenot able to distribute entanglement [33]. By producing states with >L 2, we can ensure that these cannot bewritten asmixtures of two product states.

Let us consider the following example of an initial two-qubit state [29]

∣ ∣ ( )r y y= ñá +-

pp1

4, 69AB s s 4

with ∣ ∣ ∣ ∣ ∣y ñ = ñ ñ + - ñ ñs s0 0 1 1 1s .When =s 1 2, these are theWerner-like states, (64), generated by∣F ñ+ ; in particular, the protocol can be realizedwith an initially separable state, which further imposes p 1 3.As explained in the previous paragraph, these states can be generated from aweakly correlated (L=2) state byaction of the collective dephasingmap. States with other values of s are not accessible as themap only producesstates with aβmatrix that contains a doubly degenerate eigenvalue. In the three-partite scenario consideredhere, theWerner-like state above can be generated on systemsAB if the initial state contains no correlationswithsystemC, which is prepared in amaximallymixed state r = 2C 2 . By virtue of(54) the qubitC is invariantunder the collective dephasing dynamics. The state r rÄAB C can thus be generated by collective dephasing of aweakly correlated L=2 state of zero discord for the choice of parameters =s 1 2 and, e.g., =p 1 3. For theseparameters the same state was shown to be useful for entanglement distribution [29].

Entanglement activation from uncorrelated states through collective dephasing—a three-partite scenario is alsoconsidered for the entanglement activation protocol [34, 35], where initial discord between systemsA andB isconverted into entanglement across the partition ∣AB M , whereM is an initially independent ancilla system thatinteracts via a local unitary operationwith systemB. Formally, this can be interpreted as a localmeasurementprocess of the systemB, whereM is ameasurement apparatus. Thereby, the entanglement of the combinedquantum state describing the apparatus and the systembefore readout (state reduction) of the apparatus is thendirectly linked to the discord of the initial bipartite state ofA andB [34].

An all-optical experimental realization of this protocol confirmed this general theoretical result bymodelingall three subsystemsA,B andM, as polarization qubits [36]. Based on the collective dephasing process, we cannow extend this protocol such that also initial zero-discord states or even completely uncorrelated states can beused to generate entanglement. To this end, we subject an initial three-qubit state of the form

∣ ∣ ( )r r= Ä ñá1 1 70AB M0

to a three-partite collective dephasing process in direction n, wherewe choose ∣ ñ1 as an eigenstate of the operator· sn , and rAB may be an arbitrary state of qubitsA andB.While in the previous sectionwe havemade use of the

local invariance under the dephasing process, see (54), for the trivial case of the identity operator, here weprepare the third qubit in an eigenstate of the localHamiltonian to achieve the same effect. Application of thecollective dephasingmap to the state(70) yields

( ) ( ) ∣ ∣ ( ) r r= Ä ñá1 1 . 71n nAB M0

By following the conditions provided in sections 4.1 and 4.2, initial product states of Bell-diagonal zero discordstates rAB can be easily converted into strongly correlated, non-zero discord states ( ) rn

AB . Based on theentanglement activation protocol, any local unitary operation onBM necessarilymust generate distillableentanglement across the ∣AB M partition, and the amount of generated entanglement is lower bounded by the

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New J. Phys. 18 (2016) 073010 EGCarnio et al

discord of the state ( ) rnAB [34, 35]. In the present sectionwe only considered the asymptoticmapping of the

collective dephasingmap (23) after long times, but the results hold also for intermediate times t, as described bythemap (22).

In a recent experiment [37], local noise processes were used to generate discordant states (see also [6]), whosediscordwas subsequently activated into entanglement.

Local quantum interferometry—to end this section, we briefly comment on the use of correlated separablestates in the context of local precisionmeasurements [38]. Thefield of quantummetrology is dedicated todevelopingmethods that allow to estimate unknown parameters with the highest possible precision, often byexploiting the usage of entangled states [39–41]. Consider a setup, inwhich two incoming particles are sent intotwo different arms of an interferometer before being jointlymeasured. The parameter to be estimated is a localphase shiftj, imprinted on one of the particles by an unknownHamiltonian ( )H i (with fixed, non-degeneratespectrum) through the unitary operation ( ) ( )=j

j-U ei Hi i. In aworst-case scenario, the localHamiltonianmay

commutewith the quantum state, which therefore renders any estimation of the phase shift impossible. This,however, is only possible if the quantum state has zero discord. In general, theworst-case estimation precision ofj is quantified by a particularmeasure of discord [38].

Consequently, the successful estimation of the phase in the above scenario requires the presence of non-zerodiscord between the two particles. Using the results provided in sections 4.1 and 4.2, the required discord can beeasily generated by submitting the two parties to a collective dephasing process before sending them into theinterferometer.

4.4. SummaryIn the present section, we have discussed the behaviour of the correlation rank under the action of the collectivedephasing dynamics.We have seen in section 4.1 that completely uncorrelated initial states (L=1) can betransformed into states with L=3 by a single application of collective dephasing, or into states of L=4 by adouble application, provided that themagnetic field direction is changed before the second dephasing.Weaklycorrelated states with L=2 are transformed into states with themaximumcorrelation rank L=4 formostchoices of themagnetic field direction, as discussed in section 4.2. Since any value of L 3 implies the presenceof non-zero discord, the strongly correlated states that can be conveniently generated by the collective dephasingprocess allow for direct applications in a series of tasks fromquantum information theory, as shown insection 4.3.

5. Protecting correlations under collective dephasing

5.1. Initial states of arbitrary correlation rankIn the previous sectionwe sawhow the collective nature of the ensemble averaged dephasing process, induced bya spatially homogeneous, fluctuating externalfield, can be used to generate strongly correlated quantum states,which have direct applications for specific tasks in quantum information processing.

We now extend the discussion from the initially weakly correlated states to initial states with an arbitrarycorrelation rank L 40 .We thus shift our focus from the generation of strongly correlated quantum states, tothe robustness of the correlations under the action of collective dephasing. The objective, then, is to control themagnetic field orientation, such that the preservation of these correlations under the collective dephasing isensured.

As in the previous section, we assumeBell-diagonal states (thus vanishing reduced Bloch vectors rA B, ). Thematricesβ and b - Är rA B have the same rank unless r rA B is a singular value ofβwith left- and right-singularvectors r rA A and r rB B, respectively, and only in these cases can the correlation rank be reduced by non-zeroreduced Bloch vectors [recall (7)]. The effect of the collective dephasingmap on arbitrary initial states is easilyinvestigated based on (28).

Furthermore, before application of themap (38), we employ unitary transformations to bring theβmatrixof the initial state into diagonal form ( )b = d d ddiag , ,0 1 2 3 . Its three eigenvalues then parametrize [25] the Fanoform (5) of the initial state:

· · ( ) å s sr = + Ä=

⎛⎝⎜

⎞⎠⎟e ed

1

4, 72

ii i i0 4

1

3

where { }ei i is the standard basis in 3. This transformation neither affects the correlation rank of the initial state,as discussed earlier, nor that of thefinal state, unless we start from a rank-2 state of the form (57) andwe apply amagnetic field in the direction v or w, as dealt with in section 4.2. In all other cases, there is at least one rank-1matrix of the form Äv wi i in (32) that is transformed, according to (38), into a rank-3matrix by the collectivedephasingmap.

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New J. Phys. 18 (2016) 073010 EGCarnio et al

Application of the collective dephasingmap (38) yields

⟶ {( )[ ] } ( )

åb b = - Ä + Ä + Ä=

a a b b n nd

n n2

1 2 , 73i

ii i i i i i0 1

1

32 2

n

with the normalized vectors

( )=-

-=

´

-a

e nb

e nn

n n1,

1. 74i

i i

i

ii

i2 2

Notice that all the aiʼs and biʼs belong to the plane orthogonal to n and aremutually orthogonal:· = "a b i0,i i . Thismeans that wemaywrite all orthonormal bases { }a b,i i , with = ¼i 1 3, as a rotation

about n of, e.g., { }a b,1 1 by an angleji (in this casej = 01 ):

( )j j= +a a bcos sin 75i i i1 1

( )j j= - +b a bsin cos . 76i i i1 1

By direct substitutionwe obtain

[ ]( ) ( )j jÄ + Ä = + Ä + Äa a b b a a b bcos sin , 77i i i i i i2 2

1 1 1 1

which shows that Ä + Ä = Ä + Ä "a a b b a a b b i,i i i i 1 1 1 1 . Theβmatrix of the final state, given by (73), canthus be rewritten as

{( )[ ] } ( )åb = - Ä + Ä + Ä=

a a b b n nd

n n2

1 2 . 78i

ii i1

1

32

1 1 1 12

Because { }a n b, ,1 1 is an orthonormal basis of 3, the above expression is a spectral decomposition of b1wheretwo eigenvalues appear, one of which is two-fold degenerate:

( ) ( )ål ==

n d n , 79i

i i11

32

( ) ( ) ( )ål = -=

n d n1

21 . 80

ii i2

1

32

From these expressions the invariance of the eigenvalues under cyclic permutation of the indices is evident. Theβmatrix of the state after application of themap is given by

( ) ( )[ ] ( )b l l= Ä + Ä + Än n n n a a b b , 811 1 2 1 1 1 1

wherewe could have equivalently chosen { }a b,2 2 and { }a b,3 3 , instead of { }a b,1 1 (we have used above theirequivalence under rotation of the basis about n, which is orthogonal to all of them).

To summarize, wefind

( ) ⟶ ( ( ) ( ) ( )) ( )

b b l l l= ~ n n nd d ddiag , , diag , , . 820 1 2 3 1 2 1 2

n

From the above explicit expressions for the eigenvalues we can immediately verify that the trace of theβmatrix ispreserved under the application of themap (recall the general result, (27)):

( ) ( )å å åb b= + ´ - = == = =

d n d n dtr 21

21 tr . 83

ii i

ii i

ii1

1

32

1

32

1

3

0

5.2. Geometric descriptionBell-diagonal states of two qubits allow for a simple geometric description of their correlations properties [25],whichwewill employ in the following. Recall that Bell-diagonal states are unambiguously characterized by theirβmatrix (72). Since there exists a unique unitary operator that diagonalizes thismatrix without changing thestateʼs correlation properties, we can parametrize the correlation properties of Bell-diagonal states by the threereal eigenvalues ofβ. It follows that this unitary operator defines an isomorphism thatmaps each Bell-diagonalstate to a point ( ) = Îd d d d, ,1 2 3

3. Positivity implies that the space of densitymatrices is isomorphic to atetrahedron whose vertices represent the four Bell states [25]:

∣ ∣ ( )∣ ∣ ( )∣ ∣ ( )∣ ∣ ( ) ( )

Y ñáY ~ = - - -F ñáF ~ = -F ñáF ~ = -Y ñáY ~ = -

- -

- -

+ +

+ +

B

B

B

B

1, 1, 1

1, 1, 1

1, 1, 1

1, 1, 1 . 84

0

1

2

3

In this tetrahedronwe distinguish two regions: a central octahedron with vertices = ¼e i, 1 3i , whichcontains the separable states, and the four remaining corners, containing the entangled states. Because each

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New J. Phys. 18 (2016) 073010 EGCarnio et al

corner has one of the Bell states as its vertex, we call the other entangled states in that corner ‘Bell-like’.We noticethat allB0-like entangled states have negative co-ordinates, while theBi-like states, with = ¼i 1 3, have onlythe i-th co-ordinate negative, likeBi itself.

Infigure 2we showhow the classes of states we are interested in are represented in the tetrahedron . Theorigin is themaximallymixed state r = 4m 4 , and the rotation axes of the octahedron correspond to the rank-2states. The rank-3 states lie on the squares obtained by intersecting the octahedronwith the planes orthogonal toits rotation axes. Both these sets of states (rank-2 and 3) havemeasure zero inside the tetrahedron.Moreover, asdiscussed in section 2, all entangled states have rank-4, but the converse is not true.

5.3. Geometric action of themapWenow turn to describing the action of themap (38) in this geometrical framework. In particular, the subset ofstates that is accessible by themap is defined by the conservation of the trace of theβmatrix (section 3.3) and thedouble degeneracy in the co-ordinates of the asymptotic state (section 5.1).

In the followingwe adoptGreek letters to indicate cyclic indices, i.e., { }n n n- +1, , 1 denotes an evenpermutation of { }1, 2, 3 .Whenever we apply themap (38)we transform to the reference framewhere thedirection of themagnetic field is the unit vector ne of the standard basis. This allows for themost generaldescription, which is independent of the choice of reference frame. Consequently, the set of points with twoequal co-ordinates is a plane defined as

{( ) } ( )P = Î =n n n- +d d d d d, , : , 851 2 3 1 1

while the conservation of btr defines another plane

{( ) } ( )G = Î + + =d d d d d d k, , : , 86k 1 2 3 1 2 3

where, inside the tetrahedron, - k3 1. Because the trace of theβmatrix is conserved at all times, thetrajectory of each point lies on the Gk plane defined by the initial co-ordinates. However, since the final statemustbelong to Pn , every state will asymptoticallymove to the intersection line defined by ÇG Pnk , whose existence isguaranteed by the fact that nP ^G "n k, ,k (figure 3(a)).

The position of thefinal state on the line defined by ÇG Pnk is determined by themagnetic field direction.Let us use again ( )l n1 and ( ) ( ( ))l l= -n nk 22 1 as the co-ordinates of the final states.We obtain

( ) ( ) ( ) ( )b ll

~ +-

+n n n- +n en

e ek

2. 871 1

11 1

Since k is fixed by the initial state, the co-ordinates depend only on n.We remark here that, when the initial state isB0-like, the intersection lines never cross the octahedron of

separable states, because the planes Gk are parallel to the octahedron face opposite toB0 (figure 3(a)).We deducethat the states in theB0 cornermove entirely in that corner. On the other hand, the entangled states in the othercornersmove inside their respective corner, butmay also enter the octahedron. This entails significant

Figure 2.Points of interest inside the tetrahedron of Bell-diagonal states: the red lines represent the rank-2 states (points with onenon-zero co-ordinate), the planes represent the rank-3 states (points with one vanishing co-ordinate) and the green lines represent therank-3 states that are reachable via themap (28) (points with one vanishing and two equal co-ordinates).

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New J. Phys. 18 (2016) 073010 EGCarnio et al

implications for the entanglement preservation of the initial states from the different corners, and ultimatelyenables the effect of time-invariant entanglement forB0-like states, as wewill discuss in further detail later in thismanuscript.

5.4. Loss of coherenceIn the geometric representation of Bell-diagonal states, planes of equal purity are characterized by constantvalues of

( ) ( ) ( ) = + + +d d d d1

41 , 881

222

32

which correspond to concentric spheres around the tetrahedronʼs center, themaximallymixed state.From theKraus representation(22) of the collective dephasingmap n

t ,0, it follows immediately that thepurity of thefinal state is upper bounded, for all t 0, by the initial stateʼs purity:

{ ( ) ( )} { ( ) ( )} ( )r r r rt ttr tr 0 0 . 89

Inmost cases, however, the collective dephasing tends to push states towards the center of the tetrahedron. Thepreservation of the initial purity can be achieved for time-invariant states.Whether there exist states outside ofinvariant subspaces for which the purity is preserved is unclear, but seems unlikely since the entire dynamicsmust lie on the circumference generated by the intersection of the sphere defined by = const and the plane Gk.

5.5. Impact on the correlation rank anddiscordWenow focus on the correlation rank of the final states, which directly translates into the number of non-vanishing co-ordinates of a point in the tetrahedron.

The states in theB0 corner never exhibit a vanishing co-ordinate, as displayed infigure 2.Moreover, asdiscussed in the previous section, the states in this corner never enter the octahedron, hence they never intersectthe subsets of rank-2 and rank-3 states.We conclude that the correlation rank in the states in theB0 corner isrobust and alwaysmaximal, which also implies that these states are always discordant, and cannot be createdusing local operations on classical states.

For the remaining states, i.e., those in the octahedron and in the other three corners, there is atmost oneintersection between the reachable rank-2 or rank-3 states, and the accessible final states defined by the line

ÇP Gn k, where - k1 1 for those states. Themagnetic field direction that yields such a state is found bysolving either ( )l =n 01 or ( )l =n 02 under the normalization constraint =n 1. Using theKronecker-

Figure 3.Geometric interpretation of the action of the collective dephasingmap (28) on the Bell-diagonal states (72) contained in thetetrahedron. The octahedron, inscribed into the tetrahedron, represents the separable states.

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New J. Phys. 18 (2016) 073010 EGCarnio et al

Capelli theorem [42] it is possible to show that there exists atmost a one-parameter family of directions thatsolve such equations.We conclude that, after the action of the collective dephasingmap (38), the state has almostalways a correlation rank equal to 4. This includes both scenarioswhere a state was already initially discordantand this property is preserved throughout the dephasing dynamics, as well as when the discord is generated bythe action of the collective dephasingmap, recall also the experiment reported in [6].

5.6. Protection of initial two-qubit entanglementLet ρ be a Bell-diagonal state of two qubits and ¼p p0 3 its eigenvalues in the basis of Bell states. If

( )= ¼p p pmaxmax 0 3 , then the concurrence of ρ can bewritten as [43]

( ) ( ) ( )r = -C pmax 0, 2 1 . 90max

Inside the tetrahedron , where states are described by vectors Îd 3, the concurrence has an isotropic form,

( ) ∣ ∣ ( ) å= -⎛⎝⎜

⎞⎠⎟d d

1

2max 0, 1 , 91

ii

which can be interpreted as the distance to the faces of the octahedron of separable states, parametrized by theequation ∣ ∣å =d 1i i . Following the state space from these faces to either one of the corners of the tetrahedron,the concurrence increases from zero to one.

Let us remind here that the collective dephasingmap, being separable, cannot create entanglement. In ourgeometrical framework, thismeans that points initially in the octahedron aremapped onto ÇPn .Suppose that an initial state is inside the octahedron, i.e., it has vanishing concurrence (91): ∣ ∣ å = d 1i i1

3 . Theconcurrence (91) in thefinal state, where the co-ordinates are ( )( )l n1 2 from (79) and (80), is then∣ ( )∣ ∣ ( )∣l l+n n21 2 andwe have the following chain of inequalities:

( ) ∣ ∣ ∣ ∣( ) ∣ ∣ ( ) å å å å å+ - + - == = = = =

d n d n d n d n d1 1 1, 92i

i ii

i ii

i ii

i ii

i1

32

1

32

1

32

1

32

1

3

wherewe have used the normalization condition =n 1, which implies "n i1,i2 .

Since the states in the ∣Y ñ- -corner are constrained tomove on planes parallel to the octahedron face oppositetoB0, these states remain on iso-concurrence planes at all times (comparefigures 3(a) and (b)).More specifically,as shown in [15], the points in this corner are characterized by negative co-ordinates: "d i0,i . Hence, theconcurrence (91) for these states can be rewritten as

( ) ( ) ( ) å= - - = - -d d k1

2

1

2

1

21 , 93

ii

wherewe have used b = å =d ktr i i and, for entangled states in this corner, < -k 1.In the other corners of the tetrahedron only one of the co-ordinates is negative, andwe denote it with ν:nd 0, where d represents the initial state.We then have

( ) ∣ ∣ ( ) ( ) ( ) å= - + = - - + + = - + -n

n

n n n n= -

+

- +

⎛⎝⎜

⎞⎠⎟d d d d d k d

1

21

1

21

1

21 2 . 94

ii

1

1

1 1

Let d f represent thefinal, asymptotic state of the system after collective dephasing, and let us once again denotethe negative components of d and d f with a subscript ν, i.e., we have nd 0 and nd 0f , respectively. Since theeigenvalue ( )l n2 has a double degeneracy, but the points in the ¼B B1 3 corners can only have a single negativecomponent, the negative eigenvalue after collective dephasingmust necessarily be

( ) ( )ål= =nn

n

= -

+

nd d n . 95i

i if

11

12

The concurrence of the final state then reads

( ) { }

( ) ( )

å

= - + -

= - + -

n

n

n

= -

+⎧⎨⎩⎫⎬⎭

d k d

n d

1

2max 0, 1 2

1

2max 0, 1 1 2 . 96

ii i

f f

1

12

It follows that ( ) ( ) ⟺ = = nd d n ef , i.e., the initial concurrence of states in the three triplet corners canonly be preserved if themagnetic field is chosen along the special direction that characterizes the respectivecorner.Moreover, (96) allows for an estimation of the loss of concurrence due to small deviations from ne .

Let us conclude this section by highlighting the relationship between the different types of states and therespectivemagnetic field directions: the concurrence of aBν-like state is conserved under collective dephasing ifand only if themagnetic field points in the ne direction, while inB0-like states the concurrence is always

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New J. Phys. 18 (2016) 073010 EGCarnio et al

conserved, independently of the direction of themagnetic field. Thismeans that, independently of the fielddirection, one quarter of the entangled Bell-diagonal states shows preserved concurrence, while an additionalquarter of states that depend on themagnetic field direction shows the same effect.

Let us briefly remark on the applicability of the results of the present section to states that are not Bell-diagonal, i.e., which have non-vanishing reduced Bloch vectors. To this end, let us consider a state

( )r br r, ,A BG G and the local operation τ such that [ ] ( )r t r b= 0, 0,BD G BD [44]. The operation thattransforms a general two-qubit into a Bell-diagonal state, i.e., sets the reduced Bloch vectors to zero, is a localoperation that cannot increase entanglement [44], therefore ( ) ( )r rC Cs

BDsG , where the behavior of the lower

bound ( )rC sBD under collective dephasing can be controlled, e.g., bymanipulating themagnetic field direction

as described above.Despite the possibility to achieve conservation of entanglement in a Bell-diagonal state, the concurrence of

themore general state can still decrease due to the time evolution of the reduced Bloch vectors. Perfectconservation of concurrence in a general bipartite state is consequently only possible when concurrence ispreserved in the corresponding Bell-diagonal state, andwhen additionally both initial reduced Bloch vectors areparallel to themagnetic field direction, which is the only possible case where the reduced Bloch vectors of theinitial and asymptotic state coincide, as per (37).

5.7. Time-invariant states versus time-invariant entanglementLet us review the insights gained in the previous section on the entanglement dynamics under collectivedephasing employing the geometric representation of the tetrahedron picture. The state ∣Y ñ- , represented asone of the corners of the tetrahedron, is completely unaffected by the collective dephasing process because it isan eigenstate of theHamiltonian of the system for everymagnetic field direction. The state is therefore time-

invariant. This is true also for all theWerner states ∣ ∣ ( ) Y ñáY + -- -s s144 . In the geometric framework, these

states constitute the rotation axis of the tetrahedron that passes throughB0 and the origin.For the other Bell states, as well as the respectivemixtures with the identity (Werner-like states), the previous

remarks only hold provided themagneticfield is chosen according to the association rule (68), since in that casethe respective Bell state becomes an eigenstate of theHamiltonian. This explains and generalizes theobservations reported in [45], where a dynamical evolution equivalent to the collective dephasing along the z-axis is applied to theWerner andWerner-like states. Those obtained from the ∣Y ñ states are then categorized as‘robust’, while those from ∣F ñ are ‘fragile’. In fact, as we have shown above, only theWerner state (59) is robustunder collective dephasing in anyfield direction, while the otherWerner-like (64) states are robust only when aspecific direction is chosen [15].

The preservation of entanglement within decoherence-free subspaces [46–48]—which in the present case,due to the absence of aHamiltonian evolution, coincidewith the above-mentioned time-invariant subspaces—is a rather obvious phenomenon: if the state does not evolve in time, then its properties are naturally alsoconserved. Our analysis, however, points out the non-trivial possibility of time-invariant entanglement of statesbeyond time-invariant subspaces [15], see also [49]. In fact, as pointed out above, all states in theB0 cornerremain at a level of constant concurrence, even if they do not belong to the axis of time-invariant (Werner) states.

5.8. SummaryIn the present sectionwe have studied the behavior of initially correlated quantum states under the influence ofthe collective dephasing evolution. To this end, we have extended the analysis of the correlation rank L undercollective dephasing to states of arbitrary initial correlation rank. Furthermore, we have presented a convenientgeometrical setting to describe Bell-diagonal states and their evolution under collective dephasing. Thisgeometrical interpretation allowed us to demonstrate that highly correlated (L=4) initial states retain theirstrong correlations, as quantified by L, under the collective dephasing dynamics for almost all choices of themagnetic field direction.

Combining the a geometric description of the collective dephasing dynamics with a geometric interpretationof the concurrence—an entanglementmeasure—wehave characterized afinite set of states showing time-invariant entanglement despite their incoherent evolution.

5.9. Extension to themultipartite caseSome features of bipartite states can be directly generalized to amultipartite setting. For example, the generalizedWerner states, i.e., those states that are invariant under ÄU N-operations [18, 50], are the fixed points of thecollective dephasingmap for any number of qubits [15]. In [15], the decay and the time-invariant preservation ofmultipartite entanglement properties was also provided. A compelling explanation for themechanism thatenables this phenomenon in amultipartite case is, however, presently unavailable. In the followingwe allude totwo possible approaches towards achieving this goal.

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New J. Phys. 18 (2016) 073010 EGCarnio et al

The conservation of btr in bipartite states of qubits can be related to the overlap between the state of thesystem ( )r t and the singlet state ∣ ∣Y ñáY- - [15]. This is true for any state ∣fñ that is an eigenstate of the collectivedephasingHamiltonian for an arbitrary choice of themagnetic field direction. These eigenstates are themulti-qubit singlet states [51], which exist only for an even numberN of qubits, and are a family of

! [( )!( )!]+N N N2 2 1 linearly independent states. The overlapwith each of these states is an integral ofmotion, which in the bipartite case reduces to the conservation of entanglement for allmagnetic field directions.It is however not clear whether the overlapwith themulti-qubit singlet states is related to entanglement,especially considering thatwe observe conserved entanglement properties for any—and not just for an even—number of qubits.

Another approach originates from the observation that the collective dephasingmap describes asimultaneous rotation of all qubits, hence it has a set offixed points, the rotation axis, and the (hyper-)planesorthogonal to it aremapped onto (a subset of) themselves. Thefixed points of themap are theWerner states [15],and, in the tetrahedron of Bell-diagonal states, the planes orthogonal to the rotation axis contain the states withthe same amount of entanglement. In an analogous higher-dimensional picture, the set of states orthogonal tothe family ofmultipartiteWerner statesmay lead to an interesting set of integrals ofmotion.

6. Conclusions

To summarize, we provided a detailed analysis of the impact of a collective dephasing process on the correlationproperties of bipartite states. Based on theKraus representation of the dephasing process [15], we providedconditions that enable the generation of states with high correlation rank and non-vanishing quantumdiscordfromuncorrelated or only weakly correlated states.

Using an intuitive geometric representation of the state space and the collective dynamics, we investigatedthe entanglement dynamics under collective dephasing. For initially entangled states, we provided conditionsthat ensure the complete preservation of the entanglement for all times under the dephasing dynamics.Surprisingly, this is possible even for large families of states that do not belong to time-invariant subspaces, i.e.,states that change in time due to the incoherent dynamics.While some of the results could be generalizedstraight-forwardly tomultipartite scenarios, a compelling picture describing time-invariantmultipartiteentanglement remains to be conceived.

Since collective dephasing represents one of the dominant sources of error formany experiments withtrapped atomic particles, we expect that the results derived in the present article can be readily harnessed in state-of-the-art setupswith trapped ions [6, 7] or ultracold atoms [8].

Note added—an experimental observation of time-invariant entanglement was reported in [52] aftercompletion of thismanuscript.

Acknowledgments

MGthanks theGermanNational Academic Foundation for their support. This project has received fundingfrom the EuropeanUnionʼsHorizon 2020 research and innovation programme under grant agreementNo641277.

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