Controllable Gaussian-qubit interface for extremal quantum state engineering
aa r X i v : . [ qu a n t - ph ] J u l Controllable Gaussian-qubit interface for extremal quantum state engineering
G. Adesso , S. Campbell , F. Illuminati , and M. Paternostro School of Mathematical Sciences, University of Nottingham,University Park, Nottingham NG7 2RD, United Kingdom School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom Dipartimento di Matematica e Informatica, Universit`a degli Studi di Salerno, CNR-SPIN, CNISM, Unit`a di Salerno,and INFN, Sezione di Napoli - Gruppo Collegato di Salerno, Via Ponte don Melillo, I-84084 Fisciano (SA), Italy (Dated: November 6, 2018)We study state engineering through bilinear interactions between two remote qubits and two-mode Gaussianlight fields. The attainable two-qubit states span the entire physically allowed region in the entanglement-versus-global-purity plane. Two-mode Gaussian states with maximal entanglement at fixed global and marginalentropies produce maximally entangled two-qubit states in the corresponding entropic diagram. We show that asmall set of parameters characterizing extremally entangled two-mode Gaussian states is su ffi cient to control theengineering of extremally entangled two-qubit states, which can be realized in realistic matter-light scenarios. PACS numbers: 03.67.Mn, 42.50.Dv, 42.50.Pq, 03.67.Hk
The structure of quantum correlations within a given sys-tem depends strongly on the dimension of the state spaces ofits constituents. The relation between correlations and systemdimensionality becomes particularly relevant when consider-ing states of bipartite systems whose parties are defined onHilbert spaces of di ff erent dimension. An extreme instance isthat of a bipartite compound consisting of a continuous vari-able (CV) system with infinite-dimensional state space and adiscrete system with finite D -dimensional Hilbert space. Thissituation is particularly relevant in quantum communication,where many advantages come from the use of interfaces be-tween light fields and matter-like systems [1], which are thebasis of recent important experimental demonstrations [1–3].Ground-breaking architectures for communication, such asquantum repeaters [2], rely on interfaces and their e ffi cientimplementation is based on the availability of such techno-logical primitives . A “desideratum” of any reliable interfacewould be its ability to connect systems of di ff erent dimen-sionality so as to transfer important physical features (such asentanglement) across the interfaced systems while faithfullyrespecting their inherent structure. Among the proposals forquantum interfaces put forward so far [1, 3], those aiming attransferring entanglement from a light field encoding a CVstate to a qubit system [4, 5] are appealing for implementa-tions in a number of physical settings, ranging from cavity-and circuit-quantum electrodynamics (QED) [4, 6] to polarmolecules close to superconducting resonators [7] or quantumdots and color-centers in diamonds in defect-microcavitiesand photonic crystals. Yet, a problem in dealing with mis-matched system dimensionality is the loss of any clear relationbetween the purity and entanglement properties of the CV re-source and those of an addressed two-qubit system. Control-ling that the state hierarchy properties of the fields are inher-ited by the qubits is thus di ffi cult.In this paper we discuss a simple and flexible CV-to-qubitmap, based on the use of handy Gaussian-light resources,which works as a powerful tool for quantum state-engineeringof the steady-state of two remote qubits. The rich variety of mapped two-qubit states faithfully respects the state hierar-chy induced by the degrees of local / global purities and en-tanglement, whose structure is inherited from the Gaussianresources. Extremal bipartite Gaussian states (in terms of en-tanglement and mixedness) give rise to equally extremal two-qubit states. Our results thus shed new light on the mecha-nisms at work at the interface between discrete and CV sys-tems. Pragmatically, we show that our map represents an im-portant step in the long-sought task of distributing channelsfor quantum communication [1] and is implementable withresources available in many optical labs and relying on exisit-ing mature technology in cavity- and circuit-QED [6, 8]. The map. – To keep the discussion as concrete as possi-ble, in the following we will adopt the language of cav-ity QED, although our scheme is completely general andindependent of any specific physical setting. Two remotesingle-mode cavities contain a qubit each and are coupledto a broad-band two-mode driving field prepared in an en-tangled state ̺ [4, 5]. Qubit A is coupled to cavitymode 1 via the well-known Jaynes-Cummings Hamiltonianˆ H A = ω ( ˆ q ˆ σ A + ˆ σ A ˆ p ), where { ˆ q j , ˆ p j } are the quadratures ofmode j = , σ k is the x ( y ) Pauli matrix of qubit k = A , B . A similar model holds for qubit B and mode 2. Here, {| i , | i} k are the logic states of qubit k and ω is the Rabi fre-quency. The driven cavities leak photons at rate κ and em-body local environments for the qubits. We are interested inthe control and manipulation of the entanglement and purityproperties of the two-qubit system by means of its interac-tion with the bosonic field. States of CV systems can bedescribed in terms of a (generally infinite) hierarchy of mo-ments of their quadratures. Without loss of generality, thefirst moments are hereafter set to zero as they play no rolein characterizing entanglement and mixedness. The secondmoments are used in the covariance matrix (CM) V of el-ements Tr[ ̺ { ˆ x α , ˆ x β } ] ( α, β = , ..,
4) with ˆ x = { ˆ q , ˆ p , ˆ q , ˆ p } .Gaussian states are completely specified by the knowledgeof the first and second moments [9]. The CM of any two-mode state can be brought in the form V = V C C T V ! with V = diag[ a , a ] ( V = diag[ b , b ]) and C = diag[ c + , c − ]. In or-der for ̺ to be a physical state, we must satisfy both V > V + i Ω ≥ Ω = L k = , i ˆ σ k the symplectic matrix. In the bad-cavity limit where modes 1and 2 reach their stationary state sooner than any changes inthe qubit-field subsystems and within the first Born-Markovapproximation, the state of qubits A and B evolves accordingto the master equation (ME) [5] ∂ τ ̺ AB = X j , k = d jk ( ˆ O j ̺ AB ˆ O k −{ ˆ O k ˆ O j , ̺ AB } / , (1)where ˆ O j = ˆ σ jA ⊗ B for j = , O j = A ⊗ ˆ σ j − B for j = ,
4. The Kossakowski matrix D of elements d jk reads D = γ ( V + i Ω ), where γ = ω /κ is the e ff ective qubit-fieldcoupling strength, κ is the cavity decay rate, and τ = γ t is thedimensionless time [11]. The map in Eq. (1) is completelypositive i ff D ≥ [5], which is equivalent to the uncertaintyprinciple for the field CM. Thus the mapping holds for any(Gaussian or non-Gaussian) two-mode state with bona fide CM. Here, we consider only Gaussian states (generally mixedand asymmetric), bearing in mind that our results hold also formaps driven by non-Gaussian states with the same CM [10].In Ref. [12] we study the dynamics of the two-qubitsystem. Here we deal with the features of ̺ AB ( ∞ ),which is found by setting ∂ τ ̺ AB ( τ ) = ̺ . By calling ̺ i j , kl = h i j | ̺ AB ( ∞ ) | kl i (with | i j i and | kl i states of the two-qubit basis and i , j , k , l = , ̺ , = [( ab − a − b ) z + ( a + b ) ] /δ , ̺ , = ̺ , + az − ( a + b ) ] /δ , ̺ , = ̺ , + bz − ( a + b ) ] /δ , ̺ , = ̺ , = a + b )( c − − c + ) /δ , ̺ , = ̺ , = a + b )( c − + c + ) /δ with δ = abz , ̺ , = − ̺ , − ̺ , − ̺ , , and z = ( a + b ) − c + + c − ). Degrees of entropy and entanglement .– The mixedness (orlack of purity) of the state ̺ of a D -dimensional system can bequantified by the linear entropy S L ( ̺ ) = [ D / ( D − − Tr ̺ ),ranging from 0 (pure states) to 1 (totally mixed states).For a CV Gaussian state ̺ with CM V one has S L ( ̺ ) = − / √ det V . Both for two-qubit and two-mode Gaussian states, separability is equivalent to positivityof the partially transposed density matrix [13]. The degreeof violation of such a criterion provides an entanglementmonotone, the negativity N ( ̺ AB ) = max { , k ̺ T A AB k − } [14],where T A stands for partial transposition with respect toqubit A and || · || is the trace-norm. For a two-mode Gaus-sian state ̺ one has N ( ̺ ) = max { , (1 − ˜ ν − ) / ˜ ν − } , where˜ ν − is the smallest symplectic eigenvalue of the partiallytransposed CM, ˜ ν − = (1 / √ ∆ − ( ˜ ∆ − V ) ] ,with ˜ ∆ = det V + det V − C . A two-mode Gaus-sian state is entangled if and only if ˜ ν − <
1. Weparameterize V by setting a = s + d , b = s − d and c ± = √ ( f d − h d ) − g ± √ ( f s − h d ) − g √ s − d with h d = (2 d + g )( λ +
1) and f x = x + ( g + λ − / x = d , s ) [15]. The Gaussian state (a) (b) FIG. 1: (Color online). (a) S L ( ̺ AB ) versus S L ( ̺ ) for 20000 randomstates. Darker (lighter) dots denote entangled (separable) qubit states.Points below the dashed line show states purified by our map. (b) N ( ̺ AB ) versus normalized field negativity for 15000 random states. ̺ (with purity S L ( ̺ ) = − g − ) is physical and entangledfor s ≥ , | d | ≤ s − , | d | + ≤ g ≤ s − λ ∈ [ − , s and d determine the properties of the reduced states ̺ having CM V according to S L ( ̺ k ) = − [ s + ( − k − d ] − with ( k = , g , d and s , N ( ̺ ) growsmonotonically with λ .What properties of the two-mode state are transferred tothe two-qubit system? Marginal properties are faithfully re-produced as S L ( ̺ A ( B ) ) = S L ( ̺ )[2 − S L ( ̺ )], where ̺ A ( B ) is the reduced state of qubit A ( B ). The proportional-ity between S L ( ̺ A ( B ) ) and S L ( ̺ ) entails that the state-symmetry is preserved by the map. However, one finds that S L ( ̺ AB ) = − / (3 a ) − / (3 b ) − a + b ) ξ / (3 δ ) (with ξ = [( a + b ) + c + + c − )]), which shows that the two-qubitmixedness is not a simple function of g alone, but dependsnontrivially also on s , d , and λ . In particular, in the al-lowed range of parameters, S L ( ̺ AB ) increases with the globalfield mixedness ( i.e. with g ) and its mean energy (param-eterized by s ), but decreases with λ : larger input entangle-ment at given entropy results in qubit states of higher purity.Notwithstanding the e ff ective non-unitary dynamics, when ̺ is pure (which occurs when a = b and c + = − c − = √ a − ̺ AB ( ∞ ) is also pure. In general, for a given field mixed-ness, S L ( ̺ AB ) cannot vary unconstrained. Analytically, wefind that S L ( ̺ AB ) admits tight upper and lower bounds thatdepend on the field mixedness S L ( ̺ ). This is further con-firmed by random numerical sampling [cfr. Fig. 1 (a) ]. Themaximum of S L ( ̺ AB ) at a given g is found by optimiz-ing over s , d and λ . This implies taking s ≫ , λ = − d =
0. The corresponding two-qubit states tend asymptoticallyto the Werner state ̺ WAB = p | Φ − i AB h Φ − | + (1 − p ) /
4, where | Φ − i AB = ( | i − | i ) AB / √ p = / (1 + g ), for which S max L ( ̺ AB ) = − + ( S L ( ̺ ) − − ] − . On the other hand, S min L ( ̺ AB ) at a given g is obtained for s = ( g + / λ = d = ( g − /
2. Such S min L ( ̺ AB ) is achieved by the productstates [( g − | ih | + ( g + | ih | ] / (2 g ), for which we have S min L ( ̺ AB ) = (2 / S L ( ̺ )[2 − S L ( ̺ )]. The protocol can alsobe used to realize state purification: one finds many mapped ̺ AB ’s whose mixedness is smaller than the input S L ( ̺ ) , evenfor totally mixed fields [see Fig. 1 (a) ]. Noticeably, two-qubitstates obtained from highly mixed ̺ ’s are separable whileentanglement arises in the region of moderate mixednesses.By introducing η = a + b )( c + − c − ) z ( a − b ) and µ = z ( a − b )(1 + η ) / , thetwo-qubit negativity at the steady-state reads N ( ̺ AB ) = max { , (2 /δ )[( a + b ) − δ/ + µ ] } . (2)In the physically allowed range of parameters, we have ∂ g N ( ̺ AB ) ≤ ∂ s ,λ N ( ̺ AB ) ≥ d isnon-monotonic. This implies that the two-qubit entangle-ment increases with the marginal entropies, decreases with theglobal mixedness and, at fixed global and marginal entropies,increases with λ . These are the very same patterns followedby N ( ̺ ) and therefore the map in Eq. (1) fully preservesthe qualitative structure of bipartite entanglement . Quantita-tively, however, N ( ̺ ) does not determine directly the two-qubit negativity. For a given value of N ( ̺ ), the correspond-ing ̺ AB range from separable to highly entangled. The behav-ior of N ( ̺ AB ) versus the negativity of randomly sampled CMs V is reported in Fig. 1 (b) . While N ( ̺ AB ) can vanish for ar-bitrarily entangled ̺ ’s, we find a maximum of the two-qubitentanglement N max ( ̺ AB ) = − + N ( ̺ )) + − that isachieved by pure states (both for fields and for qubits). Entanglement versus global mixedness. – Maximally entan-gled two-qubit mixed states (MEMS) are defined as thosemaximizing a given entanglement measure at any fixed valueof the global mixedness [16]. In the { S L ( ̺ AB ) , N ( ̺ AB ) } space,MEMS include the family of Werner states. The correspond-ing minimally entangled mixed states are just separable states.The Gaussian counterparts to MEMS (GMEMS) are two-mode mixed states with infinite entanglement, such as thetwo-mode squeezed thermal states, with a = b = √ g cosh(2 r ), c + = − c − = √ g sinh(2 r ) in the limit r → ∞ . We remark that,given all possible CMs, our scheme does not generate ev-ery possible two-qubit state. However, the set of ̺ AB ’s thatcan be engineered by our process does fill the entire regionof the { S L ( ̺ AB ) , N ( ̺ AB ) } diagram physically allowed to two-qubit states. This result is illustrated in Fig. 2 (a) where wereport the diagnostics of ̺ AB ’s obtained from random ̺ ’s.The upper bound to the physically allowed region includesstates ̺ WAB for which N max ( ̺ AB ) = [ − + p − S L ( ̺ AB )] / S L ( ̺ AB ) < /
9, and zero otherwise. A direct way to obtainsuch states is by maximizing λ , minimizing d and setting g = / p , which fixes the global purity of the two-mode re-source to be equal to the | Φ − i component of ̺ WAB . Interest-ingly, the field state associated to such parameters is preciselya GMEMS with g = / p . The mapped two-qubit state con-verges to the corresponding boundary state when the squeez-ing in ̺ is large. We can thus engineer MEMS of tunableentanglement / purity by adjusting purity and squeezing in V .In Ref. [12] we present an additional study of N ( ̺ AB ) againstthe marginal entropies of the resource. Entanglement versus global and marginal entropies. – Avery refined characterization of entanglement is possiblein the space of global and marginal entropies, where allthe entangled two-mode Gaussian states lie in a narrowregion bounded by Gaussian least-entangled and Gaussianmaximally-entangled mixed states (GLEMS and GMEMS, re-spectively) [1]. These are achieved, at given s , d , and g (fix-ing the entropies) for λ = − λ = (a) (b) (c) FIG. 2: (Color online). (a) N ( ̺ AB ) versus S L ( ̺ AB ) of two-qubit statesobtained from 20000 random entangled two-mode Gaussian states ̺ ’s. The upper boundary (MEMS) includes Werner states. (b) N ( ̺ AB ) versus S loc and S for states obtained using 10 random sym-metric entangled ̺ ’s. (c) The same states as in (b) and the surfacesof maximum and minimum negativity at fixed entropies. respectively. Thus, by accessing only the restricted set of pa-rameters that determine the marginal and global purities of atwo-mode Gaussian state, one can pin-down its entanglementand bound it with the corresponding GLEMS and GMEMS.Surprisingly, no exact two-qubit counterpart to this structurewas known thus far. We will now show that, via Eq. (1),the two-qubit states ̺ AB inherit and enhance the propertiesof such CV states. As λ ∈ [ − , ∂ λ N ( ̺ ) | s , d , g ≥ ̺ AB . For ease of notation, wewrite S ≡ S L ( ̺ AB ) and N q ≡N ( ̺ AB ). From Eqs. (2) it fol-lows that N q is a function of s , d , λ and g ( s , d , λ, S ) such that ∂ λ N q | s , d , S = ∂ λ N q | s , d , g + ∂ g N q | s , d ,λ ∂ λ g | s , d , S = ∂ λ N q | s , d , g + ∂ g N q | s , d ,λ ( ∂ λ S | s , d , g /∂ g S | s , d ,λ ) ≥
0. This shows the existenceof two-qubit least-entangled and maximally-entangled mixedstates (QLEMS and QMEMS, respectively), at fixed globaland marginal entropies. These are obtained by mapping ofGLEMS and GMEMS, respectively. The values of s and d are set by S L ( ̺ k ) ( k = A , B ), while g is determined by themarginal and global entropies. For symmetric QMEMS with λ = S L ( ̺ k ) ≡ S loc = − s − and global mixedness S , one has g = / [1 − S loc + √ − S + S loc (4 + S loc )]. The corresponding N max q (see Ref. [12]) fixes the upper bound for all two-qubitstates obtainable by our process and compatible with the givenentropies. An analogous analysis holds for QLEMS.The behavior of the negativity versus global and marginalentropies shows that all the mapped two-qubit states lie in aquasi-bidimensional region [see Fig. 2 (b) ]. Even for mixedstates, N AB is almost perfectly a function of the global andmarginal entropies alone. By superimposing the boundarycurves corresponding to QLEMS and QMEMS to the numeri-cal analysis in Fig. 2 (b) we see that all the randomly generatedstates ̺ AB ’s accumulate in the tight interval between maxi-mum and minimum negativity [cfr. Fig. 2 (c) ]. Numerically,the negativities of QLEMS and QMEMS di ff er by less than0 .
04 e-bits close to the separability point, while in the regionof larger entanglement they practically coincide. Thereforethe field-qubit interface defines, within the set of entangledtwo-qubit states, the analogues to two-mode GMEMS andGLEMS. Moreover, the situation typical of the Gaussian caseis enhanced in the case of qubits, since in the latter the gap be-tween maximal and minimal entanglement is even narrower.The introduced mode-qubit dynamical interface preserves thehierarchy of entangled states in the entropic space: GMEMS(GLEMS) are mapped into QMEMS (QLEMS). Furthermore, ̺ AB weakly depends on λ so that N ( ̺ AB ) can be accuratelydetermined only controlling the engineered entropies of ̺ . Practical considerations.–
Cavity QED is a natural settingfor the implementation of the proposed scheme, due to theavailability of a variety of CV resources [18]. The feedingof an optical cavity containing a trapped atom with squeezedlight (bandwith of about 12 MHz) and the correspondingcontrolled light-atom interaction have been experimentallydemonstrated [8] for ( κ, ω ) / π ≃ (70 ,
20) MHz. This givesan e ff ective coupling rate γ ≃
10 MHz. The engineeringof MEMS from approximated GMEMS produced at highsqueezing is thus feasible. The swift progress in circuit-QEDmakes also such setting appealing [6]. A qubit is embodied bya superconducting quantum interference device (SQUID) atthe charge-degeneracy point. The qubit transition-energy canbe set by an in situ magnetic flux that modulates the Joseph-son energy so as to adjust the qubit-to-light coupling. Eachqubit is integrated in a full / half-wave waveguide split by in-put / output capacitances: we can thus consider two discon-nected regions of a coplanar waveguide, joined via indepen-dent input / output capacitive lines for the injection / leakage ofthe field resource or via a large Josephson junction. The res-onator quality-factor is typically well within a range appro-priate to our scheme (10 to 10 with ω ∼ . / thermal states have been produced by Josephson pa-rameteric amplifiers embodied by large junctions [20], whichis a very promising step towards the preparation of microwavestates belonging to the classes studied here. Our scheme maybe used to entangle collections of remote matter-like qubits,so as to achieve key resources in quantum technology. Conclusions.–
We have discussed a CV-to-qubit map that en-gineers two-qubit states spanning the region of physically al-lowed quantum correlations at fixed values of local and globalentropies. It incorporates the most relevant sources of noisea ff ecting a CV-to-qubit interface. Our results assure the re-alistic possibility for non-demanding production and controlof qubit states using o ff -line preparation of CV entangled re-sources and linear local interactions. The relations found be-tween two-mode fields and two-qubit states make our schemea basic predictive tool for light-matter entanglement transferand related implementations in quantum technology. Acknowledgments.–
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QUANTITATIVE ANALYSIS OF THE DYNAMICALCV-TO-QUBIT MAP
In this Section we provide further details on the map em-bodied by the master equation (ME) in Eq. (2) of the mainpaper. Rather than discussing the steady state properties ofthe mapped two-qubit state, here we address the full dynami-cal evolution of an initial preparation of qubits A and B .When written in the computational basis {| i , | i , | i , | i} AB , ̺ AB can be partitioned as ̺ AB ( τ ) = ̺ x AB ( τ ) + ̺ o AB ( τ ), where ̺ x AB ( τ ) = ⋆ ⋆⋆ ⋆⋆ ⋆⋆ ⋆ , ̺ o AB ( τ ) = ⋆ ⋆⋆ ⋆⋆ ⋆⋆ ⋆ . (S-1)Here, the symbol ⋆ is used to denote any potentially non-zero matrix entry. Obviously, ̺ o AB ( τ ) is not a density ma-trix. It is straightforward to check that the Liouvillian ˆ L inEq. (2) of the main paper keeps ̺ x AB ( τ ) disjoint from ̺ o AB ( τ ),so that starting from a state having elements only in ̺ x AB ( τ ),we are sure that ̺ o AB ( τ ) = ∀ τ . This is clearly the case for ̺ AB (0) = | i AB h | , and this observation yields a great simpli-fication enabling us to swiftly determine the dynamics of theentanglement transferred to the qubits. Moreover, such an ini-tial condition is the most favorable one to achieve the highestpossible qubit entanglement in the model under investigation.Starting from the ME in Eq. (2) of the manuscript, itis straightforward to write down the Bloch-like equationsfor the evolution of the two-qubit density matrix elementsand solve them against the dimensionless interaction time τ for any specific assignment of the resource’s covariancematrix (CM). Fig. 3 (a) shows the typical τ -dependent be-havior of such elements, where we have used the notation ̺ i j , kl = h i j | ̺ AB ( τ ) | kl i (with | i j i and | kl i states of the two-qubitbasis and i , j , k , l = , V . Interesting information can be extractedfrom the dynamical behavior of ̺ AB ( τ ). First, by fixing s , d and g in V , one can easily see that the dynamical evolutionof the negativity of the two-qubit system keeps N ( ̺ AB ) withina very narrow range of values, as λ is varied (we remind that λ ∈ [ − , (b) and (c) of the main paper, although here we are notimposing any restriction to the local / global entropies of theGaussian resource state. Fig. 3 (b) shows, in this sense, an un-usual scenario where the di ff erence between the negativity at λ = − λ = τ as a sort of curvilinearabscissa of the negativity-versus-linear entropy functions, wecan determine the trajectories of the two-qubit density ma-trices up to the boundary curve accommodating maximallyentangled mixed states and Werner states [see Fig. 4]. A sim-ilar study can be conducted in order to infer the dynamicalmapped-state entanglement against the resource negativity (asin Fig. 5). We refer to the captions of such figures for furtherdetails on the simulations. ENTANGLEMENT VERSUS MARGINAL MIXEDNESSES.
Here we classify the bipartite entanglement in the marginal-entropy space. Such a characterization is exact for pure states.In general, bounds on mixed-state entanglement can be de-rived for all mixed states compatible with given marginals.For two qubits, the upper bound to the physical set ofstates in this diagram defines maximally entangled states atfixed marginal mixednesses (MEMMS) [2], while separable(product) states exist for any pair of marginals [3]. Forequal marginal entropies, MEMMS are pure states. Their (a)(b)
FIG. 3: (Color online). (a)
Behavior of the density matrix elementsagainst the dimensionless interaction time τ for (randomly taken) s = . , d = . , g = . , γ = . λ = (b) Dynamicalbehavior of the negativity N ( ̺ AB ) against τ for the same values as inpanel (a) but λ , which is taken to grow from − . λ = − λ = FIG. 4: (Color online). Typical trajectories of the mapped two-qubitstates ̺ AB in the { S ( ̺ AB ) , N L ( ̺ AB ) } plane. The dashed boundary curveembodies maximally entangled mixed states (MEMS) and Wernerstates ̺ WAB . The dots show the evolution of negativity and global lin-ear entropy of the two-qubit state for five di ff erent choices of theGaussian resource parameters s , g , d and λ . The dimensionless in-teraction time τ grows as indicated by the arrow. The final dot ineach trajectory indicates the corresponding steady state. The trajec-tory superimposed to the MEMS curve is for a configuration of theparameters entering V which guarantees the asymptotic mappingto a ̺ WAB state.
Gaussian counterparts have been introduced in Ref. [1] anddubbed Gaussian-MEMMS, or GMEMMS. Their CM is char-acterized by having g = | d | +
1. Correspondingly, any de-pendence of the CM on λ disappears and one simply has c ± = ± √ (1 + max { a , b } )( − + min { a , b } ) [1].A random-state investigation of the performances of ourmap in such local-entropy space shows that the mapped two-qubit steady states do not fill the whole region allowed to anytwo-qubit state with given marginals. In particular, any at-tempt to reproduce MEMMS results in unphysical parame- FIG. 5: (Color online). Typical trajectories of the mapped two-qubitstates ̺ AB in the plane spanned by the (normalized) negativity ofthe Gaussian resource N ( ̺ ) and the target qubit-state negativity N ( ̺ AB ). The boundary curve follows the functional form found inthe main paper. The dots along each vertical line show the evolution of the two-qubit negativity for random choices of the Gaussian re-source parameters s , g , d and λ . The dimensionless interaction time τ grows as indicated by the arrow (at τ =
0, obviously, each two-qubit state is separable). The final dot in each trajectory indicates thecorresponding steady state. FIG. 6: (Color online). Two-qubit negativity against local marginalentropies. Physical two-qubit states lie below the top tent-like bound-ary (MEMMS). Such states are not reproducible using Gaussianstates as these produce ̺ AB ’s lying in the inner region, whose upperboundary contains states engineered via GMEMMS. ters for the driving field. In Fig. 6 we show such a numeri-cal exploration of the negativity N ( ̺ AB ) against the marginalentropies S L ( ̺ A ) and S L ( ̺ B ). The attainable qubit state filla restricted, pyramid-like region: their entanglement is nevermaximal at given marginals, except in the pure-state case of S L ( ̺ A ) = S L ( ̺ B ). We now aim at characterizing analyticallythe upper boundary of this set. Recalling that the negativity isan increasing function of λ and a decreasing function of g (atfixed a and b ), we can conclude that the boundary is obtainedby minimizing g , which gives g = | d | +
1. Such values of theparameters are exactly those characterizing a GMEMMS. Al-though the full range of entanglement is not achievable in thespace of the marginal mixedness, our map is such that the two-qubit states endowed with the maximum achievable entangle-ment at fixed marginals are those obtained by using GMEMS.Therefore, despite being unable to reproduce MEMMS, themaximum negativity achieved by our scheme corresponds tothe images of GMEMMS. Fig. 6 provides further details onthis point. We believe this is yet another clear indication ofthe powerful and faithful mapping embodied by by our sim-ple bilocal linear interaction model.
ANALYTIC EXPRESSION FOR THE NEGATIVITY OFQMEMS
We consider the space of negativity against global andmarginal mixedness. The aim of this Section is to give anexplicit form to the negativity of two-qubit most-entangledmixed states (QMEMS). By taking the case of a resourceembodied by a Gaussian most entangled mixed states (orGMEMS) with λ = S L ( ̺ A ) = S L ( ̺ B ) = − / s (we call S L ( ̺ AB ) ≡ S theglobal linear entropy), it is straightforward to check that thenegativity of the extremal two-qubit states in this space reads N max q = " − (2 + S loc ) + q (2 + S loc ) − S + + S loc − S − S + ( S loc − q (2 + S loc ) − S ! (S-2)if 9 S + − + S loc ) S loc < S loc q − S loc + S and N max q = g and the corresponding minimal value N min q valid for two-qubitleast entangled mixed states (QLEMS). [1] G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. Lett. ,087901 (2004); Phys. Rev. A , 022318 (2004).[2] G. Adesso, F. Illuminati, and S. De Siena, Phys. Rev. A68