Lorentz invariance of entanglement classes in multipartite systems
Marcus Huber, Nicolai Friis, Andreas Gabriel, Christoph Spengler, Beatrix C. Hiesmayr
aa r X i v : . [ qu a n t - ph ] J a n Lorentz invariance of entanglement classes in multipartite systems
M. Huber , N. Friis , A. Gabriel , C. Spengler and
B. C. Hiesmayr , University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna, Austria School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United King-dom Research Center for Quantum Information, Institute of Physics, Slovak Academy of Sciences, Dubravska cesta 9,84511 Bratislava, Slovakia
PACS – Entanglement classification
PACS – Quantum systems with finite Hilbert space
PACS – Special relativity
Abstract – We analyze multipartite entanglement in systems of spin- particles from a relativis-tic perspective. General conditions which have to be met for any classification of multipartiteentanglement to be Lorentz invariant are derived, which contributes to a physical understandingof entanglement classification. We show that quantum information in a relativistic setting requiresthe partition of the Hilbert space into particles to be taken seriously. Furthermore, we study ex-emplary cases and show how the spin and momentum entanglement transforms relativistically ina multipartite setting. In quantum many-body systems multipartite entangle-ment is a key feature. It plays a central role in a broadvariety of physical processes and occurs in various physi-cal systems. In quantum information processing it facili-tates quantum computation ( e.g.,
Ref. [1]), enables mul-tiparty cryptography ( e.g.,
Refs. [2, 3]) and quantum al-gorithms ( e.g.,
Ref. [4]). It appears in quantum phasetransitions ( e.g.,
Ref. [5]) and ionization procedures ( e.g.,
Ref. [6]). Recently even biological systems have raisedquestions as to whether multipartite entanglement mightbe responsible for their astonishing transport efficiency( e.g.
Refs. [7, 8]).The structure of multiparticle entanglement is however farmore complex than the well studied bipartite case (forthat see e.g.
Ref. [9]). Only recently first tools havebeen developed to answer the question of whether or nota given multi-body quantum state exhibits multipartiteentanglement (see, for example, Refs. [10–16]). Furtherprogress was made on experimentally implementable cri-teria in multipartite systems, first for many qubits in (a)
E-mail: [email protected] (b)
E-mail: [email protected] (c)
E-mail: [email protected] (d)
E-mail: [email protected] (e)
E-mail: [email protected]
Refs. [17–19] and then for arbitrary higher dimensionalsystems in Refs. [20, 21]. These criteria answer the ques-tion whether multipartite entanglement is present, but it isknown that there is a lot of structure beyond merely beingentangled. It was shown that there exist several inequiva-lent ways in which multipartite systems can be entangled(see e.g.
Refs. [22–25]). Questions concerning the num-ber of such entanglement classes and their classificationremain unanswered in general. Furthermore, the possiblephysical implications of these entanglement classes is notfully understood, and addressing this issue will serve as amotivation for this letter.Before we continue to discuss relativistic entanglementlet us first briefly review the formal definition of genuinemultipartite entanglement in a finite dimensional Hilbertspace. Any n -partite pure state that can be written as atensor product | Ψ n i = | Ψ B i ⊗ | Ψ B i (1)with respect to some bipartition B | B is called bisepa-rable. Pure states that are not biseparable with respectto any bipartition are called genuinely multipartite entan-gled. For mixed states this generalizes in a straightforwardway. Any mixed state that can be decomposed into a con-vex sum of biseparable pure states is called biseparable.p-1uber, Friis, Gabriel, Spengler and HiesmayrAny non-biseparable mixed state is called genuinely multi-partite entangled. For details on the intricacies that mixedstate biseparability provides consult e.g. Refs. [17–21]. Fa-mous examples of genuinely multipartite entangled statesare the GHZ (Greenberger, Horne, Zeilinger) state and theW state which will be used later.Studying entanglement in a relativistic framework takesquantum information one step further. In a relativisticsetting many new features appear which are not presentin a non-relativistic framework. Although a self-containeddescription of relativistic quantum information has not yetbeen formulated, the identification of observer indepen-dent quantities ought to be crucial to such an endeavor.It has been shown that the entanglement between thespins of two particles is not Lorentz invariant (see e.g. Refs. [26–32]). In a more general approach, taking intoaccount not only the spin entanglement, but also the mo-menta of the relativistic particles, it has recently beenshown, that in bipartite systems Lorentz invariance of en-tanglement can generally only be claimed for the Hilbertspace partition into individual particles (for details seeRef. [33]). The entanglement in other partitions is ob-server dependent (i.e. it can be transferred between them).The effect of relativistic entanglement transformation isnot limited to only systems of two particles. In this paperwe show that in a relativistic multi-particle setting thedifferent classes of multipartite entanglement are indeedobserver independent. We also provide a general frame-work to identify which conditions have to be met by apossible classification of multipartite entanglement in or-der to be Lorentz invariant. Furthermore, we show howthe entanglement in different partitions changes throughLorentz transformations, while it is preserved for the par-tition into individual particles, which is thus singled outnaturally.We first start with an illustrative example of a genuinelymultipartite entangled system of three spin- particles,observed from two different reference frames. We thendiscuss in detail the effect of Lorentz transformations onthe entanglement of this exemplary system. Finally weprove that the results also hold in a general setting.Let Alice, Bob, and Charlie be inertial observes, resting ina common frame of reference, who share a quantum statewhich is separable between spins and momenta | Ψ ABC i = | ψ mom i ⊗ | φ spin i , (2)where | ψ mom i is the state of the momenta and | φ spin i isthe state of the spins. Let us further assume that theyshare a multipartite entangled spin state, e.g. the wellknown GHZ state | φ spin i = 1 √ | ↓↓↓i + | ↑↑↑i ) , (3)with which they want to perform a quantum protocol(e.g., quantum secret sharing as in Ref. [3]). Now a rela- (cid:65) (cid:66)(cid:67)(cid:82) (cid:122) (cid:121)(cid:120) (cid:49)(cid:50)(cid:48)(cid:176) (cid:49)(cid:50)(cid:48)(cid:176)(cid:49)(cid:50)(cid:48)(cid:176) Fig. 1: Scheme of particle and observer motion in the referenceframe of Alice, Bob, and Charlie tivistic observer (Robert) is moving perpendicular to theplane of their shared quantum state in the z -direction (seeFig. 1 for details). From his point of view the situationof course looks different, as he observes a Lorentz-boostedstate ρ Λ . Let us also, for the sake of simplicity and with-out loss of generality (regarding our result) assume thatwe have momentum eigenstates, i.e. P µi | ψ ( p , p , p ) i = p µi | ψ ( p , p , p ) i and only three possible momenta. Letour three sharp momenta be denoted by p A , p B and p C ,where | ~p A | = | ~p B | = | ~p C | . The Lorentz boost to Robert’sframe induces Wigner rotations of the spins, i.e. for aseparable momentum state the state (2) is transformed to | ψ (˜ p , ˜ p , ˜ p ) i ⊗ ( U local ( p , p , p , δ )) | φ spin i , (4)where U local ( p , p , p , δ ) = U ( p , δ ) ⊗ U ( p , δ ) ⊗ U ( p , δ ) , (5)are local unitary operations on the spin vector, and theWigner rotation angle δ is given bytan δ = sinh η sinh ξ cosh η + cosh ξ , (6)where η and ξ are the rapidities corresponding to thevelocities u of Robert and v of the three particles relativeto Alice’s, Bob’s and Charlie’s frame of reference, givenby tanh η = u and tanh ξ = v (for more details on Wignerrotations consult e.g. Refs. [33, 34]). The assumptionof sharp momenta, which is a common approximation(see, e.g., Ref. [28]), allows us to apply a single Wignerrotation for each particle momentum.The entanglement class of the spin state remains invari-ant, if the momentum state is separable as in this casethe spin state undergoes only a local unitary transforma-tion. So this yields the first result on the conditions ofentanglement classes in a relativistic setting:p-2orentz invariance of entanglement classes in multipartite systems Condition 1 :Different Lorentz invariant classes of multipartite en-tanglement need to be inequivalent under local uni-tary operations.That this is a necessary condition follows simply fromthe fact that even for separable momentum states theLorentz boost acts as a local unitary transformation. Soif two local unitary equivalent states were in a differententanglement class the class membership would fail to beLorentz invariant. Also it would be rather meaningless,as any local basis change (e.g. relabeling the measure-ment apparatuses) could change the classification of aninvestigated state. This condition is satisfied for all pre-viously introduced entanglement classification schemes.Those are usually defined via SLOCC (stochastic local op-erations and classical communication)-inequivalent states(see, e.g. , Ref. [22, 25]), which incorporate local unitaryoperations. Now we focus on the case where the momen-tum state is not separable, e.g. | ψ ( p , p , p ) i = X i α i | Π i ( p A p B p C ) i , (7)where Π i denotes a permutation of p A p B p C (an even per-mutation for even i and an odd permutation for odd i )and the sum is taken over all possible permutations. Inthis case the spin state transforms asΛ[ ρ spin ] = X i | α i | · (8) U local (Π i ( p A , p B , p C ) , δ ) ρ spin U † local (Π i ( p A , p B , p C ) , δ ) , where ρ spin = | φ spin ih φ spin | and Λ denotes the appropriaterepresentation of the Lorentz transformation. Equiva-lently to the bipartite case the spin state becomes moremixed and entanglement decreases as we have plotted indetail for our example state in Fig. 2. In a more generalsetting of unsharp momenta, the sum would have to bereplaced by an integral and the coefficients α i would beexchanged with an appropriate distribution function inmomentum space, (implicitly) containing all additionalinformation about the state (e.g., position and orbitalangular momentum). Now it is evident which furthercondition has to be met for a classification in order to beobserver independent: Condition 2 :Any convex combination of local-unitarily equivalentpure states defines a Lorentz invariant class of genuinemultipartite entanglement.To prove this condition it is sufficient to look at the mostgeneral setting and work out which Wigner rotations the
Fig. 2: Illustration of the violation of inequality (A.1)from Ref. [20] for the state P i ( − i √ | Π i ( p A p B p C ) i ⊗ (cid:16) cos( α ) | ↓↓↓i + sin( α ) | ↑↑↑i (cid:17) . This inequality detects genuinemultipartite entanglement and is maximally violated by theGHZ state with an arbitrary phase α . This can be seen forthe two points maximally violating the inequality, which cor-respond to the standard GHZ state and the same state withan extra relative phase of π . The plane corresponds to equal-ity and the area above this plane is detected to be genuinelymultipartite entangled. The absolute value of violation of thisinequality gives a lower bound on a measure of genuine multi-partite entanglement, which is tight for all pure GHZ states (fordetails see Ref. [35]). This figure visualizes how the amount ofgenuine multipartite entanglement decreases as a function of δ . Lorentz boost induces. The most general state in the rest-ing frame may be an arbitrarily mixed state: ρ = X i q i | Ψ i mom+spin ih Ψ i mom+spin | , (9)where | Ψ i mom+spin i = P k α ik | ψ k mom i i ⊗| φ k spin i i . In this casethe state of the spins is given by ρ spin = X i q i Tr mom ( | Ψ i mom+spin ih Ψ i mom+spin | )= X i q i X k | α ik | | φ k spin i i h φ k spin | i | {z } σ ki . (10)If we look at the state from a perspective from which aLorentz boost induces a Wigner rotation, the reduced spinstate is given asΛ[ ρ spin ] = X i q i X k | α ik | · (11) U k local ( p , p , · · · , p n , δ ) σ ki U k † local ( p , p , · · · , p n , δ ) . Certainly, if the initial spin states | φ k spin i i lies within acertain equivalence class C , then their convex combination ρ spin ∈ C as well. Defining an entanglement class byp-3uber, Friis, Gabriel, Spengler and Hiesmayr(a)(b) Fig. 3: Illustration of how the entanglement for different parti-tions transforms depending on δ . To measure entanglement weused the m-concurrence introduced in Refs. [16, 36] and opti-mized it using the approach from Ref. [37]. In (a) the entangle-ment of exemplary state (2) with momentum state (7), where α i = ( − i √ , and spin state (3) is plotted and in (b) the spinstate is replaced by a W state | W i = √ ( | ↓↓↑i + | ↓↑↓i + | ↑↓↓i ).Each graph corresponds to the partition depicted in the sameorder on the right, where the top (red) dots represent the spinsand the bottom (green) dots represent the momenta of thethree particles. Condition 1 it is evident that also Λ[ ρ spin ] ∈ C , sinceall U k local σ ki U k local † must remain within C , thus provingthat Condition 2 is necessary and sufficient for a class ofgenuine multipartite entangled states to be observer in-dependent. In the more general case of unsharp momentathe statement remains true, as the Wigner rotations areonly induced by the momentum part of the system. Inthis case the discrete sum would change into a continuousintegral over such local unitarily rotated states. Thiswould surely make the analysis harder (although withthe methods introduced in Ref. [38] it can even be doneanalytically in arbitrary dimensions), but the sufficiencyof the condition of course remains unchanged.The resulting conditions do not unambiguously de-termine the classification of multipartite entanglement.They are however a necessary criterion for physicalconsistency of any possible classes. Indeed the previouslydefined entanglement classes from Ref. [22] and the more general SLOCC classification from Ref. [25] meet thiscondition. In Ref. [39] the authors introduce an exper-imentally accessible classification scheme of multiqubitentanglement, which incorporates the one introduced inRef. [22], and is indeed also observer independent.In our exemplary figures we illustrate how the amountof entanglement changes through the Lorentz transfor-mation. In Fig. 2 we plot a lower bound of a measureof genuine multipartite entanglement, which is tight inthe case of GHZ states. It is based on the non-linearentanglement witness introduced in Ref. [20] and canbe experimentally ascertained using only few localmeasurements (in the case of three qubits only 9 localmeasurement settings, as opposed to a full state to-mography requiring 63). That it also serves as a lowerbound to an entanglement measure quantifying genuinemultipartite entanglement was shown in Ref. [35].While the classification itself does not depend on anyobserver’s knowledge of the entire state, Robert’s abilityto unambiguously determine the class of the reduced den-sity matrix of the spins may. Whereas the class remainsinvariant for all observers, the separability propertiesmay change, as depicted in Fig. 2 and Fig. 3 (a) and(b). In particular, if the momentum state is entangled,the reduced spin density matrix of a previously genuinelymultipartite entangled state may become biseparablethrough the Lorentz transformation. As long as the initialstate is separable with respect to spin and momentum,the entanglement of the reduced density matrix of thespins can never increase (only decrease in accordanceto the main theorem of Ref. [26]). Since biseparablestates can result from convex mixtures of any class ofgenuinely multipartite entangled states, Robert necessar-ily needs to also have information about the momentumstate (which allows for an unambiguous decomposition)in order to determine the class unambiguously in this case.In conclusion we have shown which conditions have tobe met for an entanglement classification scheme to beLorentz invariant. We have further argued why knowl-edge of the momentum state is helpful to a complete clas-sification of genuine multipartite entanglement of the spinstate, which is not surprising, as it is well known thatthe reduced spin density matrix does not transform co-variantly under Lorentz boosts. Nonetheless the proposedclassification scheme is retaining its Lorentz invariance ifthe momentum state is unknown, i.e. all inertial observerswill assign a given state to an entanglement class or, atmost, to one of the corresponding convex subsets of thisclass. This provides a general framework which paves theway for entanglement classification beyond n qubits and,at the same time, imposes an intuitive physical under-standing upon the distinction into different entanglementclasses.p-4orentz invariance of entanglement classes in multipartite systems ∗ ∗ ∗ We would like to thank R. A. Bertlmann, F. Hipp,S. Radic, H. Schimpf and T. Adaktylos for productivediscussions. M. H., A. G. and C. S. gratefully acknowledgethe Austrian Fund project FWF-P21947N16. A. G. issupported by the University of Vienna’s research grant.N. F. acknowledges support from EPSRC [CAF GrantNo. EP/G00496X/2 to I.Fuentes]. B. C. H. acknowlegesthe EU project QESSENCE.APPENDIXThe explicit form of inequality (A.1) from Ref. [20] writ-ten in terms of local spin observables for three spin- sys-tems reads (cid:12)(cid:12)(cid:12)(cid:12) h σ x σ x σ x − σ x σ y σ y − σ y σ x σ y − σ y σ y σ x i + 14 h σ y σ y σ y − σ x σ x σ y − σ y σ x σ x − σ x σ y σ x i (cid:12)(cid:12)(cid:12)(cid:12) − q h P +1 P +2 P − ih P − P − P +3 i− q h P +1 P − P +3 ih P − P +2 P − i− q h P − P +2 P +3 ih P − P +2 P +3 i (A.1)where P ± := (1 ± σ z )2 . (A.2) REFERENCES[1]
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