arXiv:1809.08532v2 [quant-ph] 2 Apr 2019 o elkonoeainlmaue,sc steentan- the formation. as of such even That glement measures, unknown in operational below). remain well-known entanglement for of (3) still measures Eq. systems other see higher-dimensional of lat- (also monogamy The the in 1] given is, Definition definition [46]. the [46, to dimensions Ref. according [45] finite monogamous G-concurrence are all the ter in that [8, monogamous only entanglement know is squashed entan- we distillable and 8], one-way 6] Theorem the Theorem to [42– [8, addition shareability in glement this far, about So known 44]. is less to higher-dimensional much proved for systems However, are which [7–24]. monogamous 41], be 32, [8, one-way entanglement and distillable [40] entanglement squashed 39], [19, glement ain[2,cnurne[3 4,tnl 3] negativ- Tsallis- [33], [37], negativity tangle q extended for- convex-roof 34], of 36], [33, [35, entanglement ity concurrence the are [32], include measures mation entanglement These known the monogamous. all en- sys- almost multiqubit given For tems, a monogamous. not is or measure whether tanglement determine to is entanglement first the proved states. three-qubit Wootters for [7] and relation share- monogamy quantitative Kundu, ever This [7–31] Coffman, extensively [6]. explored entangle- since been itself of has mechanics traits relation fundamental quantum ability the of of and one is ment is features and un- monogamy so-called [5] parties, key the many law is its among This shared of correlations. classical freely One like be cannot of it [2–4]. was informa- that tasks it quantum lines many years processing of recent classical resource tion in key hand, the from other as the identified departure On that “the [1]. entire one thought.” as the its recognized mechanics early quantum was enforces the of it In trait mechanics characteristic quantum theory. of quantum days of phenomena intuitive † ∗ lcrncades [email protected] address: Electronic lcrncades [email protected] address: Electronic nrp fetnlmn 3] R´enyi- [38], entanglement of entropy nipratqeto ntesuyo ooayof monogamy of study the in question important An counter- most the of one is entanglement Quantum unu nageetadnta trbt fsm particula some of attribute an not and entanglement quantum nrp fetnlmn.Mroe,w hwta h ovxr on convex also the monogamous that are formation) show monogamou of we entanglement is Moreover, entan entanglement. of matrix measures of density bipartite entropy of reduced class important the the includes of function concave s h ento fmngm ihu nqaiis recent inequalities, without 2 monogamy of definition the use 1(08] u eut rmt h ocp htmngm o monogamy that concept the promote results Our (2018)]. 81 , eso htaymaueo nageetta npr biparti pure on that entanglement of measure any that show We 2 eateto ahmtc n ttsis nttt o Qu for Institute Statistics, and Mathematics of Department 1 nttt fQatmIfrainSine colo Mathem of School Science, Information Quantum of Institute ooayo h nageeto formation of entanglement the of Monogamy nvriyo agr,Clay let 2 N,Canada 1N4, T2N Alberta Calgary, Calgary, of University hniDtn nvriy aog hni070,China 037009, Shanxi Datong, University, Datong Shanxi α nrp fentan- of entropy uGuo Yu 1, ∗ n ia Gour Gilad and he usseso opst unu ytm and system, quantum composite a S of subsystems three on ain antstsyayrlto fteform the for- of of relation entanglement mea- any the many satisfy as cannot that mation, such showing entanglement, by of raised sures was monogamy sus that quantify to impression used entan- is the that of function give property it. the a of may but mea- not itself This entanglement is glement entanglement many of for 9–17]. monogamy valid not [7, How- is sures measured. 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If h is also tanglement left for A to share just with C. concave, i.e. H Clearly, this definition captures the spirit of monogamy of entanglement, and perhaps not surprisingly, can yield h[λρ1 + (1 λ)ρ2] λh(ρ1)+(1 λ)h(ρ2) (8) a family of monogamy relations similar to (1) by replac- − ≥ − α ing E with E for some α > 0 [46]. More precisely, a for any states ρ1, ρ2, and any 0 λ 1, then EF as ≤ ≤ continuous measure E is monogamous according to this defined in (5) is an entanglement monotone. definition if and only if there exists 0 <α< such that Theorem ∞ . Let E be an entanglement monotone for which h h Eα(ρA|BC ) Eα(ρAB)+ Eα(ρAC ), (4) , as defined in (6), is strictly concave; i.e., satis- ≥ fies (8) with strict inequality whenever ρ1 = ρ2, and ABC ABC 6 for all ρ acting on the state space with fixed 0 <λ< 1. Let also EF be as in (5). Then, dim ABC = d < (see Theorem 1 inH Ref. [46]). Note ABC ABC thatH (4) can be expressed∞ in a similar form as in (2) with 1. If ρ = ψ ψ is pure and (3) holds then B | ih | B B α α 1/α has a subspace isomorphic to 1 2 and f(x, y) = (x + y ) . However, we stress here that (4) H H ⊗ H is not a special case of (2) since the exponent factor α de- up to local unitary on system B1B2, pends on the underlying dimension of the Hilbert space. ψ ABC = φ AB1 η B2C , (9) Note that there is no a priori physical reason to assume | i | i | i that the exponent factor is universal and independent on where φ AB1 AB1 and η B2C B2C are pure the dimension. | i ∈ H AC | i ∈ H In this paper, we show that almost all entanglement states. In particular, ρ is a product state (and AC monotones are monogamous on pure tripartite states, consequently E(ρ )=0), so that E is monoga- and furthermore, those that are based on convex roof ex- mous on pure tripartite states. tension [e.g., entanglement of formation, see also Eq. (5)], ρABC are also monogamous on mixed tripartite states accord- 2. If is a mixed tripartite state and E ρA|BC E ρAB ing to our definition in terms of Eq. (3) [or equivalently F ( )= F ( ), then Eq. (4)]. Our results indicate that monogamy is indeed ρABC = p ψ ψ ABC , (10) a property of entanglement and not a consequence of a x| xih x| x particular measure of entanglement. X AB A function E : R+ is called a measure of entan- SAB → where px is some probability distribution, and for glement if (1) E(σ ) = 0 for any separable density ma- { } B AB AB each x the Hilbert space has a subspace isomor- trix σ , and (2) E behaves monotonically under x x B( ) B( ) H local operations∈ S and classical communications (LOCC). phic to 1 2 such that up to local unitary on systemH B,⊗ each H pure state ψ ABC is given by That is, for any given LOCC map Φ we have | xi AB AB AB AB x x E[Φ(ρ )] E(ρ ), ρ . ABC AB( ) B( )C ψ = φ 1 η 2 . (11) ≤ ∀ ∈ S | xi | xi | xi Moreover, convex measures of entanglement that do not AC increase on average under LOCC are called entanglement In particular, the marginal state ρ is separable monotones [48]. so that EF is monogamous (on mixed tripartite Let E be a measure of entanglement on bipartite states). states. The entanglement of formation E associated F Remark 1. The condition that E in the theorem above is with E is defined by an entanglement monotone can be replaced with a weaker n condition that the measure of entanglement E satisfies AB AB EF (ρ ) min pj E( ψj ψj ), (5) E E ≡ | ih | F on all bipartite density matrices. This is due to j=1 ≤ X the fact that both Theorem 4 and Corollary 5 in Ref. [46] are still true if we assume only that E satisfies E E where the minimum is taken over all pure state decompo- ≤ F sitions of ρAB = n p ψ ψ AB. That is, E is the and E is not necessarily an entanglement monotone. j=1 j | j ih j | F convex roof extension of E. Vidal [48, Theorem 2] showed Remark 2. Part 1 of the Theorem indicates that, if E P that EF above is an entanglement monotone on mixed is an entanglement monotone with h is strictly concave, ABC AB AB bipartite states if the following concavity condition holds. then for pure state ψ , E(ρ )= EF (ρ ) provided AB AB A AB A|BC | i AB A|BC For a pure state ψ , ρ = TrB ψ ψ , de- that E( ψ ) = EF (ρ ), that is, E( ψ ) = | i A ∈ H | ih | AB | i A|BC |ABi fine the function h : R+ by E(ρ ) is equivalent to E( ψ )= EF (ρ ) in such S → | i A AB a case. Corollary 5 in Ref. [46] proved only that if h(ρ ) E( ψ ψ ). (6) A|BC AB AB AB E( ψ ) = E(ρ ), then E(ρ ) = EF (ρ ), but ≡ | ih | | i A|BC AB it is unknown whether E( ψ ) = EF (ρ ) can im- Note that since E is invariant under local unitaries we AB AB | i ply E(ρ ) = EF (ρ ) since in general we have only must have AB AB E(ρ ) EF (ρ ) (e.g., for the negativity N, we have h(UρAU †)= h(ρA) (7) N N ≤). ≤ F 3
Proof. Part 1. In Ref. [46] it was shown that if (3) holds Denoting by for a pure tripartite state ρABC ψ ψ ABC then all AB≡ | ih | n pure state decompositions of ρ must have the same B1→B 1 B1→B AB n AB Xk ukj √pj Vj , k =1, ..., n , average entanglement. Let ρ = j=1 pj ψj ψj ≡ √qk | ih | j=1 be an arbitrary pure state decomposition of ρAB with X P n = Rank(ρAB). Then, we have
n (IA XB1→B) φ AB1 = (IA W B1→B ) φ AB1 . E(ρAB) E (ρAB )= p E( ψ ψ AB), ⊗ k | i ⊗ k | i ≤ F j | j ih j | j=1 X Now, multiplying both sides of the equation above by (ρA)−1/2 (the inverse is understood to be on the support where the inequality follows from the convexity of E, of ρA), we get and the equality holds since all pure state decompositions AB of ρ have the same average entanglement. Moreover, A B1→B AB1 A B1→B AB1 (I Xk ) φ+ = (I Wk ) φ+ , since EF is an entanglement monotone, we must have ⊗ | i ⊗ | i B1→B B1→B AB A|BC A and we therefore conclude that Xk = Wk . This EF (ρ ) E( ψ ψ )= h(ρ ). B1→B ≤ | ih | means that Xk is an isometry for any choice of uni- U u U u Therefore, denoting by ρA Tr ψ ψ AB, we con- tary matrix = ( kj ). But since = ( jk) is an ar- j B j j u clude that if (3) holds then we≡ must| haveih | bitrary unitary matrix, we can take its first row, 1j j to be an arbitrary normalized vector. Hence, we{ con-} n clude that any linear combination of the isometric ma- p h(ρA)= h(ρA). trices V B1→B is proportional to an isometric matrix. j j { j } j=1 We now discuss the consequence of this property on the X B1→B form of Vj . A n A { } Given that ρ = p ρ and h is strictly concave we B1→B j=1 j j The isometries Vj can be expressed as must have P B1→B A A Vj = vjk k , (14) ρj = ρ , j =1, ..., n. (12) | ih | Xk A B B Set r Rank(ρ ) dim , and let 1 be an r- where k is an orthonormal basis of B1 , and for each ≡ ≤ B H H {| i} H B dimensional subspace of such that there exists a pure j, vkj k are some orthonormal vectors in . Con- AB1 AB1 H A {| i} H state φ with marginal on part A being ρ . sider the arbitrary linear combination cjVj . It can be | i ∈ H AB j Since all the reduced density matrices of ψj have expressed as the same marginal on system A they must{| bei related} P φ AB1 via local isometry on Bob’s side, to a purification cj vkj k uk k , uk cj vkj . A B1→B| i | ih |≡ | ih | | i≡ | i of ρ . Therefore, there exists isometries Vj such k,j k j that { } X X X Therefore, j cj Vj is proportional to an isometry if and AB A B1→B AB1 ψj = (I V ) φ , j =1, ..., n. (13) only if for all k = k′, u ′ u = 0 and u = u ′ . | i ⊗ j | i k k k k Observe thatP 6 h | i k k k k AB n AB Now, let ρ = =1 qk φk φk be another pure k | ih | ∗ state decomposition of ρ with the same number of ele- uk′ uk = cj c ′ vk′ j′ vkj . h | i j h | i ments n. For theP same reasons leading to (13), there j,j′ X exists isometries W B1→B such that k Now, for a fixed k and k′, the above equation can be ′ AB A B1→B AB1 viewed as an inner product between a vector vkk , whose φk = (I Wk ) φ , k =1, ..., n. | i ⊗ | i components are vk′ j′ vkj , and a vector c¯ c, whose h∗ ′ | i ′ ⊗ components are cj cj . Since vkk is orthogonal to any On the other hand, since both decompositions ′ AB AB vector of the form c¯ c whenever k = k , it must be pj, ψj and qk, φk correspond to the same ⊗ 6 {density| i matrix} ρAB{ , they| i must} be related by a unitary equal to the zero vector. We therefore conclude that matrix U = (ukj ) in the following way: ′ v ′ ′ v =0, k = k h k j | kj i 6 n AB AB and √qk φk = ukj √pj ψj | i | i j=1 X vkj′ vkj = djj′ =1 n h | i 6
A B1→B AB1 ′ = I ukj √pj Vj φ . for some djj which are independent of k. Note that ⊗ | i AB j=1 we can assume with no loss of generality that ρ = X 4
AB AB A|BC AB j pj ψj ψj is the spectral decomposition of ρ . Hence, if EF (ρ )= EF (ρ ) then In such| aih case,| we have P A|BC AB AB AB pxE( ψx )= EF (ρ ) pxEF (ρx ), ′ | i ≤ ψj ψj x x h | i X X
AB1 A ′ AB1 where we used the convexity of EF . On the other hand, = φ I k vk′ j′ vkj k φ h | ⊗ | ih | ih | | i since EF is a measure of entanglement for each x we k,k′ X have E( ψ A|BC) E (ρAB). Combining this with the | xi ≥ F x AB A AB equation above we conclude that = φ 1 I k v ′ v k φ 1 h | ⊗ | ih kj | kj ih | | i k ! E( ψ A|BC )= E (ρAB) , x . (19) X | xi F x ∀ AB1 A AB1 = φ I djj′ k k φ Therefore, the rest of the proof of part 2 follows from h | ⊗ | ih | | i " k !# part 1 of the theorem. AB A BX AB = φ 1 I d ′ I 1 φ 1 = δ ′ h | ⊗ jj | i jj Note that if system B in the second part of the the- orem above has dimension not greater than 3, then we holds for any given k. We now obtain that x x B( ) B( ) must have for each x that either 1 or 2 are one v ′ ′ v = δ ′ δ ′ . (15) H H h k j | kj i jj kk dimensional. We therefore get the following corollary. B Corollary. Using the same notations as in the theorem Denote span vkj . Then, the equation K ≡ {| i} ⊂B H B A|BC AB B above implies that = 1 2 for some subspace above, if EF (ρ ) = EF (ρ ) and dim 3 then K ∼ H ⊗ H ABC H ≤ B2 of B, and in particular, there exists a unitary ma- ρ is bi-separable, and in particular it admits the form H BH trix, U , relating the basis elements vkj of with the ABC A|BC AB|C B B B {| Bi} K ρ = tσ + (1 t)γ , (20) basis elements k 1 j 2 of 1 2 . We therefore − conclude that {| i | i } H ⊗ H where σA|BC is A BC separable, γAB|C is AB C separa- ble, and t [0, 1]|. In particular, if ρABC = |ψ ψ ABC B1→B B B1 B2 ∈ ABC |ABih |C V = vkj k = U k k j is a pure state, then ψ has the form φ η or j | ih | | ih | ⊗ | i | i | i | i k k ! φ A η BC . X X | i | i = U B IB1 j B2 . At last we discuss the strict concavity of the entangle- ⊗ | i ment measures so far. Many operational measures of en- Hence, tanglement such as the relative entropy of entanglement, entanglement cost, and distillable entanglement, all re- ψ AB = (IA U B) φ AB1 j B2 , j =1, ..., n. (16) | j i ⊗ | i | i duce on a bipartite pure state to the entropy of entangle- Therefore, ment given in terms of the von Neumann entropy of the reduced state, H(ρ) Tr(ρ log ρ). The von Neumann ≡ − ρAB = (IA U B)( φ φ AB1 σB2 )(IA U B)†, entropy is known to be strictly concave [49] and there- ⊗ | ih | ⊗ ⊗ fore they are all monogamous on pure tripartite states. where σB2 is some density matrix given by σB2 = The first part of the theorem above generalizes a simi- B2 AB pj j j . Therefore, ρ has a purification of the lar result that was proved in Ref. [43, the disentangling j | ih | form theorem] for the special case in which E is taken to be P the negativity. It demonstrates that many measures of (IA U B IC ) φ AB1 η B2C , (17) ⊗ ⊗ | i | i entanglement are monogamous on pure tripartite states, while their convex roof extensions as defined in (5) are where η B2 C p j B2 j C . Since all purifications | i ≡ j j | i | i monogamous even on mixed tripartite states. of ρAB in ABC are related via a local unitary on C, we P Any function that can be expressed as conclude thatH ψ ABC has the form (9) up to local uni- | ρiAC tary. Therefore is a product state and consequently H (ρ) = Tr[g(ρ)] = g(p ) , (21) E(ρAC ) = 0. This completes the proof of part 1. g j ABC j Proof of Part 2: Let px, ψx be the optimal X pure state decomposition{ of ρABC| i = } p ψ ψ ABC x x| xih x| where pj are the eigenvalues of ρ is strictly concave if such that g′′(p) < 0 for all 0
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