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arXiv:quant-ph/0209081v1 12 Sep 2002

Broken Symmetries in the Entanglement of Formation

Fabio Benatti

Dip. Fisica Teorica, Universit` a di Trieste Strada Costiera 11, I-34100 Trieste, Italy email: Benatti@Trieste.infn.it

H. Narnhofer

Inst. f¨ ur Theoretische. Physik, Vienna University, Austria email: narnh@ap.univie.ac.at

A. Uhlmann

Inst. f¨ ur Theoretische Physik, Leipzig University, Germany email: Armin.Uhlmann@itp.uni-leipzig.de

Abstract We compare some recent computations of the entanglement of formation in quantum information theory and of the entropy of a subalgebra in quantum ergodic theory. Both notions require optimization over decompositions of quantum states. We show that both functionals are strongly related for some highly symmetric density matrices. Indeed, for certain interesting regions the entanglement of formation can be expressed by the entropy of a commuting subalgebra, and the corresponding optimal decompositions can be obtained one from the other. We discuss the presence of broken symmetries in relation with the structure of the optimal decompositions.

1. INTRODUCTION Entanglement, always one of the most intriguing among quantum marvels, has lately become a powerful resource in prospective quantum information tech1

nologies [1]; measuring the entanglement content of states of multipartite quantum systems is thus of great practical importance. If a bipartite system A + B is described by a density matrix ρAB , the so-called entanglement of formation [2] is measured by E (ρAB ) := inf

j

λj S (TrB πj ) : ρAB =

j

λj πj .

(1)

In the above expression, S (ρ) := ?Trρ log ρ denotes the von Neumann entropy of the state obtained by partial trace over B and the in?mum is computed over all possible decompositions of ρ as convexly linear combinations, that is λj > 0, λj = 1, of one-dimensional projections πj of A + B . In the following we call such decompositions extremal convex decompositions of ρ to be distinguished from generic convex decompositions into mixed states. When ρAB = |ΨAB ΨAB |, the entanglement of formation gives the asymptotic ratio between the number of singlet states necessary to construct N ? 1 copies of ρAB [3]. In the following, we will compare the entanglement of formation with a particular case of a more general quantity, the “entanglement with respect to a subalgebra” or “entanglement”, for short. This latter concept is related to the so-called “entropy of a subalgebra” A contained in a reference algebra M, relative to a state ρ on M [4], Hρ (A) := S (ρ | `A) ? inf

j

λj S (ρj | `A) : ρ =

λj ρj .

j

(2)

In the above expression, the in?mum is calculated over all convexly linear decompositions of ρ into other states on M. It plays a key role in extending the classical dynamical entropy of Kolmogorov to quantum systems [5, 6, 7]. The entanglement of formation (1) can be considered a special case of (2). We shall call “optimal” those decompositions achieving the extremum in (1) and (2). Calculating either E (ρAB ) or Hρ (A) is particularly complicated. The problem has been completely solved for the entanglement of formation if HA = HB = C2 [8], and for the entropy of a subalgebra if M = M2 (C) [17, 9, 10]. So far, all other available results concern states ρAB and ρ that are highly symmetric, isotropic in [11], respectively permutation-invariant in [13]. In this paper we will discuss the previously mentioned results by comparing the two notions of entanglement sketched above. We show, that some of them are one-to-one related. To do so, we shall focus on the structure of optimal decompositions in relation to the symmetries existing in the problem and show 2

possible ways of breaking them. These symmetries form a group G and leave invariant both the state ρ and, as a set, the subalgebra A. Given extremal optimal decompositions, the G-orbits of each of their pure states consist of optimal decomposers, too. We will study the dependence of either entanglements upon the number of di?erent orbits. 2. ENTANGLEMENT In the following, we shall consider quantum systems described by algebras of operators, M, acting on ?nite or in?nite dimensional Hilbert spaces H, with states, M ? X ?→ Tr(ρ X ), represented by density matrices which we shall denote by greek letters. De?nition 2.1 Given a ?nite dimensional subalgebra A ? M, we de?ne the entanglement of the state ρ with respect to A by E ρ; M, A := inf

j

λj S (ρj | `A) : ρ =

λj ρj ,

j

(3)

where ρ = j λj ρj runs through all convexly linear decompositions of ρ with states of M, and S (ρj | `A) is the von Neumann entropy of the state ρj restricted to the subalgebra A. The entanglement (3) is convex as a function of ρ. Remarks 2.1 (i) The entanglement (3) is a convex functional over the states: E

j

?j ρj ; M, A ≤

j

?j E ρj ; M, A ,

j

?j = 1 , ?j ≥ 0 .

(4)

This follows by choosing optimal decompositions for the ρj ’s, which together provide a decomposition, not necessarily optimal, for j ?j ρj . (ii) The entanglement of formation (1) is the entanglement (3) with A, respectively B, the algebra of observables of the system A, respectively B , M = A?B and ρAB | `A = TrB ρAB . (iii) The entanglement (3) is related with the entropy of a subalgebra (2) by E (ρAB ) = S (ρAB | `A ? 1B ) ? HρAB (A ? 1B ) . (5)

Indeed, as we shall see below in Proposition 2.1, the in?mum is achieved at decompositions using pure states of M only, and it enjoys some further remarkable properties. The quantity in (5) and some techniques [13, 14] that were developed for computing (2), have recently been used to attack the question whether the 3

entanglement of formation is additive [15]. Among them, a useful result is contained in the following proposition. The idea is in [13] and, slightly extended, in [19]. We include a proof for the bene?t of the reader. Proposition 2.1 If the algebra M is ?nite dimensional then ? the entanglement E ρ; M, A is achieved at certain extremal convex decompositions ρ = j λj πj , λj > 0 which saturate (3). Such decompositions are called optimal. Every pure state, π , which appears in at least one optimal decomposition of ρ is called ρ-optimal or an optimal decomposers of ρ. ? For every ρ there is an optimal decomposition with a length not exceeding the linear dimension of M. ? The functional E . ; M, A is convexly linear on the convex hull R(ρ) of all ρ-optimal pure states: Let be ω = i αi πi , αi > 0, i αi = 1 any extremal convex decomposition where the πj are some optimal decomposers of ρ. Then E ω ; M, A = αi S (πi | `A) . (6)

i

Proof: Any mixed state ρ can be decomposed and, since the von Neumann entropy is concave on convex combinations, mixed states cannot improve (3) with respect to pure states. If M is d dimensional, compactness of the state space, extremality and compactness of the set of pure states ensure by a theorem of Caratheodory that we need not less than d and not more than d2 decomposers [10, 16]. Because of convexity (4), the functional E . ; M, A is the supremum over a?ne functionals. Thus, for every ρ there are functionals ? such that E ρ; M, A = ?(ρ), while, on generic states σ , E σ ; M, A ≥ ?(σ ). Given an optimal decomposition ρ = j λj πj it follows E ρ; M, A =

j

λj E πj ; M, A λj ?(πj ) = ?(ρ) = E ρ; M, A . (7)

≥

j

Since equality must hold in (7) and because λj > 0, while E πj ; M, A ≥ ?(ρ) by assumption, we conclude E πj ; M, A = ?(πj ) for all j . With ω ∈ R(ρ), let us now ?x this a?ne functional ? and consider the extremal decomposition 4

′ ′ ω = αk πk such that all the πi are optimal decomposers of ρ. By convexity and the preceding argument we deduce

E ω ; M, A ≤

k

′ αk E πk ; M, A =

′ αk ?(πk ) = ?(ω ) k

(8)

Thus, E ·; M, A is convexly linear on R(ρ).

However, ?(ω ) ≤ E ω ; M, A by our choice of ?, and equality holds in (8).

De?nition 2.2 We shall call the convex hull R(ρ) of the optimal decomposers of ρ a leaf with respect to the entanglement E ρ; M, A . Then, the state space appears as covered by leaves, and the entanglement itself is convexly linear above every leaf. That e?ect is referred to as the roof property of E · ; M, A , [10], i.e. E · ; M, A is a convex roof. De?nition 2.3 Given ρ on M, we shall call a group G a symmetry group with respect to E (ρ; M, A), if for all g ∈ G there exists a linear map γg : M ?→ M such that the state and the subalgebra A (as a set) are left invariant by γg , ? ? Namely, γg [ρ] = ρ, where γg [ρ](m) = Tr(ργg (m)). Proposition 2.2 If G is a symmetry group with respect to E (ρ; M, A), the leaf R(ρ) is G-invariant as a set. In particular, the action of G permutes the optimal decomposers of ρ. Proof: Let ρ = j ∈J λj ρj be an optimal decomposition with respect to ? E (ρ; M, A). Then, since γg [ρ] = ρ and γ (A) = A for g ∈ G, the decompo? (ρj ) is also optimal. Therefore, its leaf R(ρ) must contain sition ρ = j ∈J λj γg ? both the ρj ’s and the γg (ρj )’s. Based on the previous two propositions, the entropy Hρ (A) has explicitly been computed in the following cases, Case 1. [17, 9, 10] Let M be the full 2 × 2 matrix algebra M2 (C), A the subalgebra of all 2 × 2 matrices diagonal with respect to a given basis |1 , |2 , a b , 0 ≤ a ≤ 1, |b|2 ≤ a(1 ? a), any density matrix. and ρ = ? b 1?a

Case 2. [13] Let M = M3 (C), A the subalgebra of all 3 × 3 diagonal matrices with respect to the basis |1 , |2 , |3 and 1 x x ? 1? ? , x 1 x ρ(x) = ? ? 3? x x 1

? ?

?1/2 ≤ x ≤ 1 , 5

(9)

any density matrix invariant under the group of permutations of {1, 2, 3}. For future comparison with the entanglement of formation of isotropic states of d-dimensional bipartite systems studied in [11], we ?x an orthonormal basis |j ∈ Cd and consider the group G of permutations of {1, 2, . . . , d}. It turns out that any G-invariant density matrix ρ(x) over M = Md (C) can be written as ρF = 1?F 1 ? |ψ ψ | + F |ψ ψ | , d?1 (10)

1 d where |ψ = √ |j and F is the ?delity parameter d j =1 0 ≤ F := ψ |ρ(x)|ψ = (d ? 1)x + 1 ≤1. d?1 (11)

Setting s(t) := ?t log t, we have, Case 1. For all ρ, the optimal decompositions are ρ = λ|w1 w1 | + (1 ? λ)|w2 w2 | ? z2 z1 ? , b = z1 z2 , , |w2 = |w1 = ? z1 z2 |z1 |2 = (1 + 1 1 ? 4|b|2 )/2 = 1 ? |z2 |2 , λ = ?1 + 2

? ?

(12) (13) 2a ? 1 ? . (14) 1 ? 4|b|2

The corresponding entanglement is E (ρ; M2 (C), A) = s(|z1 |2 ) + s(|z2 |2 ). If ρ = ρF is permutation-invariant, that is, if a = 1/2, b = x/2 F = (1 + x)/2, the entanglement reads E (ρF ; M2 (C), A) = s 1 + 2 F (1 ? F ) 2 +s 1 ? 2 F (1 ? F ) 2 . (15)

Case 2. Given the group G of permutations of {1, 2, 3}, let V , V 2 implement unitarily the subgroup G0 of cyclic permutations. Then, any G-invariant state ρF can be written 1 1 1 ρF = |w w | + V |w w |V ?1 + V 2 |w w |V ?2 , 3 3 3 where a + 2b cos θ ? 1? |w = ? a ? 2b cos(θ ? π/3) ? ? , ? 3 a ? 2b cos(θ + π/3) 6

? ?

(16)

a=

√

3F ,

b=

3 (1 ? F ) . 2

(17)

The structure of optimal decompositions depends on the convexity of

3

S (F ) := min

θ ∈[0,2π ]

j =1

s (| w j (F ; θ )| 2 ) .

(18)

For F ≥ F ? := (2x? + 1)/3, x? = ?0.4150234, the minimum is achieved at a single extremal G0 -orbit generated by the vectors ? ? ? √ ? F + 2(1 ? F ) a + 2b ? ? 1 ?√ 1? ? = √ ? F ? (1 ? F )/2 ? |w = ? a ? b (19) ? ? ? ? 3 3 √ a?b F ? (1 ? F )/2

For each 0 < F < F ? , there are two di?erent orbit-generating vectors, |w± (F ) , whose G0 -orbits provide di?erent optimal decomposers for (18), and which form together one orbit of the full permutation group G. They are a + 2b cos αF ? 1? ? |w± (F ) = ? a ? 2b cos(π/3 ? αF ) ? ? , 3 a ? 2b cos(π/3 ± αF )

? ?

(20)

where the angle αF varies with 0 < F < F ? . Finally, for F = 0, αF = ?π/6, the minimum is achieved again at a single 1 G-orbit containing the vector, |w0 = √ (1, 0, ?1). As the 6 vectors coincide 2 pairwise up to a sign, the states form a single optimal decomposition of length 3. In [13]), it is shown that the above vectors give optimal decompositions as long the function S (F ) is convex. Numerically, this is the case for all F ≤ 8/9. The corresponding entanglement is E (ρF ; M3 (C), A) = s 2 ? F + 2 2F (1 ? F )

+ 2s

3 1 + F ? 2 2F (1 ? F ) 6

.

(21)

for ?delities F ? ≤ F ≤ 8/9. For F = 0 the entanglement equals log 2. We have only numerical results within the interval 0 < F < F ? , [14], re?ecting that the exact dependence of the angle αF in (20) as a function of F is unknown. Remark 2.2 Permutation-invariant states as in (10) can be written as averages over the unitaries Uπ implementing the permutation group G, ρF = 1 d!

?1 Uπ |φ φ|Uπ ,

(22) 7

π

if and only if | ψ |φ |2 = F , where |ψ is the vector in (11). Necessity comes from the fact that Uπ |ψ = |ψ . Su?ciency: The identity 1 and |ψ ψ | form a basis for all possible contributions to the averages (22). In view of the structure of the optimal decomposers discussed above, we introduce a notion of regularity with respect to a subgroup of a symmetry group, as follows. De?nition 2.4 Given a symmetry group G with respect to E (ρ; M, A), we shall call a leaf R(ρ) regular of order n with respect to a subgroup H ? G, if ? there exist n pure states ρ ?j ∈ R(ρ) such that γh [? ρj ] = ρ ?j for all h ∈ H , whereas ? the convex span of the orbits γg [? ρj ] is the whole of R(ρ).

g ∈G

We illustrate the previous de?nitions with some examples. Example 2.1 Let M be a full d × d matrix algebra on Cd and A ? M did agonal with respect to a chosen orthonormal basis {|j }d j =1 in C . Let ρ be a symmetric density matrix, j |ρ|k = k |ρ|j . Then, with respect to the chosen representation, the transposition T respects both the state and the subalgebra A. Also, R(ρ) is regular with respect to G = H = {id, T }, the order of regularity depending on the state ρ. In fact, let π = |ψ ψ | ∈ R(ρ), then, because of Proposition 2.2, T (π ) = π ′ = |ψ ′ ψ ′| ∈ R(ρ), too. If π = π ′ , we may consider the state ω = π/2 + π ′ /2. which, by Proposition 2.1, is already optimally decomposed. Also, E (ω ; M , A ) = S (π | `A) = S (ω | `A) . Instead, the decomposition 1 ? Re( ψ |ψ ′ ) 1 + Re( ψ |ψ ′ ) π+ + π? , ω = 2 2 |ψ ± ψ ′ ψ ± ψ ′ | π± = 2(1 ± Re( ψ |ψ ′ ) 1 + Re( ψ |ψ ′ 2 1 ? Re( ψ |ψ ′ + 2 ) ) where (24) (25) (23)

need not be optimal. However, the concavity of the von Neumann entropy yields E (ω ; M , A ) ≤ S (π+ | `A) S (π? | `A) ≤ (S (ω | `A) . (26)

It thus follows from (23) that π | `A = π± | `A, whence the components ψ (i), ψ ′ (i) of ψ and ψ ′ must coincide apart from an overall phase. Thus, π = π ′ and the T -symmetry cannot be broken. 8

Example 2.2 Let M = A ? B, with A and B isomorphic and σ : A ?→ B the algebraic exchange of the two of them. If ρ is a state on M such that ρ ? (σ ?1 ? σ ) = ρ, in general, σ ?1 ? σ does not belong to any subgroup of regularity of ρ; indeed, if A (and thus B) is a d-dimensional matrix algebra and {|? } is an orthonormal basis in the corresponding Hilbert space HA (and thus also in HB ), the density matrix 1 1 ρAB := |1 1| ? |2 2| + |2 2| ? |1 1| , 2 2 (27)

is such that Tr ρ(σ ?1 ? σ )(X ? Y ) = Tr ρ(X ? Y ) . Also, ρAB is already optimally decomposed, E (ρAB ; A, M) = 0 is achieved with the decomposers |1 1|?|2 2| and |2 2|?|1 1|, which, however, are not invariant under σ ?1 ? σ . Example 2.3 Let M = A ? B, with A and B both d × d full matrix algebras. We ?x the same orthonormal basis {|? } in both Hilbert spaces HA,B and consider the one-parameter group U of unitaries Ut :=

j,k

eit(hj ?hk ) |j j | ? |k k | .

(28)

The density matrix ρAB := j,k Rjk |j k | ? |j k |, , R = [Rjk ] ≥ 0, TrR = 1, is √ √ U -invariant; moreover, ρAB = j,k ( R)jk | |j k |?|j k |, so that the operators √ √ ρAB M ρAB , M ∈ M, have the same matrix structure as ρAB . Choosing positive Mj ≥ 0, j ∈ J , such that j ∈J Mj = 1, ρAB decomposes into √ √ ρAB Mj ρAB . (29) ρAB = Tr(ρAB Mj ) Tr(ρAB Mj ) j ∈J Since it is also true that every mixed state ρ on M can be written as in (29) by means of a suitable positive Mj , (29) indeed exhausts all possible decompositions of ρAB . Thus, the decomposers πj of ρAB which are optimal with respect to E (ρAB ; M, A), have the same structure of ρAB and are then U -invariant. Hence, the group U is a group of symmetries of ρAB with respect to entanglement and the leaf R(ρAB ) is regular with respect to H ≡ U , its order depending on which further symmetries are enjoyed by ρAB . Example 2.4 Let M = M2 (C), A as in Case 1, and ρF a permutationinvariant state. The leaf R(ρF ) is the orbit of the group G of permutations of {1, 2}. This follows from the form of the optimal vectors (12) in such a case: 9

Example 2.5 Let M = Md (C) and ρF a permutation-invariant state. Then, for F ? ≤ F and F belonging to the convexity region of S (F ) in (18), the structure of the optimal vectors (19) ensures that the leaf R(ρF ) is regular of order 1 for the subgroup H of permutations {2, 3} → {3, 2}. However, at the point F = F ? such a H -invariant vector bifurcates into the two optimal ones (20). Thus regularity with respect to the subgroup H is broken and remains broken for 0 < F < F ? . At F = 0 optimal vector states of di?erent G0 orbits degenerate pairwise into a single one, and one of them is H -invariant, while the corresponding vector changes its sign. In the last two examples, for all F when d = 2, and for F greater than the bifurcation values F ? in the convexity region of S (F ) in (18), when d = 3, the leaf R(ρF ) of a permutation-invariant ρF is generated by the orbit under the subgroup G0 of cyclic permutations V j |w , j = 0, 1, 2. The vector |w is invariant under a unique transposition out of G. This structure is indeed more general as will be showed in the next two propositions. Proposition 2.3 Let A ? M = Md (C) be chosen as in Example 2.1 and the density matrix ρF be invariant with respect to the permutation group G. If the leaf R(ρF ) with respect to A is generated by exactly one G0 -orbit of a normalized vector state |w ∈ Cd , with G0 ? G the subgroup of cyclic permutations, then the entanglement is 1 ? pF E (ρF ; Md (C, A) = s(pF ) + (d ? 1)s d?1 √ 2 F + (d ? 1)(1 ? F ) . pF := d (30) (31)

z2 z1 , with z1,2 = 1/2(1 ± 2 F (1 ? F )). It is regular , |w2 = z1 z2 of order 1 with respect to rotations with elements from A. |w1 =

Remarks 2.3 (i) The assumption of the previous proposition amounts to ask R(ρF ) to be regular of order 1 with respect to the subgroup H ? G of permutations on {2, 3, · · · , d}. Indeed, the leaf is G-invariant, so that the d states |φj = V j |w , j = 0, 1, . . . , d ? 1, obtained via cyclic permutations, must be invariant under the remaining (d ? 1)! permutations This is possible only if d ? 1 of the d components of the optimal vector |w are equal. 10

(ii) If |w has three di?erent components, then the decompositions (22) contain at least d(d ? 1) di?erent terms. (iii) In section 3 we will show that, upon identi?cation of pF with the quantity γ (F ) in [11], the entanglement of formation calculated there is given by (31) and (30) in a range F ?? ≥ F > 1/d. The upper limit F ?? is a particular bifurcation point which was discovered in [11] and that will be reinterpreted accordingly within the framework of this work. 1 d?1 j Proof: By hypothesis, ρF = V |w w |V ?j is an optimal decomposition d j =0 with entanglement

d

E (ρF ; Md (C), A) =

j =1

s | j |w |2

.

(32)

Also, taking into account Remark 2.2 and 2.3, and decomposing |w = √ √ ⊥ F |ψ + ε 1 ? F |w1 = α |1 + β

d j =2

|j ,

√ √ ⊥ where ε is a pure phase, it follows that |w1 = ( d|1 ? |ψ )/ d ? 1 and √ 1?F 1 √ |w = √ F + ε (1 ? F )(d ? 1) |1 + F ?ε d?1 d With ξ := 2Re(ε), the right hand side of (32) reads S (ξ ) = s(p(ξ )) + (d ? 1)s( p (ξ ) = 1 ? p (ξ ) ), d?1 F + (1 ? F )(d ? 1) + ξ F (1 ? F )(d ? 1)

d j =2

|j

.

. d It achieves its minimum at the maximum value of p that is for ε = 1, from which the result follows. Indeed, as we show below, |w must be real. If remark 2.3(i) applies we always get a local extremum. Either by direct calculation or relying on [13] one concludes ? = 1. We now relax the hypothesis of the previous proposition and allow for more than one G0 -orbit to be optimal for the entanglement of ρF with respect to the subalgebra A, that is we allow the leaf R(ρF ) to be generated by more than one G0 -orbit. Proposition 2.4 Let A ? M = Md (C) be chosen as in Example 2.1. If the density matrix ρF is invariant with respect to the permutation group G and its 11

entanglement with respect to A can be achieved at an optimal decompositions consisting of one G0 -orbits of normalized vector states |w ∈ Cd , with G0 ? G the subgroup of cyclic permutations, then we have three possibilities 1 d ? |w = √ |k in which case F = 1 and ρF = |ψ ψ |; d k=1 ? |w is real with 1 component equal to a1 and d ? 1 real components all equal to a2 = a1 ; ? |w is real with 2 components a1 = a3 and d ? 2 components all equal to a3 di?erent from both a1 and a2 . To prove the result we need a preliminary Lemma 2.1 The vector |w whose G0 -orbit is optimal can be chosen real. Proof: Let vk , k = 1, 2, . . . , d, be the components of |w with respect to the 1 d |k . The assumption is that chosen orthonormal basis {|k } and |ψ = √ d k=1 1 d?1 j V |w w |V ?j ; from normalization it follows that the components of ρF = d j =0 |w must satisfy

d k =1 d

|wk | = 1 ,

2

wk

k =1

2

d

=1?

? w? wj = dF . ?=k =1

(33)

Further, in order to implement optimality and achieve E (ρF ; M, A), we minimize

d

S (w, λ, ?) := ?

k =1

|wk |2 log |wk |2 + λ

d k =1

|wk |2 + ? wk =

? w? wk , ?=k

(34)

d

with Lagrange multipliers λ, ?. Setting v :=

k =1

√ dF eiθ , equating to zero

the derivative of (34) with respect to wj and multiplying by wj we get ?|wj |2 log |wj |2 + (λ ? 1)|wk |2 + ?(v ? wj ? |wj |2 ) = 0 . Therefore, the quantity v ? wj ? and thus, after summing over j , also ?, must be real, whence, necessarily wj = eiθ vj , with vj ∈ R, for all j . The result follows by eliminating the overall phase.

12

Proof (of Proposition 2.4): According to the previous Lemma, we choose |w real and proceed to minimize

d

S (w, λ, ?) := ?

2 2 wk log wk +λ

d k =1

2 wk +?

d

wk .

k =1

(35)

k =1

Because of convexity, the function g (x) := ?x log x2 intersects the straight line f (x) := 2(1 ? λ)x ? ? in at most three points on [?1, 1]. Therefore, the d solutions to 2 ?2wk log wk ? 2wk + 2λwk + ? = 0 , can have at most three di?erent real values, ai , i = 1, 2, 3. We denote by ni the number of times they appear among the components and consider the functional

3 3 3

S (a; nλ, ?, ν ) := ?

ni a2 i

i=1

log a2 i

+λ

i=1

ni a2 i

+?

i=1

ni ai ,

(36)

where we treat the ni ’s as continuous variables constrained by n1 + n2 + n3 = d. Minimizing (36) yields the following equations ni (ai log a2 i + ai ? λai ? ?) = 0 ,

2 2 ?a2 i log ai + λai + ?ai + ν ,

i = 1, 2 , 3

(37) (38)

i = 1, 2 , 3 .

2 It follows that, if ni > 0, i = 1, 2, 3, then, 3 i=1 (?ai + 2ν + 2ai ) = 0, i = 1, 2, 3, and thus a = b = c. This case corresponds to ρF =1 = |ψ ψ |, a pure state, with null entanglement with respect to A. Therefore, if there are three di?erent intersections, the minimum entanglement is reached at the boundary values of ni , i = 1, 2, 3, that is, without loss of generality, at n1 = n2 = 1 and n3 = d ? 2. If there are two intersections, that is if, without loss of generality, n3 = 0 and a1 = a2 = a3 , then, from (37,38), we calculate ? = ?2(a1 + a2 ), ? = a1 a2 and deduce the equality a2 2 2 + a a log a2 ? a =0. 1 2 1 2 a2 1

For ?xed a1 , because of their convexity properties, the two functions f (x) := a1 x a2 and g (x) := ? intersect at x = a1 , but, at no other points. Therefore, log 1 2 x x a1 the entanglement is again minimal at the boundary, that is at , say n1 = 1 and n2 = d ? 1. Remark 2.4 Lagrange multipliers have been used in [11] in order to calculate the entanglement of formation of isotropic states of bipartite quantum systems, 13

where it is shown that, when F > 1/d, the optimal decomposers have only two di?erent components. We shall relate those results to ours in the following section, where we also discuss the fact, discovered in [11], stating there is a bifurcation point F ?? such that the entanglement of formation is linear in F between F ?? and F = 1. Proposition 2.4 shows that when the vector |w has only two di?erent components, then we reduce to the case discussed in Proposition 2.3. Instead, when |w has three di?erent components, which is possible in a range of values of F , then we have more than one optimal decompositions. If d = 3 one gets at least two. Notice that these results are obtained under the hypothesis that G0 -orbits of vectors |w provide optimal decompositions for the entanglement of ρF with respect to the subalgebra A. This fact is linked to the convexity of the function (18), which, as observed in the discussion of Case 2, fail in a neighborhood of F = 1: If F ≥ F ?? one needs two orbits: the optimal orbit for F = F ?? and the singlet for F = 1, just as observed in [11]. Consequently, for F ?? < F < 1 no G0 -orbits can be optimal.

3. ENTANGLEMENT AND ENTANGLEMENT OF FORMATION In this section we establish a one-to-one correspondence between the results of the previous section, in particular proposition 2.3, and the entanglement of formation of highly symmetric states as examined in [11]. This concerns mainly the region (1/d) ≤ F . From [11] we learned the existence of the bifurcation point F ?? . On the other hand, our results in the region (1/d) < F ≤ F ?? can be converted into those found by Terhal and Volbrecht. Indeed, the value of the entanglement of formation will be proved to be just (30). To this end we consider the tensor product M := A ? B of the full d × d matrix algebra, denoted by A, with a copy, B, of itself. We ?x an orthonormal basis {|j } of Cd and given any density matrix, that is a state on A, ρA =

j,k

Rjk |j k | ,

R = [Rjk ] ≥ 0 ,

TrR = 1 ,

(39)

we embed it as D [ρA ] into the state space of M according to the following De?nition 3.1 Let D be the linear map associating matrix units |j k | of A with matrix units {|j k | ?|j k | of M. We shall refer to it as the doubling map. It transforms states ρA on A into states on M = A ? B of the form 14

ρA ?→ D [ρA ] :=

j,k

Rjk |j k | ? |j k | ,

(40)

Remark 3.1 This yields the class of density matrices in Example 2.3, which we shall refer to as diagonal class (with respect to the chosen basis). On the given diagonal class the doubling map can be inverted D ?1 : ρAB =

j,k

Rj,k |j k | ? |j k | ?→ ρA =

j,k

Rj,k |j k | .

(41)

The argument developed in Example 2.3 ensures that decompositions of ρA can be mapped onto decompositions of D [ρA ]. Vice versa, decompositions of ρAB provide decompositions for the diagonal class of ρA by applying D ?1 . Moreover, if A0 ? A denotes the subalgebra of diagonal matrices in the given, ?xed representation, then S (ρ | `A0 ) = S (D [ρA ] | `A). Therefore: The entanglement is preserved by D , in the sense that E (ρA ; A, A0) = E (D [ρA ]; A ? B, A) . (42)

In [11] the entanglement of formation has been calculated for the isotropic states ωF = 1?F (1AB ? |Ψ Ψ|) + F |Ψ Ψ| . d2 ? 1 (43)

In the above expression 1AB is the identity for the algebra A ? B and 1 |Ψ = √ |j ? |j . d j =1 (44)

Remark 3.2 The isotropic states are invariant under the group G of all uni? where a|U |b = a|U ? |b ? , taries of the form U ? U ? F U ?1 ? U ? ?1 = ω F . U ? Uω (45)

As in Remark 2.2, it follows that ωF can be expressed as the following average with respect to the Haar measure dG U , ωF =

G

? |Φ Φ|U ?1 ? U ? ?1 , dG U U ? U

(46)

if and only if F = Ψ|ωF |Ψ = | Ψ|Φ |2. We compare the isotropic state ωF with the doubling of ρF in (10),

15

D [ρF ] =

1?F D [1A ] ? D [|ψ ψ ]| + F D[|ψ ψ |] d?1 1?F d = |j j | ? |j j | ? |Ψ Ψ]| + F |Ψ Ψ| . d ? 1 j =1 1 d!

(47)

Proposition 3.1 Let F > 1/d and consider the decomposition ωF =

?1 ?1 Uπ ? Uπ |Φ Φ|Uπ ? Uπ

π

by means of the unitaries Uπ that implement the permutation group G. If the latter is optimal for the entanglement of formation E (ωF ) with |Φ Φ| in the diagonal space, then E (ωF ) = E (ρF , A, A0). Proof: The d! unitaries Uπ form a subgroup G ? G of the group of unitaries in Remark 3.2; they implement the permutation of the chosen basis {|j ? |j } of the diagonal space. Then, Ψ|ωF |Ψ = Ψ|D [ρF ]|Ψ = F and D [ρF ] = 1 d!

π ?1 ?1 Uπ ? Uπ |Φ Φ|Uπ ? Uπ .

If |Φ Φ| is optimal for ωF , it turns out from Proposition 2.2 that the decom? |Φ Φ|U ?1 ? U ? ?1 are optimal, too. Thus the result follows from poseres U ? U Proposition 2.1. Remarks 3.3 (i) If F > 1/d the isotropic state ωF is entangled. When F ≤ 1/d it becomes separable. There exist several proofs of this fact, e.g.[18]. (ii) In view of Remark 2.3(ii), the previous proposition establishes a link between our results and those of [11]. In [11] a new symmetry breaking bifurcation point was observed at F = 8/9 when d = 3. The doubling map makes it correspond to a bifurcation point within case 2 of the previous section at the same value of F The numerical analysis in [14] missed it, the needed accuracy being of the order of 10?4 . In both cases the leaves R(ωF ), respectively R(ρF ), are identical for all F within F ?? = 8/9 < F < 1. This unique leaf is generated by the optimal decompositions of ω8/9 respectively ρ8/9 , which form one orbit, and by the pure state ω1 given by (44) respectively ρ1 . The latter orbits are singlets. (iii) The entanglement of ρ1 and ρ8/9 that generate the leaf discussed in the previous remark do not coincide, E (ρ1 ; M, A) = ln 3 , E (ρ8/9 ; M, A) = ln 3 ? 16 1 ln 2 . 3 (48)

We shall now relate the remark above to another observation which again relate entanglement of di?erent algebras with one another. From Case 1 in section 2, we know that vectors of the form x y and

y , with x2 + y 2 = 1 generate the leaf of some state ρ2 on M2 (C). These x 2-dimensional vectors can be embedded in C3 as follows, |w1 x ? √ ? ? =? ? y/√2 ? , y/ 2

? ?

|w2

With them we construct the density matrix in M3 (C) of the form ρ ?3 = λ|w1 w1 | + (1 ? λ)|w2 a b b ? ? ? . b c c w2 | = ? ? ? b c c

? ?

y ? √ ? ? =? ? x/√2 ? . x/ 2

?

?

(49)

(50)

It is easy to check that powers of ρ ?3 have the ? same ? structure which is thus inu ? ? √ √ ? herited by ρ ?3 . It thus follows that ρ ?3 |φ = ? ? v ? for any |φ . The discussion v of Example 2.3 assures and that the optimal decomposers of ρ ?3 with respect to the entanglement E (? ρ; M3 (C), A3 ), with A3 the maximally Abelian subalgebra x in the chosen representation, have again the same form. But then, being y y optimal with respect to E (ρ2 ; M2 (C), A2 ), (50) is itself an optimal and x decomposition of ρ ?3 with respect to E (? ρ3 ; M3 (C), A3). According to the discussion at the beginning of this section, it also follows that the doubling map y |w1 → |W1 = x|1 ? |1 + √ |2 ? |2 + |3 ? |3 2 x |w2 → |W2 = y |1 ? |1 + √ |2 ? |2 + |3 ? |3 2 (51) , (52)

provides optimal decomposers, too. In particular, for given x, y on the unit circle the pure states |Wj Wj |, j = 1, 2, generate a leaf of the entanglement of formation functional on which it is convexly linear. √ Moreover, for x = 1/ 3 and y = 2/3, we get |W1 = |Ψ , with ?delity F = | Ψ|W1 |2 = 1, and |W2 = |Φ8/9 with ?delity F = | Ψ|W2 |2 = 8/9, indicating a reason for the bifurcation value F = 8/9. 17

√ One observes that (51) and (52) become identical for x = y = 1/ 2 so that the doubling map gets the vector 1 1 |W3 = √ |1 ? |1 + |2 ? |2 + |3 ? |3 2 2 which has ?delity F = | Ψ|W3 |2 = 1 + 2 2 8 = p + (1 ? p) , 3 9 √ 7 0 < p = 3 6 ? < 1 . (54) 2 , (53)

Let us now consider the state ρF = p|Ψ Ψ| + (1 ? p)|Φ8/9 Φ8/9 | . (55)

By using (48), it can be shown that its entanglement E (ρF ) is larger than pE (ρ(1)) + (1 ? p)E (ρ(8/9)) for 0 < p < 1. This implies that convexity of S (F ) in (32) is lost for F > F ?? in accordance with the discussion above. We ?nally note that one can extend (49) to all dimensions larger than two. Indeed, let z1 , z2 denote the components of a unit vector in two dimensions. By similar arguments one proves that the leaves of case 1 of the previous section are mapped onto certain leaves belonging to the entanglement of formation in d + 1 dimensions by the embeddings z1 z2 √ d+1 |jj ?→ z1 |00 + (z2 / d)

j =2

(56)

? ? In particular, the embeddings of {z1 , z2 } and {z2 , z1 } form an optimal pair with respect to the entanglement of formation. One further observes in the special √ case z1 = 1/ d + 1 the embeddings (56) are the totally symmetric vector Ψ in d + 1 dimensions and

d |11 + d+1

d+1 1 |jj d(d + 1) j =2

(57)

Its ?delity reads F = 4d/(1 + d)2 , and we see as above

?? ?2 Fd +1 = 4d(d + 1)

(58)

i. e. the bifurcation value given in [11] for d + 1 > 2. 4. CONCLUSIONS We have studied in several examples the entanglement de?ned by a maximal commuting subalgebra of a full matrix algebra, and in its relation to the entanglement of formation. Apart from its actual numerical value, what is interesting 18

is the structure of both entanglement functionals upon the space of states, and their separation into di?erent leaves. To some extent these leaves can be found by applying group theoretical considerations. They show a rich structure with varying stability under the groups under consideration, Since the same group appears in di?erent algebraic contexts, it can be shown that the decompositions of states on di?erent algebras can be related. This helps to control the optimal decompositions and to understand their variety. This new technique is shown at work in several examples: The doubling map relates two quite di?erent lines of research which had been considered almost independently up to now. In particular we have a further proof of the entanglement of formation results for isotropic states of Terhal and Volbrecht in the region (1/n) ≤ F ≤ F ?? , [11]. Another embedding map veri?es their bifurcation point F ?? close to F = 1 as a footprint of a symmetry-breaking in two dimensions. It belongs to class of maps which change entanglement but not the leaves. The leaves are respected because the entanglements di?er just by a convexly linear function. It should be clear that we only provide some distinguished ?rst examples of our embedding procedures which can connect various entanglement problems and, evidently, other ones which are de?ned via convex or concave roofs, for example general entanglement monotones or Holevo (1-shot) capacities.

19

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