Peccei-Quinn Transformations and Black Holes : Orbit Transmutations and Entanglement Generation

In a recent paper [Mod. Phys. Lett. A30 (2015) 1550104], the black-hole/qubit correspondence (BHQC) was exploited to define"black hole quantum circuits"allowing for a change of the supersymmetry-preserving features of electromagnetic charge configurations supporting the black hole solution, thus switching from one U-duality orbit to another, or - equivalently - from an element of the corresponding Freudenthal Triple System with a definite rank, to another one. On the supergravity side of BHQC, such quantum gates relate to particular symplectic transformations acting on the black hole charges; namely, such transformations cannot belong to the U-duality group, otherwise switching among orbits would be impossible. In this paper, we consider a particular class of such symplectic transformations, namely the ones belonging to the so-called Peccei-Quinn symplectic group, introduced some time ago within the study of very special Kaehler geometries of the vector multiplets' scalar manifolds in N = 2 supergravity in D = 4 space-time dimensions.

dimensions [42,32], and exceptional groups E 7 and E 6 were respectively related to the tripartite entanglement of seven qubits [43] and to the bipartite entanglement of three qutrits [44]. Recently, "BH quantum gates" were formulated within the BHQC in [45]. The Freudenthal triple system (FTS) [46,47,48] plays a key role in BHQC; these are algebraic systems, whose elements can be grouped into different ranks, ranging from zero to four [49,50,51]. In the classification of tripartite entanglement, the ranks 0, 1, 2a, 2b, 2c, 3 and 4 are associated respectively with null, separable a − b − c, biseparable a − bc, b − ac and c − ab, W and GHZ states [28]. By means of a set of local operations and classical communications (LOCC), the state can be operated without generating a new class of entanglement.
The inter-relation between equivalent states in the same class is achieved by the possibility of interconverting one to another by means of the group of local unitaries (LU) in the set of LOCC [52]. Rather than declare equivalence when states are deterministically related to each other by LOCC, one may require only that they may be transformed into one another with some non-zero probability of success; this can be achieved by the so-called stochastic LOCC (SLOCC) [53]. Within the SLOCC paradigm, two states are identified if there is a non-zero probability that one can be converted into the other and vice versa; one can trace back the origin of such a generalization by the physically-motivated fact that any set of SLOCC equivalent states may be used to perform the same non-classical operations, only with varying likelihoods of success. In terms of FTS, with SLOCC, one is varying the representative element with the same entanglement class, corresponding to a well-defined FTS rank. As analyzed in [29], SLOCC equivalence classes of three qubit states are stratified into an entanglement hierarchy [53], which can be consistently reproduced in terms of FTS, or equivalently in terms of orbits of its automorphism group.
Going beyond the SLOCC paradigm within the BHQC, [45] investigated the possibility of establishing quantum circuits (realizing certain classes of quantum gate operations) capable of switching, within the same FTS, from one rank class to another one. On the BH (supergravity) side of BHQC, this essentially amounts to switching from one U -duality orbit to another one; in the present treatment, this phenomenon will be named orbit transmutation. In FTS terms, this amounts to switching from an FTS element with a definite rank to another FTS element with another rank, lying in another orbit of the FTS automorphism group. In this context, specific quantum circuits generating Bell states, quantum teleportation and GHZ states can be clearly associated with the protocols of actions on the charge state configurations of EBHs [45].
On the BH (supergravity) side of BHQC, the aforementioned "BH quantum gates" are related to a particular kind of symplectic transformations (which are the most general transformations acting linearly on BH electromagnetic charges 2 ). Indeed, "BH quantum gate" symplectic transformations generally allow for the switching from the representative of one U -duality orbit to a representative of another U -duality orbit; the most interesting cases are the ones in which a change of FTS rank occurs, i.e., in which a transition between different levels of the entanglement hierarchy takes place. Since the U -duality orbit structure (namely, the stratification of the charge representation space R under the non-transitive action of the U -duality group G 4 ) is invariant under G 4 itself, it is clear that such symplectic transformations cannot belong to the U -duality group, otherwise switching among orbits would be impossible. Thus, the "BH quantum gate" transformations generally belong to the pseudo-Riemannian, non-symmetric coset 3 .
where 2n + 2 denotes the dimension of the charge representation space R.
In the present paper, we consider a particular class of "BH quantum gate" symplectic transformations, namely the ones belonging to the so-called Peccei-Quinn (PQ) symplectic group, introduced in [56] in the study of very special (i.e., cubic) Kähler geometries in N = 2 supergravity 4 . In such a framework, the special Kähler geometry of the vector multiplets' scalar manifold is based on a holomorphic prepotential function F in the symplectic sections X Λ . At least within models with symmetric scalar manifolds, all symplectic transformations belonging to the coset Equation (1.1) and, thus, a fortiori the PQ symplectic transformations, change the value of unique quartic invariant polynomial I 4 of the charge representation space R, and consequently, they change the EBH Bekenstein-Hawking entropy. This is of particular importance in the theory of EBH attractors, because such a phenomenon allows for orbit transmutations, namely for transitions between different charge orbits in the stratification of R, with a corresponding change of the supersymmetry-preserving features of the corresponding EBH charge configurations [56].
The plan of the paper is as follows.
In Section 2, we recall the definition of the Peccei-Quinn symplectic group and introduce the corresponding operator in the Hilbert space of EBH charge configuration states. Then, in Section 3, we introduce some examples of "large" and "small" EBH charge configurations, whose transformation under the PQ operator is discussed in Section 4, with particular emphasis on the occurrence of different types of orbit transmutations.
After a brief interlude dedicated to state superposition in Section 5, in Section 6, we start the analysis of general mechanisms of entanglement generation on EBH charge states, by defining the entanglement PQ operator and pointing out the necessity of switching to a complex ground field (a already stressed within BHQC; see, e.g., [28]) within the indistinguishability assumption. Some final comments are contained in Section 7.
The group P Q (n + 1) and its Lie algebra pq n+1 were introduced in [56] in the study of very special Kähler geometries. However, interestingly, matrices with structure as W (2.1), and thus belonging to P Q (n + 1), made their appearance also in other contexts. For example, W can be regarded as a particular case of the quantum perturbative duality transformations in supersymmetric Yang-Mills theories coupled to supergravity [58]; in such a framework, Equation (2.1) defines the structure of quantum perturbative monodromy matrices in heterotic string compactifications with classical U -duality group SL(2, R) × SO(2, n + 2).
In N = 2 (ungauged) supergravity theories in D = 4 space-time dimensions, n denotes the number of (Abelian) vector multiplets coupled to the gravity multiplet. By virtue of a theorem due to Dynkin [59], the U -duality group G 4 is embedded into the symplectic group Sp(2n + 2, R) in a maximal (and generally non-symmetric) way, as follows: where R is the symplectic (i.e., anti-self-conjugated) representation of G 4 in which the electric and magnetic charges of the (E)BH sit. Consequently, one can define a dyonic 5 charge vector state |Q as follows 6 : where: is a symplectic basis of the Hilbert space realization of R. It is instructive to report here the holomorphic sections' state associated with a given holomorphic prepotential F (X): where F i (X) := ∂F (X)/∂X i .
A general operator in the Hilbert space of EBH state configurations can be defined as follows: However, in the present treatment, we will focus on the class of Peccei-Quinn (PQ) operators, related to the aforementioned PQ symplectic transformations. From Equations (2.1) and (2.2), one can define the PQ operator O P Q associated with the PQ transformation W as follows: corresponding to: ∀i, j = 1, .., n : in the parametrization of the general operator (2.9).
The PQ-transformations of States (2.6) and (2.8) formally read as: 3 Some "Large" and "Small" Configurations Within the considered symplectic frame, the general charge configuration state (2.6) is associated with the following, unique [61, 62] G 4 -invariant (cf., e.g., [4,15,63] and the refrences therein): where: As anticipated in Section 1, the Bekenstein-Hawking EBH entropy formula can be recast in terms of such an invariant as: Below, we consider some noteworthy EBH charge configurations, which also provide well-defined representatives of the corresponding U -duality orbits, i.e., examples of FTS elements of all possible ranks (4, 3, 2, 1).

1.1]
In the case with both p 0 and q 0 non-vanishing, one obtains a dyonic "large" configuration (corresponding to the maximal rank= 4 element in the related FTS), whereas for p 0 or q 0 vanishing, one has a "small" configuration (namely: 1.2] for p 0 = 0: |Q eKK and 1.3] for q 0 = 0: |Q mKK , such that S(Q eKK ) = S(Q mKK ) = 0, corresponding to minimal rank= 1 elements in the FTS).

2.1]
In the case with both p 0 and I 3 (q) non-vanishing, one obtains a dyonic "large" configuration (corresponding to the maximal rank= 4 element in the related FTS, i.e., to an element in the same duality orbit of the "large" dyonic KK configuration). As an example of "small" configurations (S(Q E ) = 0), one may set p 0 = 0; this latter case further splits into three subcases: 2.2] I 3 (q) = 0, corresponding to a rank-three FTS element: |Q 3E ; 2.3] I 3 (q) = 0, but ∂I 3 (q)/∂q i = 0 for at least some i, corresponding to a rank-two FTS element: |Q 2E ; 2.4] ∂I 3 (q)/∂q i = 0 ∀i, but q i = 0 for at least some i, corresponding to a rank-one FTS element (i.e., in the same duality orbit of the small KK configurations): |Q 1E .

Peccei-Quinn Orbit Transmutations
As introduced in Section 2, a PQ symplectic transformation is given by the action of the corresponding operator O P Q (2.10). On the charge state (2.6), the action (2.12) can be explicated as follows: where:q As a consequence, PQ operators leave the EBH magnetic charges invariant, while changing the electric ones [56]: Let us now analyze the effects of the application ofÔ P Q on the various U -orbit representatives/FTS elements presented in Section 3 (considering before the "large" and then the "small" ones).

1.1] "Large" KK configuration :
Consequently, after a PQ transformation, the original KK dyonic EBH state is changed to an electric state. The value of the quartic invariant, and thus of the Bekenstein-Hawking EBH entropy, generally changes: Thus, depending on whether: 2.1] "Large" electric configuration : Correspondingly, the quartic invariant changes as follows: and there are three possibilities: ii : I 4 (Q ′′ E + Q ρeKK ) < 0; (4.13) iii : I 4 (Q ′′ E + Q ρeKK ) = 0. (4.14) In Cases iand ii, the resulting FTS element has rank four , and thus, it corresponds to a "large" EBH, whereas in Case iii, the resulting FTS has rank three, corresponding to a "small" EBH. Therefore, depending on the sign of p 0 I 3 (q), the action ofÔ P Q may result in various types of orbit transmutations: a change of orbit representative within the same U -orbit (at most, generally with a change of value of I 4 ) or a change of U -orbit, with preservation of the FTS rank = 4 (namely, a change of sign, and generally, of value of I 4 ), or (in Case iii) a change from a "large" U -orbit (rank-four) to a "small" U -orbit (rank-three).
3.1] "Large" magnetic configuration : where: and: and there are three possibilities: ii : iii : In Cases i and ii, the resulting FTS element has rank four, and thus, it corresponds to a "large" EBH, whereas in Case iii, the resulting FTS has rank three, corresponding to a "small" EBH. Thus, depending on the sign of q 0 I 3 (p), the action ofÔ P Q may once again result in various types of orbit transmutations: a change of orbit representative within the same U -orbit (at most, generally with a change of value of I 4 ), or a change of U -orbit, with preservation of the FTS rank = 4 (namely, a change of sign, and generally, of value of I 4 ), or (in Case iii) a change from a "large" U -orbit (rank-four) to a "small" U -orbit (rank-three).
rank Q E = 2 ⇔ n i,j,k=1 d ijkq iqjqk = 0, but n j,k=1 d ijkq jqk = 0 for at least some i, (4.25) and there are three possibilities (i-iii) as above (note that the term {I 3 (p), I 3 (q)} may or may not be vanishing).
and there are three possibilities (i-iii) as above.
and there are three possibilities: In Cases i and ii, the resulting FTS element has rank four, and thus, it corresponds to a "large" EBH, whereas in Case iii, the resulting FTS has rank three, corresponding to a "small" EBH. Thus, in this case, the action ofÔ P Q may once again result in various types of orbit transmutations: a change from a "small" U -orbit (rank-three) to a "large" U -orbit (rank-four, with positive or negative I 4 ), or (in Case iii) a preservation of the rank= 3 of the original FTS element, but generally with a change of orbit representative (depending on the real form of the theory under consideration, this may result necessarily in remaining in the same rank-three orbit, as in N = 8 supergravity, or possibly in switching to another rank-three orbit, as in N = 2 supergravity with symmetric, very special Kähler vector multiplets' scalar manifolds [6,5] and there are three possibilities i-iii as above (note that the term {I 3 (p), I 3 (q)} may or may not be vanishing).
and there are three possibilities i-iii as above.
and there are three possibilities: i : I 4 Q 2M +Q E > 0; (4.46) ii : I 4 Q 2M +Q E < 0; (4.47) iii : I 4 Q 2M +Q E = 0. (4.48) In Cases i and ii, the resulting FTS element has rank four, and thus, it corresponds to a "large" EBH, whereas in Case iii, the resulting FTS has rank three, corresponding to a "small" EBH. Thus, in this case, the action ofÔ P Q may once again result in various types of orbit transmutations: a change from a "small" U -orbit (rank-two) to a "large" U -orbit (rank-four, with positive or negative I 4 ), or (in Case iii) a change from a "small" rank-two orbit to a "small" rank-three orbit. (4.50) and there are three possibilities (note that the term {I 3 (p), I 3 (q)} may or may not be vanishing): ii : iii : Once again, in Cases i and ii, the resulting FTS element has rank four, and thus, it corresponds to a "large" EBH, whereas in Case iii, the resulting FTS has one of the possible ranks 3, 2, 1, corresponding to a "small" EBH. Thus, in this case that the action ofÔ P Q may once again result in various types of orbit transmutations: a change from a "small" U -orbit (rank-two) to a "large" U -orbit (rank-four, with positive or negative I 4 ), or (in Case iii) the change from a "small" U -orbit (rank-two) to "small" orbit, of possible rank 3, 2 ,1. In the case of rank= 2, generally a change of orbit representative takes place (as above, depending on the real form of the theory under consideration, this may result necessarily in remaining in the same rank-two orbit, or possibly in switching to another rank-two orbit [6,5]).
rank Q E = 1 ⇔ n j,k=1 d ijkq jqk = 0 ∀i, butq i = 0 for at least some i, (4.54) 55) and there are two possibilities: ii : In Case i, the resulting FTS element has rank four (with negative I 4 ), and thus, it corresponds to a "large" EBH, whereas in Case ii, the resulting FTS has one of the possible ranks 3, 2, 1, corresponding to a "small" EBH. Thus, in this case, the action ofÔ P Q may once again result in various types of orbit transmutations: a change from a "small" U -orbit (rank-two) to a "large" U -orbit (rank-four, with negative I 4 ), or (in Case ii) the change from a "small" U -orbit (rank-two) to "small" orbit, of possible rank 3, 2, 1. For the case of rank= 2, generally a change of orbit representative takes place (as before, depending on the real form of the theory under consideration, this may result necessarily in remaining at the same rank-two orbit, or possibly in switching to another rank-two orbit [6,5]).
and there are two possibilities: i : I 4 Q 1M +Q E < 0; (4.63) ii : In Case i the resulting FTS element has rank four (with negative I 4 ), and thus, it corresponds to a "large" EBH, whereas in Case ii, the resulting FTS has one of the possible ranks 3, 2, 1, corresponding to a "small" EBH. Thus, in this case, the action ofÔ P Q results in various types of orbit transmutations: a change from a "small" U -orbit (rank-one) to a "large" U -orbit (rank-four, with positive or negative I 4 ), or (in Case ii) the change from a "small" U -orbit (rank-one) to "small" orbit, of possible rank 3, 2, 1; in the case of rank= 1, generally a change of orbit representative takes place, and regardless of the real form of the theory under consideration, one necessarily remains in the same rank-one orbit, which is always unique [6,5].
rank Q E = 2 ⇔ ii : In Case i, the resulting FTS element has rank four, and thus, it corresponds to a "large" EBH, whereas in Case ii, the resulting FTS has possible ranks 3, 2, 1, corresponding to a "small" EBH. Thus, in this case, the action ofÔ P Q results in various types of orbit transmutations: a change from a "small" U -orbit (rankone) to a "large" U -orbit (rank-four, with negative I 4 ), or (in Case ii) the change from a "small" U -orbit (rank-one) to "small" orbit, of possible rank 3, 2, 1; in the case of rank= 1, once again, one necessarily remains in the same rank-one orbit [6,5].
and there are two possibilities: ii : In Case i, the resulting FTS element has rank four (with negative I 4 ), and thus, it corresponds to a "large" EBH, whereas in Case ii, the resulting FTS has possible rank 3, 2, 1, corresponding to a "small" EBH.
Thus, in this case, the action ofÔ P Q results in various types of orbit transmutations: a change from a "small" U -orbit (rank-one) to a "large" U -orbit (rank-four, with negative I 4 ), or (in Case ii), the change from a "small" U -orbit (rank-one) to "small" orbit, of possible rank 3, 2, 1; in the case of rank= 1, once again, one necessarily remains in the same rank-one orbit [6, 5].
1.3] "Small" rank-1 magnetic KK configuration : where |Q ′ E | q0=0 is the q 0 = 0 limit of |Q ′ E given by (4.5). Thus, in this case, the Peccei-Quinn symplectic transformation generates a ρ-dependent graviphoton electric charge and c i -dependent electric charges. These latter in Type II compactifications correspond to a stack of D2 branes depending on the components of the second Chern class c 2 of the Calabi-Yau three-fold. The corresponding transformation of I4reads: Thus, depending on the sign of the r.h.s. of I 4 ( |Q ′ E | q0=0 ), a "large" (I 4 > 0: BPS, or non-BPS Z H = 0), a "small" (I 4 = 0:BPS or non-BPS), or a "large" non-BPS Z H = 0 (I 4 < 0) BH charge configuration is generated. 1.2] "Small" rank-1 electric KK configuration : (4.75)

Superpositions
In the Hilbert space representation, the superposition of EBH charge configurations allows one to deal with the whole charge scenario for the state configurations of EBHs. A quite general class of superpositions can be written in the following form: where α 1 , ..., α 11 ∈ C (see the next section for a discussion of the necessary emergence of complex parameters).

Entanglement PQ Operators and Complexification
Generalized PQ operators can be defined as tensor products of PQ operators (2.10). The simplest (i.e., two-fold) generalized PQ operator is defined as:Ô (2) P Q :=Ô P Q ⊗Ô ′ P Q , (6.1) where (recalling Equation (2.10)): For instance, the action ofÔ (2) P Q (with c i = 0) on a composed configuration of two KK "large" EBH states (which can also be interpreted as the state of a two-centered EBH in which each center is characterized by a KK "large" charge configuration) is given by (recall Equation (4.5)): where we introduced: |αβ := |α ⊗ |β , α, β = 0, 1 (6.5) as the basis of the Hilbert space of two "large" KK EBHs (or, equivalently, of a two-centered EBH in which each center is characterized by a KK "large" charge configuration).
However, in order to generate entanglement in EBH systems through PQ symplectic transformations and thus realize "EBH quantum circuits" and "EBH quantum gates" in the context of BHQC [45], one can employ simpler composed PQ operators, namely: which we name the (two-fold) entangled PQ operator;Î denotes the identity operator and α, β ∈ C.
As an example, let us again consider a composed configuration of two KK "large" EBH states. The action ofÊ P Q (6.6) (with c i = 0) is given by: In the limit of indistinguishability of the KK EBH states, one can generate different states of the composed KK two-fold Hilbert space by a suitable choice of the complex parameters α and β; indeed, in such a limit, Equation (6.7) simplifies down to: (|Q KK ⊗ |Q KK ) = (α + β) p 0 2 |00 + (α + β) p 0 q 0 + αρ p 0 2 01 + (α + β) p 0 q 0 + βρ p 0 2 |10 + (α + β) q 0 q 0 + ρp 0 |11 . (6.8) Thence, for instance, the conditions: (α + β) p 0 q 0 + βρ p 0 2 = 0 (6.9) solved by: yield a non-normalized Bell (GHZ) state: on which the further normalization conditions: have a solution only when considering the complexification R C of the G 4 -representation space of the EBH electromagnetic charges. As observed, e.g., in [28], this is a crucial step in order to implement a quantum mechanical computation (such as the ones exploited by the "EBH quantum circuits" and "EBH quantum gates" [45]), since in QIT, all parameters are generally complex and not real (as instead, the electric and magnetic charges of an EBH are). This clearly implies that the PQ symplectic transformations themselves should be considered on a complex ground field C, since they act on the complex vector space R C , on which G 4 (C) acts linearly, but non-transitively. Thus, by choosing 7 : (|Q KK ⊗ |Q KK ) = 1 √ 2 (|00 + |11 ) , (6.14) namely, one can generate a normalized Bell (GHZ) state acting with the entanglement PQ operator on a suitable, complexified "large" KK EBH state.
However, we anticipate that when relaxing the indistinguishability assumption (i.e., when resorting to a multicentered EBH picture), much more freedom is introduced in the above entanglement generation procedure, and more general (and less constrained) solutions are obtained .
Clearly, the presence of other EBH charge configurations/states (such as the ones considered in Sections 3 and 4) allows for a wealth of entangled EBH states. A thorough analysis of PQ entanglement generation on EBH states will be carried in a forthcoming paper.

Conclusions
Within the black-hole/qubit correspondence, in the present paper, we have paved the way to a consistent generation of entanglement in EBH systems, by means of an operator implementation of the so-called Peccei-Quinn (PQ) symplectic transformations.
Given an EBH state described by its supporting electromagnetic charge configuration (i.e., by an element, with a well-defined associated rank, of the corresponding Freudenthal triple system), the action of the PQ operator can result in various types of orbit transmutations, namely in various mechanisms of switching among different U -duality orbits.
In the indistinguishability assumption, the need for complexification of the U -duality representation space R (resulting in a complexification of the PQ transformations, as well) and the unavoidable occurrence of phases among pure EBH states has been pointed out; however, we anticipate that if one resorts to the multi-centered EBH space-time picture and relaxes the indistinguishability condition, at least some of such limits can be circumvented.