N-level ghost-spins and entanglement

Ghost-spins, 2-level spin-like variables with indefinite norm have been studied in previous work. Here we explore various $N$-level generalizations of ghost-spins. First we discuss a flavoured generalization comprising $N$ copies of the ghost-spin system, as well as certain ghost-spin chains which in the continuum limit lead to 2-dim $bc$-ghost CFTs with $O(N)$ flavour symmetry. Then we explore a symplectic generalization that involves antisymmetric inner products, and finally a ghost-spin system exhibiting $N$ irreducible levels. We also study entanglement properties. In all these cases, we show the existence of positive norm"correlated ghost-spin"states in two copies of ghost-spin ensembles obtained by entangling identical ghost-spins from each copy: these exhibit positive entanglement entropy.


Introduction
Ghost-spin systems [1,2,3] and their patterns of entanglement are interesting from multiple points of view. Some of these possible applications involve ghost sectors in theories with gauge symmetry, while others pertain to conjectures involving de Sitter physics and dS/CF T dualities [4,5,6,7,8,9,10,11,12,13,14] including higher spin versions. Ghost-spins are 2-level spinlike variables but with indefinite norm [1]: thus in some ways, they are best regarded as simple quantum mechanical toy models for theories with negative norm states. Ghost-spin systems have interesting entanglement patterns by virtue of this indefinite norm. If one considers ghost spin systems with even number of ghost spins then it is possible to find subspaces in the Hilbert space where one obtains positive entanglement entropy for positive norm states [1,2].
The situation is not so fortuitous in the case of odd number of ghost spins. However, several interesting physical systems which contain indefinite norm states seem to admit even numbers of such states. Taking the study of the ghost spins further, it was shown in [3] that appropriate ghost-spin chains in the continuum limit give rise to the bc-ghost CFTs in two dimensions. This suggests that they may be regarded as microscopic building blocks for ghost-CFTs and perhaps more general non-unitary theories.
The explorations of the ghost-spin system so far have used 2-state spin-like variables with indefinite inner product. It is interesting to study generalizations of ghost-spins with flavour degrees of freedom assigned to them. These flavour quantum numbers may be relevant for applications to non-abelian gauge theories. However, the ghost spin system with flavours by themselves is an interesting set up worthy of exploration. Once we allow for flavour symmetry, tions of holographic entanglement entropy [18,19,20,21]. In contrast with a single spin which has ↑ | ↑ = 1 = ↓ | ↓ , a single ghost-spin is defined as a 2-state spin variable with indefinite inner product ↑ | ↓ = 1 = ↓ | ↑ , ↑ | ↑ = 0 = ↓ | ↓ . (2.1) A general state ψ + |+ + ψ − |− thus has norm |ψ + | 2 − |ψ − | 2 , which is not positive definite.
Now consider restricting to the subspace These states of "correlated ghost-spins" comprise entanglement between two copies of identical states: they can be seen to be strictly positive norm, with a positive reduced density matrix (2.3) and positive entanglement. In [15], a picture of de Sitter space as a thermofield double type state (with de Sitter entropy then emerging as the entanglement entropy) was discussed based on such correlated ghost-spin states in two copies of ghost-CFTs at the future and past boundaries of dS 4 in the static coordinatization. For ensembles with an even number of ghost-spins, such correlated ghost-spin states always exist comprising positive norm subsectors, as argued in [2], where ensembles of ghost-spins were developed further with regard to their entanglement properties. Odd ghost-spins were found to behave differently: for instance, |ψ = This is not positive definite for n odd (even if ψ|ψ > 0). Ensembles of ghost-spins and spins were also found to exhibit interesting entanglement patterns.
In [3], certain 1-dim ghost-spin chains with specific nearest-neighbour interactions were found to yield bc-ghost CFTs in the continuum limit, i.e. these ghost-spin chains are in the same universality class as those ghost-CFTs. We will not review this here since this will effectively be encompassed in a related detailed description later.
N-level ghost-spins: In this paper, we generalize the 2-level ghost-spin reviewed above to N-levels by considering various generalizations as outlined below: • O(N) symmetry flavour generalization of the bc-ghost system: These are essentially N copies of the 2-level ghost-spin system. It is then possible to find appropriate ghost-spin chains which lead to a generalization of the bc-ghost system but with internal O(N) flavour indices. A simple generalization of this case involves the flavours having a spin-glass type interaction with coupling J AB which is in general non-vanishing for A = B, In flavour space, this thus encodes possibly nonlocal flavour couplings. Taking the J AB matrix to be real and symmetric allows diagonalization and in that diagonal basis, this can be reduced to the above O(N) flavoured case.
• N-levels with symplectic-like structure: (2.9) These have a symplectic structure built into the inner product, which was in part motivated by 3-dim ghost-CFTs of symplectic fermions [22,23] that have been discussed in the conjectured duals to higher spin dS 4 [7].
This case is slightly different from the previous cases in that the elemental ghost-spins are not 2-level anymore (with flavour indices), but irreducibly N-level.
In all these cases representing N-level generalizations of ghost-spin ensembles, we will argue that correlated ghost-spin states exist comprising a uniformly positive norm subspace of states with the interpretation of entanglement between two copies of the state space. This will be the main point of the paper.
It is possible to find operator realizations consistent with some of these inner products above. The first case essentially comprises N copies of the bc-operator algebra, We can then define ghost-spin-chain Hamiltonians with nearest neighbour hopping type interactions but with the flavours decoupled, Based on the fact that the continuum limit for the single flavour case is the familiar bc-ghost CFT [3], the continuum limit can be argued to be flavoured generalizations of bc-ghost CFTs, with the flavour contractions exhibiting O(N) symmetry. We will discuss this in detail in sec. 3.
The symplectic inner products above are consistent with the operator algebra as we will discuss in sec. 5. The continuum limit is less clear in this case, although there are indications that these may be related to logarithmic CFTs [24,25,26,27,28,29,30].

N -level ghost-spins with O(N ) flavour symmetry
In this section, we consider an N-level generalization of ghost-spins with O(N) symmetry among the N internal flavour indices, defined by (2.7), i.e. we have the elemental inner products A, B = 1, 2, . . . , N , This is essentially N copies of the 2-level ghost-spin reviewed in sec. 2. In the second line, we have defined a convenient basis where the inner product is diagonal: this makes manifest the negative norm basis states.
Consider first a single ghost-spin with N flavours. The general configuration is defined by specifying the simultaneuous configurations for each of the N flavours so the general state comprising various basis states |s i is i.e. in the first basis state, the first flavour is ↑ 1 , the second is ↑ 2 , third being ↑ 3 and so on, and likewise for the other basis states. It is important to note that the |s i are really direct product states over the various flavour components: although we have written them as column vectors for convenience of notation (especially in light of the discussion later on multiple ghost-spins), the inner products between these states is not a dot product between two column vectors.
Instead we define the inner products between the configurations |s i as where i, j = 1, 2, . . . , 2 N label the configurations, A 1 , B 1 , · · · = 1, 2, . . . , N label the flavours and ǫ A 1 A 2 ...A N is the totally symmetric tensor with ǫ 12...N = 1 and ǫ A 1 A 2 ...A N vanishes if any two labels are the same. In other words, ǫ A 1 A 2 ...A N is simply a book-keeping device for ensuring that each elemental state |s A j in |s j is paired with another corresponding elemental state in s i |. To illustrate how this works, let us consider a simple example of a single ghost-spin with N = 2 flavours: the distinct configurations of this system are described by the basis states Then the inner product (3.3) simplifies to where we have used ǫ 12 = 1 = ǫ 21 . Using the elemental inner products in (3.1) gives Writing this out explicitly, we have The other inner products vanish. Based on these inner products for the basis states, we can write the norm for the generic state as Appropriate pairs of states can be used to define a new basis of positive and negative norm states: s 1 ± s 4 , s 2 ± s 3 and so on have norm ±2 respectively. Likewise for N = 4 flavours, the 2 4 = 16 configurations |s i are 1 and the inner product (3.3) is (3.10) Using (3.1), this simplifies to noting that there are 4! non-zero ǫ A 1 A 2 A 3 A 4 components, which can succinctly be written as with other inner products vanishing. Similarly, for general N, the inner product (3.3) is Thus two generic states have inner product 1 These states can be written in terms of a basis which is manifestly O(N ) invariant. Namely Where, 0 is the trivial representation, vec corresponds to the vector representation and adj is the adjoint(antisymmetric) representation of O(N ). To see the relation between the |t and the |s bases, consider for example, the states in (3.9): these get organised into representations of O(4) as 1, 4, 6, 4, 1 with the states |s 1 and |s 16 forming two singlets, |s 2 to |s 5 (|s 12 to |s 15 ) belonging to the vector representation and |s 6 to |s 11 to the adjoint representations of O(4). We will, however, continue using the basis given in (3.2) for convenience.

Ghost-spin chain for the bc-CFT with O(N ) symmetry
We want to now study infinite ghost-spin chains along the lines of those in [3], but with additional flavour structure respecting the O(N) flavour symmetry we have been discussing so far. In effect, this amounts to N flavour copies of ghost-spins at each lattice site. Thus we define two species of N-component commuting spin variables, σ A bn and σ A cn at each lattice site n satisfying These commuting spin-variables are Hermitian and their action on ghost-spin states is where there is no summation over the flavour index B. For multiple ghost-spin states in the chain, the σ A bn operator acts to lower the A th -flavour state within the ghost-spin configuration at site n, and likewise σ A cn is the corresponding raising operator. To illustrate this explicitly, consider two ghost-spins with N = 2 flavours: then The ghost-spin chain is then defined by a Hamiltonian encoding interactions between the ghost-spins at various lattice sites. Since the flavours do not mix, the interaction Hamiltonian here is simply a straightforward generalization involving a decoupled sum over various flavours of the single flavour one in [3]. So consider a 1-dim ghost-spin chain with a "hopping" type interaction Hamiltonian where n, n + 1, n − 1 label nearest label lattice sites in the chain. The action on a nearest neighbour pair of ghost-spins at lattice sites (n, n + 1) is given as The N flavours are decoupled and the interaction between ghost-spins at two neighbouring lattice sites is through the same flavour at the two sites. Thus we can follow the analysis in [3] flavour-by-flavour.
Towards constructing the continuum limit of the ghost-spin chain (3.18), we note that the σ A b,c operators commute at neighbouring lattice sites as in the single flavour case. Thus we define two species of N-component fermionic operators satisfying the anti-commutation relations which anti-commute at different lattice sites i, j also. The action of these fermionic operators on ghost-spin states is This is obtained by constructing a flavoured generalization of the Jordan-Wigner transformation in [3] for the commuting spin variables we can check that the operators a A bn , a A cn are hermitian. Now substituting the Jordan-Wigner transformation (3.22) in the ghost-spin Hamiltonian in the commuting spin variables (3.18) gives We see that this Hamiltonian for the 1-dimensional chain of N-level ghost-spins with O(N) symmetry breaks up as a decoupled sum of N copies of the Hamiltonian for 2-level ghostspins in [3]. Then following the analysis there for each flavour independently and taking the continuum limit, we can show that we obtain N copies of bc-ghost CFTs with O(N) flavour symmetry. This finally gives which is essentially the operator L 0 for a bc-ghost CFT enjoying O(N)-flavour symmetry, with and a corresponding anti-holomorphic part. Further details are similar to [3], except with multiple flavours.

Correlated ghost-spin states and entanglement
We now return to ghost-spin ensembles and their entanglement properties. Along the lines in (3.2) for enumerating states in the ↑, ↓-basis, we can clearly use the |± i -basis to define the basis states |s i there: the advantage in the |± i -basis is that positive/negative norm states are easier to identify manifestly. For a single ghost-spin with N = 2 flavours, we have then We remind the reader that although we are using column vectors for notational convenience, these are really direct product states: the inner product (3.3) here gives and this is an orthonormal basis of positive and negative norm states. A generic state then has Thus states made from |s 2 , |s 3 alone have negative norm. It is straightforward to write down similar basis states for arbitrary N flavours. For N flavours, there are 2 N basis states |s i . The inner products are Since this is a diagonal basis now, we have s i |s i = ±1 respectively when there is an even or odd number of |− A elemental ghost-spins in |s i . This is exemplified in the N = 2 case (3.30) above.
Now let us consider two ghost-spins. The states can be made from the 2 2N basis states |s i |s j obtained by tensor products of the single ghost-spin states. The inner products between them are Then the general state and its norm are with two products over the flavour components of the two basis states. Tracing over say the second ghost-spin in this state leads to a subsystem comprising the single remaining ghost-spin, with reduced density matrix defined as The entanglement entropy of this reduced density matrix can then be calculated using the formulation in [1, 2, 3]: we will see this below. Restricting attention for simplicity to N = 2 flavours, we can use the four basis states (3.29). Then the 2 ghost-spin states can be described using the 16 basis states |s i,j ≡ |s i |s j , or more explicitly, The inner products (3.33) can then be seen to give the norms e.g.  This is a sum over |ψ s i ,s j | 2 weighted by ±1 depending on the sign of the norm of |s i,j . A particularly interesting subset of states are what we call "correlated states", generalizing the discussion in [2]. These are of the form |ψ s j ,s j | 2 > 0 , (3.39) and are necessarily positive norm, even though some of the individual basis states are negative norm. The basis states |s i |s i here are of the form and we see explicitly that this subspace of correlated states is obtained by entangling some configuration for the first ghost-spin with an identical configuration for the second ghost-spin.
Thus we have only 4 states which span the correlated ghost-spin subspace. It is clear that these are necessarily positive norm since there is an even number of minus ghost-spins (any odd number in each column is doubled). Note that this is a smaller subspace than that comprising all positive norm states which simply need to have an even number of minus signs: e.g. the basis state There are 2 N basis states |s i so this subspace of correlated states is 2 N -dimensional, somewhat smaller than the 2 2N -dimensional space of all states.
These correlated ghost-spin states entangle identical ghost-spins between the two sets of ghost-spins. These states necessarily encode positive entanglement since any sublinear combination of the norm is still positive definite (one way to see this is to note that this can be mapped to an auxiliary system of ordinary spins, which has no minus signs and is entirely positive norm). More explicitly, for a state of the form |ψ = ψ s I ,s I |s I |s I + ψ s J ,s J |s J |s J made of the states |s I,I , |s J,J , the reduced density matrix (3.35) can be taken to construct a mixed-index reduced density matrix as in [1,2,3], which then makes explicit the contraction structure with respect to the ghost-spin inner product metric (incorporating the signs for negative norm states). To be explicit, consider |ψ = ψ s 1 ,s 1 |s 1 |s 1 + ψ s 2 ,s 2 |s 2 |s 2 noting that |s 1 and |s 2 are positive and negative norm respectively. Then The mixed-index reduced density matrix here is (ρ A ) s i s j = γ s j ,s k (ρ A ) s i ,s k = s j |s k (ρ A ) s i ,s k , where the metric γ s i ,s j = s i |s j is defined by the inner products (3.30). This description can be generalized to all such states, and indeed to all 2-ghost-spin states (3.38): in this case, it can be shown along the lines of the single flavour case that more general positive norm subsectors exist. In general however, the state space has many branches of negative norm states, and the reduced density matrix in general has negative eigenvalues with a complex entanglement entropy correspondingly (as was already the case in the single flavour case).
These states can be generalized to any even number of ghost-spins. For odd numbers of ghost-spins however, this structure does not prevail: there are states that are positive norm but the reduced density matrix continues to have negative eigenvalues so that the entanglement entropy is not positive.
It is now interesting to consider two copies of ghost-spin chains and consider correlated ghost-spin states representing entanglement between the two chains. This is motivated by the discussion and picture in [15] of dS 4 as dual to a thermofield-double type entangled state in two copies CF T F × CF T P of the ghost-CFT at I + and I − (reviewed in the Discussion in sec. 6). So let us consider GC 1 × GC 2 where each GC represents a ghost-spin chain whose continuum limit Then correlated entangled states in GC 1 × GC 2 can be represented as Now the states |σ include the ground state as well as excited states. If we restrict to the ground states alone, then since the flavours are all decoupled from each other, the ground states |σ comprise a 2 N -dimensional subspace noting that each |s n at lattice site n in |σ has 2 N possibilities. Thus tracing over the second ghost-spin chain copy GC 2 , we obtain the entanglement entropy of |ψ restricting to ground states |σ g as We have used g |ψ σg,σg | 2 = 1 from normalization and imposed maximal entanglement, which equates all the coefficients giving |ψ σg,σg | 2 = 1 2 N . Thus the entanglement entropy scale as the number of flavours N.

Symmetric, spin-glass type, inner products
Here we briefly mention a generalization of the O(N) flavoured case but with the various flavours talking to each other, with a spin glass type coupling. We define the elemental inner products (2.8) using a symmetric form J AB and take the inner products between the configurations |s i to be (3.3).
Let us consider first N = 2 flavours: then the states are as in (3.4) and the inner products are s i |s j = s 1 i |s 1 j s 2 i |s 2 j + s 1 i |s 2 j s 2 i |s 1 j . (3.45) Using the elemental inner products (2.8), the non-zero inner products are The metric in the space of |s i 's is real, symmetric and its determinant is (det J)(J 11 J 22 + We will not dwell more on this, although this may be worth investigating further.

Symplectic inner products
We want to study "symplectically flavoured" ghost-spins. We introduce the symplectic structure by defining the elemental inner products with an anitsymmetric matrix : where Ω AB is a symplectic form, which is antisymmetric, i.e. Ω AB = −Ω BA . We will take the only nonzero elements as For a single symplectically flavoured ghost-spin there are 2 2N distinct configurations comprising the basis states |s 1 , . . . , |s 2 2N and a generic state is We define inner products s j | s i between the basis states as Let us consider first a single ghost-spin with 2 flavours (N = 1): then the distinct configurations comprise the four basis states i.e. in |s 1 , the first flavour is ↑ 1 and the second flavour is ↑ 2 , and likewise for |s 2 , |s 3 , |s 4 . The non-zero elemental inner products in (4.1) are and the inners products (4.4) between the configurations simplify to Then non-zero inner products for the configurations (3.4) are These give a real, symmetric and non-singular metric in the space of configurations |s i . Based on these inner products, we can write the norm for the generic state as Since these are the only non-zero elemental inner products, the inner products (4.4) for the basis states (3.9) reduce to whereÃ k = |ΩÃ k A l |A l , e.g. for A 1 = 1,Ã 1 = 2, for A 2 = 4,Ã 2 = 3, etc. As there are only 4! such non-zero terms in the above inner product, it becomes The non-zero inner products between |s i 's computed using this formula can be written compactly as where |s j is defined such that if the A-th flavour entry in |s i is ↑ A (or ↓ A ) then theÃ-th flavour entry in |s j is ↓Ã (or ↑Ã ) forÃ = |ΩÃ A |A. We see that the metric s i |s j is real, symmetric and non-singular.
We can generalize this to 2N flavours, where the non-zero elemental inner products are Then the inner products (4.4) become Using the elemental inner products, we see that the non-zero inner products are s i |s j = 1, with |s j as defined earlier.
Thus the norm of a generic state |ψ = ψ s i |s i is Along the same lines, the general inner product between any two states is  It is worth noting that the symplectic invariance is at the level of the elemental ghost-spin which is invariant under symplectic transformations. To see this explicitly, consider two flavours (N = 1) for simplicity. Then a symplectic transformation by a real pseudo-orthogonal matrix where Ω is the symplectic form with where we have used R CA Ω AB R BD = R T ΩR = Ω.
The elemental inner products (4.6) are consistent with (and motivated by) an operator algebra alongwith states, defined as (these arise in theories of symplectic fermions [25]) where | ↑ and | ↓ are ghost-spin states with ↑ | ↓ = 1 = ↓ | ↑ . The hermiticity of c and real Ω AB givesK † = −K i.e.K is anti-hermitian. This anti-Hermitian operator leads which implies that for ↑ |K| ↓ = 1, ↓ |K| ↑ = −1. Using these we get the elemental inner products as It is then possible to construct ghost-spin chains with nearest neighbour interactions between operators at neighbouring lattice sites, somewhat similar to the ghost-spin chain for the bc-ghost CFTs. However the continuum limit is less clear in this case, in part due to technical difficulties such as the construction of the Jordan-Wigner transformation to obtain fermionic versions of the σ b,c operators above which anticommute with each other (the σ b,c are bosonic spin-like operators commuting at neighbouring lattice sites while anticommuting at the same site). Note however that the case with N = 1 has structure similar to that appearing in the theory of anticommuting scalars: this is a logarithmic CFT in 2-dimensions [24,25,26,27,28,29,30].
So perhaps the continuum limit here gives symplectic fermions Ω AB ∂φ A ∂φ B : we hope to explore this further.
Then the generic state and its norm in both bases are To illustrate this, let us consider N = 3. The generic state and its norm are In some sense, this is a ghost-spin generalization of the N-level spins that arise in the Heisenberg spin chain: perhaps appropriate interaction Hamiltonians for ghost-spin chains on a 1-dim lattice can be studied along those lines.
Correlated ghost-spins and entanglement: We want to construct correlated ghost-spin states analogous to the discussion in sec. 3.2. So consider a system of two ghost-spins with N irreducible levels. The orthonormal basis for this system is where each |α is a single ghost-spin basis state in (5.2). A generic state |ψ = ψ αβ |αβ has a norm ψ|ψ = η ακ η βλ (ψ αβ ) * ψ κλ , which can be expanded as We see a manifest division between the positive and negative norm subspaces. For N = 3, we can see this explicitly as For general N, by tracing over the second ghost-spin, the reduced density matrix is The mixed index reduced density matrix is (ρ A ) α β = η βκ (ρ A ) κα = η βκ η λλ ψ κλ (ψ αλ ) * .
• If x < 0, the norm of |ψ can be positive or negative.

Discussion
We have constructed N-level generalizations of the 2-level ghost-spins in [1,2,3]. These include (i) a flavoured generalization comprising N copies of the ghost-spin system and corresponding ghost-spin chains which lead to 2-dim bc-ghost CFTs with O(N)) flavour symmetry, (ii) a spinglass type coupling in flavour space, (iii) a symplectic generalization involving antisymmetric inner products between the elemental ghost-spins, and (iv) an irreducible ghost-spin system with N internal levels. We have studied entanglement properties in these cases: among other things, these show the existence of positive norm states in two copies of ghost-spin ensembles obtained by entangling identical ghost-spins from each copy: these are akin to the correlated ghost-spin states in [2,3], and exhibit positive entanglement.
We now describe briefly some of the motivations from dS/CF T , in particular [15], for the studies here. Generalizations of gauge/gravity duality for de Sitter space or dS/CF T involve conjectured dual hypothetical Euclidean non-unitary CFTs living on the future boundary I + [4,5,6]. Using the dictionary Ψ dS = Z CF T [6], where Ψ dS is the late-time Hartle-Hawking wavefunction of the universe with appropriate boundary conditions and Z CF T the dual CFT partition function, the dual CFT d energy-momentum tensor correlators reveal central charge (effectively analytic continuations from AdS/CF T ). This is real and negative in dS 4 , with C 3 ∼ − R 2 dS G 4 so that dS 4 /CF T 3 is reminiscent of ghost-like nonunitary theories. Bulk expectation values are of the form ϕ k ϕ k ′ ∼ Dϕ ϕ k ϕ k ′ |Ψ dS | 2 . This involves the probability |Ψ dS | 2 = Ψ * dS Ψ dS , which suggests that bulk de Sitter physics involves two copies of the dual CFT -CF T F × CF T P on the future and past boundaries. This is unlike in AdS/CF T where Z bulk = Z bndry implies boundary correlators can be obtained as a limit of bulk ones. In the dS case, while Z CF T = Ψ dS of a single dual CFT copy at I + can be used to obtain boundary correlators (e.g. for operators O k dual to modes ϕ k ), bulk observables require |Ψ dS | 2 , the bulk probability, and so two copies of the dual Z CF T .
This dovetails with the structure of extremal surfaces and entanglement as we see below.
In AdS, surfaces anchored at one end of a subsystem dip into the bulk radial direction and then begin to return to the boundary at turning points. In dS, the boundary at I + is spacelike and surfaces dip into the time direction which ends up making their structure quite different, as studied in [16]. For instance, considering the dS Poincare slicing ds 2 = R 2 dS τ 2 (−dτ 2 + dx 2 i ) , a strip subsystem on some boundary Euclidean time w = const slice of I + with width along x gives a bulk extremal surface x(τ ) described byẋ 2 ≡ ( dx dτ ) 2 = B 2 τ 2d−2 1+B 2 τ 2d−2 (B 2 > 0). w, x can be taken as any of the x i (so the boundary Euclidean time slice is not sacrosanct). Compared with the AdS case, the denominator here crucially has a relative minus sign. Thus there is no real turning point here where the surface starting at I + begins to turn back towards I + : this requires |ẋ| → ∞ while here |ẋ| ≤ 1. There are also complex extremal surfaces however, which exhibit turning points: these end up amounting to analytic continuation from the AdS Ryu-Takayanagi surfaces. While their interpretation is not entirely conclusive, in dS 4 they have negative area, consistent with the negative central charge in dS 4 /CF T 3 as mentioned above.
Since surfaces starting at I + do not turn back, it is then interesting to ask if they could instead stretch all the way to the past boundary I − . In [15], connected codim-2 extremal surfaces in the static patch coordinatization of de Sitter space were found stretching from I + to I − passing through the vicinity of the bifurcation region with divergent area l 2 4G 4 1 ǫ , where ǫ = ǫc l is the dimensionless ultraviolet cutoff and the coefficient scales as de Sitter entropy. To elaborate a little, the static patch coordinatization can be recast as ds 2 = l 2 τ 2 − dτ 2 1−τ 2 +(1−τ 2 )dw 2 +dΩ 2 d−1 , with the future/past universes F/P parametrized by 0 ≤ τ ≤ 1 with horizons at τ = 1, while the Northern/Southern diamonds N/S have 1 < τ ≤ ∞. The boundaries at τ = 0 are now of the form R w × S d−1 , resembling the Poincare slicing locally. Setting up the extremization for codim-2 surfaces on boundary Euclidean time slices can be carried out: on S d−1 equatorial planes for instance we obtainẇ 2 = B 2 τ 2d−2 1−τ 2 +B 2 τ 2d−2 . The minus sign here, reflecting the horizons, makes the structure of these surfaces interesting, drawing parallels with the AdS extremization (we refer to [15] for further details). The limit B → 0 gives surfaces passing through the vicinity of the bifurcation region as stated above, with the width ∆w approaching all of I ± .
These connected surfaces stretching between I ± are akin to rotated versions of the connected surfaces of Hartman, Maldacena [31] in the AdS black hole. This led to the speculation there that dS 4 is approximately dual to an entangled thermofield-double type state of the form |ψ = ψ i F n ,i P n |i F n |i P n (6.1) akin to the thermofield double [32] dual to AdS black holes. Here ψ i F n ,i P n are coefficients entangling a generic ghost-spin |i F n from CF T F at I + with an identical one |i P n from CF T P at I − . The constituent states are schematically continuum versions of N level ghost-spins, with N related to dS 4 entropy l 2 4G 4 . Since bulk time evolution maps configurations at I − to those at I + [5], we have the schematic map |i P n → S[i P n , i F n ]|i P n ≡ |i F n where S[i P n , i F n ] is the operator representing bulk time evolution (note that this is a bulk object that is to be distinguished from operators in the CF T representing boundary Euclidean time evolution). If states |i P at I − map faithfully and completely to states |i P at I + , then S is expected to be a unitary operator (this is also vindicated by the fact that exchanging I ± is a bulk symmetry). This suggests that the entangled states (6.1) are unitarily equivalent to similar maximally entangled states |ψ = ψ i F n ,i F n |i F n |i F n in two CF T F copies of the ghost-CFT solely at I + . The state (6.1) is akin to a correlated ghost-spin state with an even number of ghost-spins, as discussed in [2,3].
It necessarily has positive norm i F n ,i F n γ i F n ,i F n γ i F n ,i F n ψ i F n ,i F n (ψ i F n ,i F n ) * → in |ψ i F n ,i F n | 2 , since we are entangling identical states i F n and i P n : thus it has positive entanglement, as in [2,3]. Since each constituent state |i F,P n is N-level, i.e. with N internal degrees of freedom, the entanglement entropy scales as N ∼ l 2 G 4 . The toy models in sec. 3.2 of correlated ghost-spin states (3.43) and their entanglement entropy (3.44) are of this form, written explicitly. The state (6.1) is akin to the thermofield double dual to the eternal AdS black hole [32]. This suggests the speculation that 4-dim de Sitter space is perhaps approximately dual to CF T F × CF T P in the entangled state (6.1) and the generalized entanglement entropy of the latter scales as de Sitter entropy.
(See [33] for another approach to de Sitter entropy based on the dS/dS correspondence [34].) The investigations in this paper on N-level generalizations of ghost-spins are geared towards constructing microscopic ghost-spin states that reflect the N-level internal structure which might ultimately give rise in appropriate continuum limits to theories such as the Sp(N) ghost-CFT dual to higher spin dS 4 (see also the recent work [14]). As we have seen, the N-level generalizations here do admit positive norm subsectors of the form of the correlated ghost-spin states indicated in (6.1).
As mentioned in the Introduction, the ghost-spin system has possible applications in gauge theories. The continuum limit of a d-dim ghost-spin system (as in [3] for the 2-dim case) with flavour quantum numbers may be relevant for studying entanglement in gauge theories in a covariant setting. A better understanding of the ghost-spin system and its coupling to ordinary spin systems as in [2,3] generalized to d-dimensions would be an ideal sandbox for understanding covariant formulations of subregion entanglement in gauge theories.
We have been thinking of ghost-spins as microscopic building blocks for ghost-like CFTs, and perhaps more general non-unitary CFTs. The discussions in this paper on ghost-spin chains have recovered 2-dim bc-ghost CFTs with flavour symmetries. The obvious generalization to 3 dimensions of the 2-dim case discussed in sec.3 (and in [3] for the single flavour case) has nearest neighbour hopping-type interactions of the elemental form h ∼ nn ′ σ A b, n σ A c, n ′ . This contains three σ b and three σ c operators at each lattice site, with possible flavour indices reflecting internal flavour symmetries. It would be interesting to study such 3-dim ghost-spin chains towards obtaining 3-dim ghost-CFTs in the continuum limit: so far, we have encountered conceptual difficulties as well as technical ones. We hope to report on this in the future.

A Orthonormal basis for N -level irreducible ghost-spin
We give the transformations which transform the defining basis for N-level irreducible ghostspin to an orthonormal basis. For even N-level ghost-spin, the transformations to orthonormal basis are |e i − 2(|e 5 + |e 6 ) , . . .
For odd N + 1-level ghost-spin (where N is even), the transformations to orthonormal basis as . . .
To illustrate these transformation, we write them explicitly for N = 3 and N = 4. For a 3-level ghost-spin, the orthonormal basis are