Emergent Geometry from Entanglement Structure
351
2.2 Exact Examples
For any general pure quantum state, computation of its entanglement adjacency
matrix requires knowledge of the entanglement of all possible bipartitions of the
state. The number of such bipartitions increases exponentially with the size of the
system. Hence, even for moderate system size, estimation of the entanglement adjacency matrix demands lots of computational effort. However, if the quantum state
possesses certain properties such that all entropies can be computed analytically,
corresponding entanglement adjacency matrix can be obtained straightforwardly.
Below we discuss few such cases, where the elements of the entanglement adjacency
matrix can be computed exactly.
I. Dimer model.—We start with the dimer state, which can be mathematically
expressed as |Ψ = |i 1 j 1 ⊗ |i 2 j 2 ⊗ · · · ⊗ |i k j k , with k =
N
2 for even
N and |i l j l =
1
√
2
(| ↑ i l ↓ j l − | ↓ i l ↑ j l Such states belong to the family
of valence bond states and appear as approximate ground states of certain
strongly inhomogeneous free-fermionic model. Here, from the configuration
of the state, one can observe that all the single-site entropies become S(ρ i ) =
log(2), ∀i ∈ N. On the other hand, for the two-site blocks, if i, j form a dimer
S(ρ ij ) = 0, and S(ρ ij ) = 2 log(2), otherwise. Therefore, the elements of the
entanglement adjacency matrix possess non-zero values only when i, j form
singlet, given byJ ij = log(2). In this case, one can find that the geometry
revealed by the entanglement adjacency matrix is a mere restriction of the onedimensional adjacency matrix representing the Hamiltonian.
II. Rainbow state.—Another important member of the family of valence bond
states we consider in our work is the rainbow state, which is also the ground
state of a local Hamiltonian [8–15]. Here, dimer are established among
symmetric qubits with respect to the center: i k = k, j k = N + 1 − k. The
state exhibits volume-law scaling of entanglement entropy with the increase
of the system size. In this case also, all the single-site entropies become
S(ρ i ) = log(2). Whereas, the non-zero values of the entanglement entropies
can be obtained only for ρ ij such that i+j = N +1, given by S(ρ ij ) = 2 log(2).
As a result, we get J ij = log(2), for i + j = N + 1 and zero otherwise.
Interestingly, one can note that in this case, the entanglement adjacency matrix
is not emerging as a restriction on the adjacency matrix representing the
Hamiltonian. In other words, an observer trying to determine the geometry
from the distribution of the entanglement will not find the correct geometry of
the Hamiltonian.
III. GHZ state.—A different case we consider here is the N -party GHZ state,
expressed as |GH Z =
1
√
2
(|0 N | + |1 N ). In this case, the entropy values
of all the bipartitions, irrespective of the number of sites, become identical,
given by log(2). Hence, all the J ij ’s become same. As a result, to represent the
block entropies using our formalism, we consider J ij = 0 ∀i, j ∈ N and put
351
2.2 Exact Examples
For any general pure quantum state, computation of its entanglement adjacency
matrix requires knowledge of the entanglement of all possible bipartitions of the
state. The number of such bipartitions increases exponentially with the size of the
system. Hence, even for moderate system size, estimation of the entanglement adjacency matrix demands lots of computational effort. However, if the quantum state
possesses certain properties such that all entropies can be computed analytically,
corresponding entanglement adjacency matrix can be obtained straightforwardly.
Below we discuss few such cases, where the elements of the entanglement adjacency
matrix can be computed exactly.
I. Dimer model.—We start with the dimer state, which can be mathematically
expressed as |Ψ = |i 1 j 1 ⊗ |i 2 j 2 ⊗ · · · ⊗ |i k j k , with k =
N
2 for even
N and |i l j l =
1
√
2
(| ↑ i l ↓ j l − | ↓ i l ↑ j l Such states belong to the family
of valence bond states and appear as approximate ground states of certain
strongly inhomogeneous free-fermionic model. Here, from the configuration
of the state, one can observe that all the single-site entropies become S(ρ i ) =
log(2), ∀i ∈ N. On the other hand, for the two-site blocks, if i, j form a dimer
S(ρ ij ) = 0, and S(ρ ij ) = 2 log(2), otherwise. Therefore, the elements of the
entanglement adjacency matrix possess non-zero values only when i, j form
singlet, given byJ ij = log(2). In this case, one can find that the geometry
revealed by the entanglement adjacency matrix is a mere restriction of the onedimensional adjacency matrix representing the Hamiltonian.
II. Rainbow state.—Another important member of the family of valence bond
states we consider in our work is the rainbow state, which is also the ground
state of a local Hamiltonian [8–15]. Here, dimer are established among
symmetric qubits with respect to the center: i k = k, j k = N + 1 − k. The
state exhibits volume-law scaling of entanglement entropy with the increase
of the system size. In this case also, all the single-site entropies become
S(ρ i ) = log(2). Whereas, the non-zero values of the entanglement entropies
can be obtained only for ρ ij such that i+j = N +1, given by S(ρ ij ) = 2 log(2).
As a result, we get J ij = log(2), for i + j = N + 1 and zero otherwise.
Interestingly, one can note that in this case, the entanglement adjacency matrix
is not emerging as a restriction on the adjacency matrix representing the
Hamiltonian. In other words, an observer trying to determine the geometry
from the distribution of the entanglement will not find the correct geometry of
the Hamiltonian.
III. GHZ state.—A different case we consider here is the N -party GHZ state,
expressed as |GH Z =
1
√
2
(|0 N | + |1 N ). In this case, the entropy values
of all the bipartitions, irrespective of the number of sites, become identical,
given by log(2). Hence, all the J ij ’s become same. As a result, to represent the
block entropies using our formalism, we consider J ij = 0 ∀i, j ∈ N and put