By the Majorana representation, for any $d > 1$ there is a one-one correspondence between a quantum state of dimension $d$ and $d-1$ qubits represented as $d-1$ points in the Bloch sphere. Using the theory of symmetry class of tensors, we present a simple scheme for constructing $d-1$ points on the Bloch sphere and the corresponding $d-1$ qubits representing a $d$ -dimensional quantum state. Additionally, we demonstrate how the inner product of two $d$ -dimensional quantum states can be expressed as a permanent of a matrix related to their $(d-1)$ -qubit state representations. Extension of the result to mixed states is also considered.By the Majorana representation, for any $d > 1$ there is a one-one correspondence between a quantum state of dimension $d$ and $d-1$ qubits represented as $d-1$ points in the Bloch sphere. Using the theory of symmetry class of tensors, we present a simple scheme for constructing $d-1$ points on the Bloch sphere and the corresponding $d-1$ qubits representing a $d$-dimensional quantum state. Additionally, we demonstrate how the inner product of two $d$-dimensional quantum states can be expressed as a permanent of a matrix related to their $(d-1)$-qubit state representations. Extension of the result to mixed states is also considered.

arXiv:2308.14765Aug 27, 20230 citations

Three-tangle of a general three-qubit state in the representation of Majorana stars

Chon‐Fai Kam, Ren‐Bao Liu

Majorana stars, the $2j$ spin coherent states that are orthogonal to a spin- $j$ state, offer a visualization of general quantum states and may disclose deep structures in quantum states and their evolutions. In particular, the genuine tripartite entanglement---the three-tangle of a symmetric three-qubit state, which can be mapped to a spin-3/2 state---is measured by the normalized product of the distance between the Majorana stars. However, the Majorana representation cannot be applied to general nonsymmetric $n$ -qubit states. We show that, after a series of $\mathrm{SL}(2,\mathbb{C})$ transformations, nonsymmetric three-qubit states can be transformed to symmetric three-qubit states, while at the same time the three-tangle is unchanged. Thus the genuine tripartite entanglement of general three-qubit states has the geometric representation of the associated Majorana stars. The symmetrization and hence the Majorana star representation of certain genuine high-order entanglement for more qubits are possible for some special states. In general cases, however, the constraints on the symmetrization may prevent the Majorana star representation of the genuine entanglement.Majorana stars, the $2j$ spin coherent states that are orthogonal to a spin-$j$ state, offer a visualization of general quantum states and may disclose deep structures in quantum states and their evolutions. In particular, the genuine tripartite entanglement---the three-tangle of a symmetric three-qubit state, which can be mapped to a spin-3/2 state---is measured by the normalized product of the distance between the Majorana stars. However, the Majorana representation cannot be applied to general nonsymmetric $n$-qubit states. We show that, after a series of $\mathrm{SL}(2,\mathbb{C})$ transformations, nonsymmetric three-qubit states can be transformed to symmetric three-qubit states, while at the same time the three-tangle is unchanged. Thus the genuine tripartite entanglement of general three-qubit states has the geometric representation of the associated Majorana stars. The symmetrization and hence the Majorana star representation of certain genuine high-order entanglement for more qubits are possible for some special states. In general cases, however, the constraints on the symmetrization may prevent the Majorana star representation of the genuine entanglement.

arXiv:1902.00684Physical Review AMar 13, 20209 citations

By the Majorana representation, for any

d > 1

there is a one-one correspondence between a quantum state of dimension

d

and

d-1

qubits represented as

d-1

points in the Bloch sphere. Using the theory of symmetry class of tensors, we present a simple scheme for constructing

d-1

points on the Bloch sphere and the corresponding

d-1

qubits representing a

d

-dimensional quantum state. Additionally, we demonstrate how the inner product of two

d

-dimensional quantum states can be expressed as a permanent of a matrix related to their

(d-1)

-qubit state representations. Extension of the result to mixed states is also considered.

Majorana stars, the

2j

spin coherent states that are orthogonal to a spin-

j

state, offer a visualization of general quantum states and may disclose deep structures in quantum states and their evolutions. In particular, the genuine tripartite entanglement---the three-tangle of a symmetric three-qubit state, which can be mapped to a spin-3/2 state---is measured by the normalized product of the distance between the Majorana stars. However, the Majorana representation cannot be applied to general nonsymmetric

n

-qubit states. We show that, after a series of

\mathrm{SL}(2,\mathbb{C})

transformations, nonsymmetric three-qubit states can be transformed to symmetric three-qubit states, while at the same time the three-tangle is unchanged. Thus the genuine tripartite entanglement of general three-qubit states has the geometric representation of the associated Majorana stars. The symmetrization and hence the Majorana star representation of certain genuine high-order entanglement for more qubits are possible for some special states. In general cases, however, the constraints on the symmetrization may prevent the Majorana star representation of the genuine entanglement.