Density matrices for mixed state qubits, parametrized by the Bloch vector in the open unit ball of the Euclidean 3-space, are well known in quantum information and computation theory. By presenting new identities for the qubit density matrix we indicate its intimate relationship with M\"obius addition and scalar multiplication. The latter, in turn, form the algebraic setting for the Poincar\'e ball model of hyperbolic geometry so that, as a result, the qubit density matrix is linked to hyperbolic geometry.Density matrices for mixed state qubits, parametrized by the Bloch vector in the open unit ball of the Euclidean 3-space, are well known in quantum information and computation theory. By presenting new identities for the qubit density matrix we indicate its intimate relationship with M\"obius addition and scalar multiplication. The latter, in turn, form the algebraic setting for the Poincar\'e ball model of hyperbolic geometry so that, as a result, the qubit density matrix is linked to hyperbolic geometry.

Quantum Information and ComputationOct 1, 20025 citations

Effective rationality for local unitary invariants of mixed states of two qubits

Luca Candelori, Vladimir Chernyak, John R. Klein +1 more

Abstract We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine algebraically independent polynomial invariants. We do so by constructing a relative section, in the sense of invariant theory, whose Weyl group is a finite abelian group. From this construction, we are able to give explicit expressions for the generating invariants in terms of the Bloch matrix representation of mixed states of two qubits. We also prove similar rationality results for the local unitary invariants of symmetrically mixed states of two qubits. We also provide a sketch of how to generalize our results to the case of an arbitrary number of qubits. Our results apply to both complex-valued and real-valued invariants.Abstract We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine algebraically independent polynomial invariants. We do so by constructing a relative section, in the sense of invariant theory, whose Weyl group is a finite abelian group. From this construction, we are able to give explicit expressions for the generating invariants in terms of the Bloch matrix representation of mixed states of two qubits. We also prove similar rationality results for the local unitary invariants of symmetrically mixed states of two qubits. We also provide a sketch of how to generalize our results to the case of an arbitrary number of qubits. Our results apply to both complex-valued and real-valued invariants.

arXiv:2305.16178Journal of Physics A Mathematical and TheoreticalMar 7, 20241 citation

THE GEOMETRY OF QUANTUM STATES

Julian Schwinger

World Scientific series in 20th century physicsMay 1, 20003 citations

Holographic entanglement negativity for disjoint subsystems in $\mathrm{AdS_{d+1}/CFT_d}$ Holographic entanglement negativity for disjoint subsystems in $\mathrm{AdS_{d+1}/CFT_d}$

Jaydeep Kumar Basak, Himanshu Parihar, Boudhayan Paul +1 more

We propose a construction to compute the holographic entanglement negativity for bipartite mixed state configurations of two disjoint subsystems in higher dimensional conformal field theories (CFT $_d$ s) dual to bulk AdS $_{d+1}$ geometries. Our proposal follows from the corresponding AdS $_3$ /CFT $_2$ scenario and involves an algebraic sum of the areas of bulk RT surfaces for certain combinations of subsystems. Utilizing our construction we compute the holographic entanglement negativity at the leading order through a perturbative expansion, for such bipartite mixed states of two disjoint subsystems with long rectangular strip geometries in CFT $_d$ s dual to bulk pure AdS $_{d+1}$ geometries and AdS $_{d+1}$ -Schwarzschild black holes.We propose a construction to compute the holographic entanglement negativity for bipartite mixed state configurations of two disjoint subsystems in higher dimensional conformal field theories (CFT$_d$s) dual to bulk AdS$_{d+1}$ geometries. Our proposal follows from the corresponding AdS$_3$/CFT$_2$ scenario and involves an algebraic sum of the areas of bulk RT surfaces for certain combinations of subsystems. Utilizing our construction we compute the holographic entanglement negativity at the leading order through a perturbative expansion, for such bipartite mixed states of two disjoint subsystems with long rectangular strip geometries in CFT$_d$s dual to bulk pure AdS$_{d+1}$ geometries and AdS$_{d+1}$-Schwarzschild black holes.

arXiv:2001.10534Jan 28, 20209 citations

Holographic entanglement negativity for disjoint subsystems in

\mathrm{AdS_{d+1}/CFT_d}

We propose a construction to compute the holographic entanglement negativity for bipartite mixed state configurations of two disjoint subsystems in higher dimensional conformal field theories (CFT

_d

s) dual to bulk AdS

_{d+1}

geometries. Our proposal follows from the corresponding AdS

_3

/CFT

_2

scenario and involves an algebraic sum of the areas of bulk RT surfaces for certain combinations of subsystems. Utilizing our construction we compute the holographic entanglement negativity at the leading order through a perturbative expansion, for such bipartite mixed states of two disjoint subsystems with long rectangular strip geometries in CFT

_d

s dual to bulk pure AdS

_{d+1}

geometries and AdS

_{d+1}

-Schwarzschild black holes.