We represent a two-qubit density matrix in the basis of Pauli matrix tensor products, with the coefficients constituting a Bloch matrix, analogous to the single qubit Bloch vector. We find the quantum state positivity requirements on the Bloch matrix components, leading to three important inequalities, allowing us to parametrize and visualize the two-qubit state space. Applying the singular value decomposition naturally separates the degrees of freedom to local and nonlocal, and simplifies the positivity inequalities. It also allows us to geometrically represent a state as two entangled Bloch spheres with superimposed correlation axes. It is shown that unitary transformations, local or nonlocal, have simple interpretations as axis rotations or mixing of certain degrees of freedom. The nonlocal unitary invariants of the state are then derived in terms of local unitary invariants. The positive partial transpose criterion for entanglement is generalized, and interpreted as a reflection, or a change of a single sign. The formalism is used to characterize maximally entangled states, and generalize two qubit isotropic and Werner states.We represent a two-qubit density matrix in the basis of Pauli matrix tensor products, with the coefficients constituting a Bloch matrix, analogous to the single qubit Bloch vector. We find the quantum state positivity requirements on the Bloch matrix components, leading to three important inequalities, allowing us to parametrize and visualize the two-qubit state space. Applying the singular value decomposition naturally separates the degrees of freedom to local and nonlocal, and simplifies the positivity inequalities. It also allows us to geometrically represent a state as two entangled Bloch spheres with superimposed correlation axes. It is shown that unitary transformations, local or nonlocal, have simple interpretations as axis rotations or mixing of certain degrees of freedom. The nonlocal unitary invariants of the state are then derived in terms of local unitary invariants. The positive partial transpose criterion for entanglement is generalized, and interpreted as a reflection, or a change of a single sign. The formalism is used to characterize maximally entangled states, and generalize two qubit isotropic and Werner states.

arXiv:1602.01548Physical Review AJun 20, 2016107 citations

Density Matrix: Entangled States

Belal Ehsan Baaquie

Springer eBooksDec 15, 20120 citations

The Minimum Distance of PPT Bound Entangled States from the Maximally Mixed State

Shreya Banerjee, Aryaman A. Patel, Prasanta Kumar Panigrahi

Using a geometric measure of entanglement quantification based on Euclidean distance of the Hermitian matrices, we obtain the minimum distance between a bipartite bound entangled $n$ - qudit density matrix and the maximally mixed state.This minimum distance for which entangled density matrices necessarily have positive partial transpose (PPT) is obtained as $\frac{1}{\sqrt{\sqrt{d^n(d^n-1)}+1}}$ , which is also a lower limit for the existence of 1-distillable entangled states. The separable states necessarily lie within a minimum distance of $\frac{R}{1+d^{n-1}}$ from the Identity,where R is the radius of the closed ball homeomorphic to the set of density matrices, which is lesser than the limit for the limit for PPT bound entangled states. Furthermore an alternate proof on the non-emptiness of the PPT bound entangled states has also been given.Using a geometric measure of entanglement quantification based on Euclidean distance of the Hermitian matrices, we obtain the minimum distance between a bipartite bound entangled $n$- qudit density matrix and the maximally mixed state.This minimum distance for which entangled density matrices necessarily have positive partial transpose (PPT) is obtained as $\frac{1}{\sqrt{\sqrt{d^n(d^n-1)}+1}}$, which is also a lower limit for the existence of 1-distillable entangled states. The separable states necessarily lie within a minimum distance of $\frac{R}{1+d^{n-1}}$ from the Identity,where R is the radius of the closed ball homeomorphic to the set of density matrices, which is lesser than the limit for the limit for PPT bound entangled states. Furthermore an alternate proof on the non-emptiness of the PPT bound entangled states has also been given.

arXiv:1708.03885Aug 13, 20170 citations

Quantum States and the Density Matrix

WORLD SCIENTIFIC (EUROPE) eBooksNov 8, 20170 citations

The Density Matrix and the Interpretation of Quantum Theory

O. J. E. Maroney

Bulletin of the American Physical SocietyMar 11, 20080 citations

Using a geometric measure of entanglement quantification based on Euclidean distance of the Hermitian matrices, we obtain the minimum distance between a bipartite bound entangled

n

- qudit density matrix and the maximally mixed state.This minimum distance for which entangled density matrices necessarily have positive partial transpose (PPT) is obtained as

\frac{1}{\sqrt{\sqrt{d^n(d^n-1)}+1}}

, which is also a lower limit for the existence of 1-distillable entangled states. The separable states necessarily lie within a minimum distance of

\frac{R}{1+d^{n-1}}

from the Identity,where R is the radius of the closed ball homeomorphic to the set of density matrices, which is lesser than the limit for the limit for PPT bound entangled states. Furthermore an alternate proof on the non-emptiness of the PPT bound entangled states has also been given.