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A Bell basis with phases chosen so that local special-unitary transformations of the two qubits are represented by real four-dimensional rotations. For one standard ordering it is ${\Phi }_{+},i{\Phi }_{-},i{\Psi }_{+},{\Psi }_{-}$. Permuting these vectors changes generator conventions but preserves the invariant concurrence $|{\sum }_j{b}_j^2|$.^[ref]

The specified Bell-state phases giveThereforeThe sum contains complex squares, not squared moduli. Define $q={\sum }_j{b}_j^2$ and $\mathcal{C}=|q|$. Normalization gives ${\sum }_j|b_j{|}^2=1$, so the triangle inequality ensures $0\le \mathcal{C}\le 1$.

The characteristic polynomial is ${\lambda }^2-\lambda +{\mathcal{C}}^2/4$. HenceHere logarithms are natural and $0\ln 0$ means its limiting value zero. Using logarithms to base two gives entropy in bits instead. Product states have $\mathcal{C}=0$, spectrum $(1,0)$ and entropy zero; maximally entangled states have $\mathcal{C}=1$, spectrum $(1/2,1/2)$ and entropy $\ln 2$.

This phased Bell basis is a magic basis, and $\mathcal{C}$ is the pure-state concurrence. The paper's ordering is a permutation of the usual magic-basis ordering, which leaves $|{\sum }_j{b}_j^2|$ unchanged.^[ref]

In the printed phased Bell basis, the actions on its four kets areEach matrix is purely imaginary and antisymmetric. Let $T$ have these four basis kets as columns in the computational basis. The infinitesimal generators of the standard rotation $U_A=\exp [-i\theta \mathbf{n}\cdot {\mathbf{\sigma }}_A/2]$ areUsing the displayed actions gives the phase-convention-specific identitiesThey are real antisymmetric matrices. A finite $SU(2)$ rotation on $A$ therefore acts on the magic-basis coefficient vector as $b\mapsto Ob$, with $O\in SO(4)$. ConsequentlyIt preserves concurrence and hence preserves the reduced eigenvalues and entanglement entropy. A general $U(2)$ operation adds a common phase to $b$; that changes $q$ by a phase of twice the angle but leaves $|q|$ unchanged. The real-rotation description of local transformations is a characteristic property of the magic basis.^[ref]

As an independent direct check, a local unitary sends ${\rho }_A$ to $U_A{\rho }_A{U}_A^{†}$, which is a unitary similarity transformation and cannot change its spectrum. This local-unitary invariance of entanglement entropy also explains why merely rotating one subsystem cannot create entanglement.

A theoretical physicist who coauthored the 1997 paper introducing concurrence with William Wootters. The calculation uses a specially phased Bell basis to connect the coefficients of a two-qubit quantum state to its quantum entanglement. It provided an exact result for a class of mixed states, making entanglement quantification more concrete than a qualitative claim that two systems are correlated.^[ref]

A theoretical physicist who, with Scott Hill, developed a calculable measure of quantum entanglement for pairs of qubits. Their 1997 work introduced the concurrence approach and obtained an exact preparation-cost formula for mixed states of rank at most two, with evidence for a broader extension. This turned an optimization over pure-state decompositions into a practical calculation for an important class of small quantum systems.^[ref]