Time reversal and $n$-qubit canonical decompositions

On pure states of $n$ quantum bits, the concurrence entanglement monotone returns the norm of the inner product of a pure state with its spin-flip. The monotone vanishes for $n$ odd, but for $n$ even there is an explicit formula for its value on mixed states, i.e., a closed-form expression computes the minimum over all ensemble decompositions of a given density. For $n$ even a matrix decomposition $ν=k1ak2$ of the unitary group is explicitly computable and allows for study of the monotone’s dynamics. The side factors $k1$ and $k2$ of this concurrence canonical decomposition (CCD) are concurrence symmetries, so the dynamics reduce to consideration of the $a$ factor. This unitary $a$ phases a basis of entangled states, and the concurrence dynamics of $u$ are determined by these relative phases. In this work, we provide an explicit numerical algorithm computing $ν=k1ak2$ for $n$ odd. Further, in the odd case we lift the monotone to a two-argument function. The concurrence capacity of $ν$ according to the double argument lift may be nontrivial for $n$ odd and reduces to the usual concurrence capacity in the literature for $n$ even. The generalization may also be studied using the CCD, leading again to maximal capacity for most unitaries. The capacity of $ν⊗I2$ is at least that of $ν$⁠, so odd-qubit capacities have implications for even-qubit entanglement. The generalizations require considering the spin-flip as a time reversal symmetry operator in Wigner’s axiomatization, and the original Lie algebra homomorphism defining the CCD may be restated entirely in terms of this time reversal. The polar decomposition related to the CCD then writes any unitary evolution as the product of a time-symmetric and time-antisymmetric evolution with respect to the spin-flip. En route we observe a Kramers’ nondegeneracy: the existence of a nondegenerate eigenstate of any time reversal symmetric $n$-qubit Hamiltonian demands (i) $n$ even and (ii) maximal concurrence of said eigenstate. We provide examples of how to apply this work to study the kinematics and dynamics of entanglement in spin chain Hamiltonians.