On pure states of 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 odd, but for 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 even a matrix decomposition of the unitary group is explicitly computable and allows for study of the monotone’s dynamics. The side factors and of this concurrence canonical decomposition (CCD) are concurrence symmetries, so the dynamics reduce to consideration of the factor. This unitary phases a basis of entangled states, and the concurrence dynamics of are determined by these relative phases. In this work, we provide an explicit numerical algorithm computing for odd. Further, in the odd case we lift the monotone to a two-argument function. The concurrence capacity of according to the double argument lift may be nontrivial for odd and reduces to the usual concurrence capacity in the literature for even. The generalization may also be studied using the CCD, leading again to maximal capacity for most unitaries. The capacity of is at least that of , 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 -qubit Hamiltonian demands (i) 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.
S.S.B. acknowledges a National Research Council postdoctoral fellowship, G.K.B. acknowledges support through DARPA QuIST, and DPO acknowledges NSF Grant No. CCR-0204084.
NIST disclaimer. Certain commercial equipment or instruments may be identified in this paper to specify experimental procedures. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology.
II. BACKGROUND AND PRIOR WORK
III. ODD-QUBIT CONCURRENCE CAPACITIES
A. Double-argument capacities generalize single-argument capacities
C. Parity-independent concurrence spectra
D. A convex hull argument in odd qubits
IV. AN ALGORITHM COMPUTING THE ODD-QUBIT CCD
A. Algorithm for the standard AII decomposition,
B. Symplectic diagonalization
V. TIME REVERSAL, THE CCD, AND KRAMERS’ NONDEGENERACY
A. Spin-flips as time reversal symmetry operators
B. Time reversal and the CCD Cartan involution
C. A time reversal polar decomposition
D. Kramers’ nondegeneracy
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