I generalize the concept of balancedness to qudits with arbitrary dimension d . It is an extension of the concept of balancedness by Osterloh and Siewert (2010 New J. Phys. 12 075025 ). At first, I define maximally entangled states as being the stochastic states (with local reduced density matrices for a d -dimensional local Hilbert space) that are not product states and show that every so-defined maximal genuinely multi-qudit entangled state is balanced. Furthermore, all irreducibly balanced states are genuinely multi-qudit entangled and are locally equivalent with respect to SL( d ) transformations (i.e. the local filtering transformations) to a maximally entangled state. In particular the concept given here gives a condition that all maximal genuinely multi-qudit entangled states for general local Hilbert space dimension d have to satisfy. A general genuinely multi-qudit entangled state is an element of the partly balanced SU( d )-orbits.
Abstract Collective spin operators for symmetric multi-quDit (namely identical D -level atom) systems generate a U( D ) symmetry. We explore generalizations to arbitrary D of SU(2)-spin coherent states and their adaptation to parity (multi-component Schrödinger cats), together with multi-mode extensions of NOON states. We write level, one- and two-quDit reduced density matrices of symmetric N -quDit states, expressed in the last two cases in terms of collective U( D )-spin operator expectation values. Then, we evaluate level and particle entanglement for symmetric multi-quDit states with linear and von Neumann entropies of the corresponding reduced density matrices. In particular, we analyze the numerical and variational ground state of Lipkin–Meshkov–Glick models of 3-level identical atoms. We also propose an extension of the concept of SU(2)-spin squeezing to SU( D ) and relate it to pairwise D -level atom entanglement. Squeezing parameters and entanglement entropies are good markers that characterize the different quantum phases, and their corresponding critical points, that take place in these interacting D -level atom models.
Temporary occupation of qutrit levels can reduce the width and depth required for binary circuit logic in quantum computing. We introduce an efficient intermediate-qutrit decomposition of multi-controlled Toffoli gates with binary-subspace inputs and outputs. For balanced control widths
n=2h−1, the proposed decomposition is evaluated hierarchically through a tree, allowing independent subtree computations to proceed in parallel and reducing the depth from linear to logarithmic. The balanced low-depth construction requires
6n+3 logical
P9 injections and
4n−3 clean ancillary qutrits. It matches the direct
P9 count of a recursively extended Clifford+
P9 baseline derived from the state-of-the-art work while using asymptotically one-quarter as many clean ancillas and replacing linear depth with logarithmic depth. For arbitrary control widths, a sequential extension preserves the
6n+3 P9 count but has depth
O(logm+n−m) for a balanced core
m=2h−1≤n, which is linear in the worst case over
n. By reducing the resource overhead of a widely used reversible primitive, the proposed decomposition provides a practical building block for the design and compilation of more resource-efficient quantum algorithms in the fault-tolerant regime.