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arXiv:quant-ph/0503052 · Journal of Mathematical Physics

Minimum orbit dimension for local unitary action on n-qubit pure states

David W. Lyons, Scott N. Walck

Published Oct 1, 2005• 9 citations
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Abstract

The group of local unitary transformations partitions the space of n-qubit quantum states into orbits, each of which is a differentiable manifold of some dimension. We prove that all orbits of the n-qubit quantum state space have dimension greater than or equal to 3n∕2 for n even and greater than or equal to (3n+1)∕2 for n odd. This lower bound on orbit dimension is sharp, since n-qubit states composed of products of singlets achieve these lowest orbit dimensions.

DOI
10.1063/1.2048327

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The group of local unitary transformations partitions the space of n-qubit quantum states into orbits, each of which is a differentiable manifold of some dimension. We prove that all orbits of the n-qubit quantum state space have dimension greater than or equal to 3n∕2 for n even and greater than or equal to (3n+1)∕2 for n odd. This lower bound on orbit dimension is sharp, since n-qubit states composed of products of singlets achieve these lowest orbit dimensions.
The group of local unitary transformations acts on the space of n-qubit pure states, decomposing it into orbits. In a previous paper we proved that a product of singlet states (together with an unentangled qubit for a system with an odd number of qubits) achieves the smallest possible orbit dimension, equal to 3n/2 for n even and (3n + 1)/2 for n odd, where n is the number of qubits. In this paper we show that any state with minimum orbit dimension must be of this form, and furthermore, such states are classified up to local unitary equivalence by the sets of pairs of qubits entangled in singlets.
Local unitary invariants allow one to test whether multipartite states are equivalent up to local basis changes. Equivalently, they specify the geometry of the obtained by factoring out local unitary action from the state space. This space is of interest because of its intimate relationship to entanglement. Unfortunately, the dimension of the orbit space grows exponentially with the number of subsystems, and the number of invariants needed to characterise orbits grows at least as fast. This makes the study of entanglement via local unitary invariants seem very daunting. I point out here that there is a simplifying principle: Invariants fall into families related by the tracing-out of subsystems, and these families grow exponentially with the number of subsystems. In particular, in the case of pure qubit systems, there is a family whose size is about half the dimension of orbit space. These invariants are closely related to cumulants and to multipartite separability. Members of the family have been repeatedly discovered in the literature, but the fact that they are related to cumulants and constitute a family has apparently not been observed.