Entanglement and the power of one qubit

Abstract

The ``power of one qubit'' refers to a computational model that has access to only one pure bit of quantum information, along with $n$ qubits in the totally mixed state. This model, though not as powerful as a pure-state quantum computer, is capable of performing some computational tasks exponentially faster than any known classical algorithm. One such task is to estimate with fixed accuracy the normalized trace of a unitary operator that can be implemented efficiently in a quantum circuit. We show that circuits of this type generally lead to entangled states, and we investigate the amount of entanglement possible in such circuits, as measured by the multiplicative negativity. We show that the multiplicative negativity is bounded by a constant, independent of $n$, for all bipartite divisions of the $n+1$ qubits, and so becomes, when $n$ is large, a vanishingly small fraction of the maximum possible multiplicative negativity for roughly equal divisions. This suggests that the global nature of entanglement is a more important resource for quantum computation than the magnitude of the entanglement.