arXiv subjects
Publications and source records attributed to Sergei Bravyi.
We introduce the notion of the unextendible maximally entangled basis (UMEB), a set of orthonormal maximally entangled states in d \times d consisting of fewer that d^2 vectors which have no additional maximally entangled vectors orthogonal to all of them. We prove that UMEBs don't not exist for d=2 and give an explicit constructions for a 6-member UMEB with d=3 and a 12-member UMEB with d=4.
We consider a model of quantum computation in which the set of elementary operations is limited to Clifford unitaries, the creation of the state $|0\rangle$ computational basis. In addition, we allow the creation of a one-qubit ancilla in a mixed state $ρ$, which should be regarded as a parameter of the model. Our goal is to determine for which $ρ$ universal quantum computation (UQC) can be efficiently simulated. To answer this question, we construct purification protocols that consume several copies of $ρ$ and produce a single output qubit with higher polarization. The protocols allow one to increase the polarization only along certain "magic" directions. If the polarization of $ρ$ along a magic direction exceeds a threshold value (about 65%), the purification asymptotically yields a pure state, which we call a magic state. We show that the Clifford group operations combined with magic states preparation are sufficient for UQC. The connection of our results with the Gottesman-Knill theorem is discussed.
Consider a system consisting of $n$ $d$-dimensional quantum particles and arbitrary pure state $Ψ$ of the whole system. Suppose we simultaneously perform complete von Neumann measurements on each particle. One can ask: what is the minimal possible value $S[Ψ]$ of the entropy of outcomes joint probability distribution? We show that $S[Ψ]$ coincides with entanglement entropy for bipartite states. We compute $S[Ψ]$ for two sample multipartite states: the hexacode state ($n=6, d=2$) and determinant states ($n=d$). The generalization of determinant states to the case $d<n$ is considered.