Sharing polarization within quantum subspaces

Given an ensemble of $n$ spins, at least some of which are partially polarized, we investigate the sharing of this polarization within a subspace of $k$ spins. We assume that the sharing results in a pseudopure state, characterized by a single purity parameter which we call the bias. As a concrete example we consider ensembles of spin-$1∕2$ nuclei in liquid-state nuclear magnetic resonance systems. The shared bias levels are compared with some current entanglement bounds to determine whether the reduced subspaces can give rise to entangled states.

Figure 1

The effect of sharing the polarization of $p$ pure [(a)–(e)] or partially pure [(f)–(i)] spins on quantum subspaces. Starting with $p$ pure spins (a), we can pick $k⩽p$ pure spins (b) and extract all the available polarization. The polarization can also be spread onto a bigger subspace while retaining the entanglability (c). Sharing the polarization within even higher-dimensional subspaces decreases the resulting bias and therefore the possibility of entanglement [(d) and (e)]. As the size of the extracted subspace increases, the bias drops exponentially. Smaller biases are shown by lighter tones of grey. For $p$ partially polarized spins (f), we can concentrate the bias onto smaller subspaces (g) or spread it onto bigger subspaces [(h) and (i)], with a corresponding decrease in the bias reducing the likelihood of entanglement.

Figure 2

(Color online) Sharing the polarization of two and four pure spins. The upper shaded region is $\mathbf{E}$ and the lower shaded region is $\mathbf{S}$, whereas $\mathbf{E}\mathbf{S}$ is in between. The border between two regions belongs to the less entangled region. The regions are intended to guide the eye and are only shown for two or more spins, where the concept of entanglement is valid.

Figure 3

(Color online) Sharing the polarization of impure spins. The regions $\mathbf{E}$, $\mathbf{E}\mathbf{S}$, and $\mathbf{S}$ are highlighted in the same way as in Fig. 2. The figure demonstrates polarization sharing for five pure and five impure spins. The effect of sharing from a single weakly polarized spin is also shown; when shared over six or more spins, the state moves from being provably separable into the region where it is unknown whether or not it may be entangled. Note, however, that states currently placed in $\mathbf{E}\mathbf{S}$ may subsequently move to $\mathbf{S}$ if lower bounds are further tightened.