Certifying quantum memories with coherence

Quantum memories are an important building block for quantum information processing. Ideally, these memories preserve the quantum properties of the input. We present general criteria for measures to evaluate the quality of quantum memories. Then we introduce a quality measure based on coherence satisfying these criteria, which we characterize in detail for the qubit case. The measure can be estimated from sparse experimental data and may be generalized to characterize other building blocks, such as quantum gates and teleportation schemes.

Figure 1

The image of the Bloch sphere (red area) of single-qubit maps is an ellipsoid (blue area) with semiaxes ${\lambda }_i$, displaced by $\vec{\kappa }$.

Figure 2

Left: Projection of the ellipsoid in the 1-2 direction to obtain upper bounds on the measures. Red dots indicate the points of the image of maximally coherent states in some basis which touch the boundary of the projected ellipse. Right: Projection of the ellipsoid in the direction of $\vec{\kappa }$. The semiaxes of the projection are bounded by the semiaxes of the ellipsoid.

Figure 3

Ellipsoid representations of the M&P channels that maximize the different quality measures. The displacement vector $\vec{\kappa }$ is depicted by the dotted black line; the semiaxes, by the blue and green lines. Left: The M&P channel maximizing $Q_{-}$ maps to a disk of radius $\frac{1}{\sqrt{5}}$, displaced by $\frac{1}{\sqrt{5}}$. Right: The M&P channel maximizing $Q_0$ and $Q_{+}$, mapping to a straight line of length $\frac{2}{\sqrt{2}}$, displaced by $\frac{1}{\sqrt{2}}$.

Figure 4

Lower bound (solid blue line) and allowed values above this bound (in blue) for the quality measure $Q_{-}$, given that in certain directions a coherence of at least $c$ is measured. The upper bound of $\frac{1}{\sqrt{5}}$ for M&P channels is displayed by the dashed orange line.