Min-entropy and quantum key distribution: Nonzero key rates for “small” numbers of signals

We calculate an achievable secret key rate for quantum key distribution with a finite number of signals by evaluating the quantum conditional min-entropy explicitly. The min-entropy for a classical random variable is the negative logarithm of the maximal value in its probability distribution. The quantum conditional min-entropy can be expressed in terms of the guessing probability, which we calculate for $d$-dimensional systems. We compare these key rates to previous approaches using the von Neumann entropy and find nonzero key rates for a smaller number of signals. Furthermore, we improve the secret key rates by modifying the parameter estimation step. Both improvements taken together lead to nonzero key rates for only ${10}^4$–${10}^5$ signals. An interesting conclusion can also be drawn from the additivity of the min-entropy and its relation to the guessing probability: for a set of symmetric tensor product states, the optimal minimum-error discrimination (MED) measurement is the optimal MED measurement on each subsystem.

Figure 1

Comparison of the key rates [calculated via the von Neumann entropy; see Eqs. (29) and (32)] using different parameter estimations for asymmetric BB84 protocol; ${\varepsilon }_{}={10}^{-9}$, $Q=5\text{\%}$; squares (red), CPOVM; triangles (black), IPOVM (see Sec. 3 for explanations).

Figure 2

Comparison of the key rates [calculated via the von Neumann entropy; see Eqs. (29) and (31)] using different parameter estimations for asymmetric six-state protocol; ${\varepsilon }_{}={10}^{-9},Q=5\text{\%}$; squares (red), CPOVM; triangles (black), IPOVM (see Sec. 3 for explanations).

Figure 3

Threshold value $N_0$ (number of signals, where the key rate becomes nonzero) vs QBER $Q$ with $\varepsilon ={10}^{-9}$ and ${\varepsilon }_{\mathrm{EC}}={10}^{-10}$; triangles (red), BB84 protocol; squares (black), six-state protocol; filled, min-entropy [Eq. (34)]; open, von Neumann entropy with CPOVM approach [Eq. (29)]; dashed line, von Neumann entropy [Eq. (29)] with IPOVM approach (see Sec. 3 for explanations).

Figure 4

Key rates with $d$-dimensional conditional von Neumann entropy [Eq. (35)] plotted vs scaled total number of signals for a fixed error rate $Q=5\%$. This is analogous to [15], where a different scale was used for the axes.

Figure 5

Key rates with $d$-dimensional conditional von Neumann entropy [Eq. (35)] plotted vs scaled total number of signals for a fixed error rate $Q=5\%$ (magnification of Fig. 4).

Figure 6

Threshold value $N_0$ (number of signals, where the key rate is positive) vs QBER with $\varepsilon ={10}^{-9}$ and ${\varepsilon }_{\mathrm{EC}}={10}^{-10}$ for different dimensions $d\in \{2,3,7,17\}$. Dashed line, min-entropy [Eq. (36)]; straight line, von Neumann entropy [Eq. (35)].