Hybrid quantum network design against unauthorized secret-key generation, and its memory cost

A significant number of servers that constitute the Internet are to provide private data via private communication channels to mutually anonymous registered users. Such are the servers of banks, hospitals that provide cloud storage and many others. Replacing communication channels by maximally entangled states is a promising idea for the quantum-secured Internet (QI). While it is an important idea for large distances secure communication, for the case of the mentioned class of servers pure entanglement based solution is not only unnecessary but also opens a threat. A crack stimulating a node to generate secure connections via entanglement swapping between two hackers can cause uncontrolled consumption of resources. Turning into positive a recently proven no-go result by S. Bäuml et al. [Nat. Commun. 6, 6908 (2015)], we propose a natural countermeasure against this threat. The solution bases on connections between hub-nodes and end-users realized with states that contain secure key but do not allow for swapping of this key. We then focus on the study of the quantum memory cost of such a scheme and prove a fundamental lower bound on its memory overhead. In particular, we show that to avoid the possibility of entanglement swapping, it is necessary to store at least twice as much memory than it is the case in standard quantum-repeater-based network design. For schemes employing either states with positive partial transposition that approximates certain privates states or private states hardly distinguishable from their attacked versions, we derive much tighter lower bounds on required memory. Our considerations yield upper bounds on a two-way repeater rate for states with positive partial transposition (PPT), which approximates strictly irreducible private states. As a byproduct, we provide a lower bound on the trace distance between PPT and private states, shown previously only for private bits.

Figure 1

Structure of the proposed hybrid network: thin green lines connect end-users with hubs; in these, only classical data can be transferred. Thick orange lines connect hubs being routing nodes of the network; these connections allow for passing a quantum state. Shaded lines connect two end-users that communicate securely with the hub node only classical data. Selected region denotes single hub of our interest.

Figure 2

The main idea of an attack: (a) the hub shares entangled pairs with end-users $E_i$, in particular, with Adam and Eve. Adam and Eve can attack the hub via quantum malware which performs for them entanglement swapping (b) Adam and Eve share an entangled pair after successful attack.

Figure 3

The main idea of the countermeasure: (a) hub shares bound entangled states (green lines) with end-users $E_i$ (each having at least 1 bit of key), in particular, with Adam and Eve (shaded lines). No LOCC malware can efficiently swap the key. (b) Adam and Eve can not share a state (shaded red line) with non-negligible amount of key (compare [15]).

Figure 4

Plots of lower bounds on percentage of memory overhead from theorem t16, with different values of $d_k$.

Figure 5

Upper bounds on quantum key repeater rates. Dotted blue line: introduced here in eqn. (20), solid orange line: special case for $I_{\mathrm{coh}}=0$, dashed violet line: [15], and dotted-dashed green line: only for some states [19]. Shaded region corresponds to a combination of bounds.

Figure 6

Plots of lower bounds on percentage of memory overhead from theorem t21, with different values of $d_k$.

Figure 7

Plots of lower bound on gap between $\eta$ and $\theta$ for the scheme in theorem t21.

Figure 8

Upper bounds on key repeater rate from proposition t23. Domains are constrained with $\epsilon \ge \frac{1}{2(d_s+1)}$ condition.

Figure 9

Plot of lower bound on percentage of memory overhead from theorem t24.

Figure 10

Plots of lower bounds on gap between $\eta$ and $\theta$ for the scheme in theorem t24. The case $d_s=64$ is a setting with lowest dimension for obtaining positive lower bound on gap under $\epsilon \ge \frac{1}{2(d_s+1)}$ condition.

Figure 11

Upper bounds on key repeater rate from proposition t29. We extract from condition $\frac{d_k-1}{d_s+d_k\left(d_k-1\right)}\le \epsilon$ minimal value of $d_s=\lceil \frac{d_k-1-\epsilon d_k(d_k-1)}{\epsilon }\rceil$ yielding best value for upper bound.

Figure 12

Plots of lower bound on percentage of memory overhead from theorem t30, with different values of $d_k$.

Figure 13

Plots of lower bounds on gap between $\eta$ and $\theta$ for the scheme in theorem t30. For obtaining possibly optimal value of upper bounds we attribute with $d_s$ its minimal possible value of $d_s=\lceil \frac{d_k-1-\epsilon d_k(d_k-1)}{\epsilon }\rceil$.