Possibility, impossibility, and cheat sensitivity of quantum-bit string commitment

Unconditionally secure nonrelativistic bit commitment is known to be impossible in both the classical and the quantum worlds. But when committing to a string of $n$ bits at once, how far can we stretch the quantum limits? In this paper, we introduce a framework for quantum schemes where Alice commits a string of $n$ bits to Bob in such a way that she can only cheat on $a$ bits and Bob can learn at most $b$ bits of information before the reveal phase. Our results are twofold: we show by an explicit construction that in the traditional approach, where the reveal and guess probabilities form the security criteria, no good schemes can exist: $a+b$ is at least $n$. If, however, we use a more liberal criterion of security, the accessible information, we construct schemes where $a=4{\text{log}}_{2}n+O\left(1\right)$ and $b=4$, which is impossible classically. We furthermore present a cheat-sensitive quantum bit string commitment protocol for which we give an explicit tradeoff between Bob’s ability to gain information about the committed string, and the probability of him being detected cheating.

Figure 1

(Color online) Moving from $y$ to $y^{\prime }$.

Figure 2

Execution of CS-Bob-LOCKCOM with honest Alice on the left and cheating Bob on the right. Time flows downwards.