Unconditionally secure nonrelativistic bit commitment is known to be impossible in both the classical and the quantum world. However, when committing to a string of $n$ bits at once, how far can we stretch the quantum limits? In this Letter, we introduce a framework of 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 $a=4{log}_{2}n+O(1)$ and $b=4$, which is impossible classically. Our findings significantly extend known no-go results for quantum bit commitment.

• Received 12 October 2005

DOI:https://doi.org/10.1103/PhysRevLett.97.250501