Security of Quantum Bit String Commitment Depends on the Information Measure

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.