Quantum information distance

Abstract

In this paper, we give a definition for quantum information distance. In the classical setting, information distance between two classical strings is developed based on classical Kolmogorov complexity. It is defined as the length of a shortest transition program between these two strings in a universal Turing machine. We define the quantum information distance based on Berthiaume et al.’s quantum Kolmogorov complexity. The quantum information distance between qubit strings is defined as the length of the shortest quantum transition program between these two qubit strings in a universal quantum Turing machine. We show that our definition of quantum information distance is invariant under the choice of the underlying quantum Turing machine.

Second quantised information distance

“…First, we recall the concept of indeterminate and determinate length quantum states, which have been discussed in the lossless quantum data compression [19,20]. Next, we review the concepts of quantum Kolmogorov complexity [6] and quantum information distance [7]. Then we review the concept of second quantised Kolmogorov complexity [9].…”

“…These concepts are invariant [6,7]. Let T be a universal quantum Turing machine, then for any quantum Turing machine Q, there exists a constant c Q (dependent only on Q) such that…”

“…The descriptions in quantum Kolmogorov complexity [6] and quantum information distance [7] have determinate lengths. Rogers and Vedral [9] use indeterminate length quantum states to define second quantised Kolmogorov complexity, which is described as the minimum average length of its quantum description on a quantum machine U…”

“…First, quantum information distance [7] and second quantised information distance have the following relationship. Then we come to the problem: why we need a new information distance between quantum states.…”

“…Berthiaume et al [6] defined the quantum Kolmogorov complexity of a state as the length of its shortest quantum description. Building upon their quantum Kolmogorov complexities [5,6], Dai [7,8] developed two quantum information distances, which are, respectively, based on classical and quantum descriptions. However, some quantum states have indeterminate lengths because of their superposition.…”