The Kolmogorov complexity of a physical state is the minimal physical resources required to reproduce that state. We define a second quantized quantum Turing machine and use it to define second quantized Kolmogorov complexity. There are two advantages to our approach -our measure of second quantized Kolmogorov complexity is closer to physical reality and unlike other quantum Kolmogorov complexities it is continuous. We give examples where second quantized and quantum Kolmogorov complexity differ.PACS numbers: 03.67.LxQuantum physics, as far as we know, is the most accurate and universal description of all physical phenomena in the universe. If we therefore wish to speak about the complexity of some processes or some physical states in nature, we need to use quantum physics as our best available theory. An important question is (in very simple terms), given a physical system in some state, how difficult is it for us to reproduce it. If we wish to have a universal measure of this difficulty (which applies to all systems and states) a way to proceed is to follow the prescription of Kolmogorov. First, we define a universal computer (which is capable of simulating all other computers) and then we look for the shortest input (another physical state) to this computer that reproduces as the output the desired physical state. This way, all the complexity is defined with respect to the same universal computer, and we can thus speak about universal complexity. The universal computer (such as a universal Turing machine) needs to be fully quantum mechanical in order to capture all the possibilities available in nature. In this letter, we define a fully quantized Turing machine which we use to define a fully quantum Kolmogorov complexity and apply it to a number of different problems. Our approach is different from others in that we consider indeterminate length input programs whose expected length is our measure of complexity. Others, whose work is discussed in detail below, have perhaps avoided our approach, as allowing programs in superposition can lead to a superposition of halting times. Since traditionally computation is viewed as giving a deterministic output after a fixed amount of time, having a superposition of halting times seems to contradict the very concept of computation. We, on the other hand, view the superposition of different length inputs which can lead to a superposition of halting times as a necessity dictated by a fully quantum mechanical description of nature. Moreover allowing superpositions of different programs can lead to a program which has on average a smaller Kolmogorov complexity than when programs are restricted to having a variable but determinate length (we give examples of this in this letter). Since we want to know what is physically the shortest input which produces a given output, we need allow the computation with superpositions of different length programs. This letter is organized as follows. First we discuss the concept of a Turing machine and review previous work. Then we define the...

We describe a method for lossless quantum compression if the output of the information source is not known. We compute the best possible compression rate, minimizing the expected base length of the output quantum bit string ͑the base length of a quantum string is the maximal length in the superposition͒. This complements work by Schumacher and Westmoreland who calculated the corresponding rate for minimizing the output's average length. Our compressed code words are prefix-free indeterminate-length quantum bit strings which can be concatenated in the case of multiple sources. Therefore, we generalize the known theory of prefix-free quantum codes to the case where strings have indeterminate length. Moreover, we describe a communication model which allows the lossless transmission of the compressed code words. The benefit of compression is then the reduction of transmission errors in the presence of noise.

The Kolmogorov complexity of a string is the length of its shortest description. We define a second quantised Kolmogorov complexity where the length of a description is defined to be the average length of its superposition. We discuss this complexity's basic properties. We define the corresponding prefix complexity and show that the inequalities obeyed by this prefix complexity are also obeyed by von Neumann entropy.