• Corpus ID: 17405115

Quantum Bit Strings and Prefix-Free Hilbert Spaces

  title={Quantum Bit Strings and Prefix-Free Hilbert Spaces},
  author={Markus M{\"u}ller and Caroline Rogers},
We give a mathematical framework for manipulating indeterminate-length quantum bit strings. In particular, we define prefixes, fragments, tensor products and concatenation of such strings of qubits, and study their properties and relationships. The results are then used to define prefix-free Hilbert spaces in a more general way than in previous work, without assuming the existence of a basis of length eigenstates. We prove a quantum analogue of the Kraft inequality, illustrate the results with… 

Figures from this paper

It is shown that classical and quantum Kolmogorov complexity of binary strings agree up to an additive constant, and it follows that quantum complexity is an extension of classical complexity to the domain of quantum states.
Optimality in Quantum Data Compression using Dynamical Entropy
This article studies lossless compression of strings of pure quantum states of indeterminate-length quantum codes which were introduced by Schumacher and Westmoreland and discusses the notion of quantum stochastic ensembles, allowing it to be proved that the optimal average codeword length via lossless coding is equal to the quantum dynamical entropy of the associated Quantum dynamical system.
Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum Rényi entropy
It is shown that by invoking an exponential average length, related to an exponential penalization over large codewords, the quantum Rényi entropies arise as the natural quantities relating the optimal encoding schemes with the source description, playing an analogous role to that of von Neumann entropy.
On Gács' quantum algorithmic entropy
An infinite dimensional modification of lower-semicomputability of density operators by G\'acs is defined and it is shown that universal operator has certain nontrivial form if it exists.
Second quantized Kolmogorov complexity
This work defines a second quantized Kolmogorov complexity where the length of a description is defined to be the average length of its superposition and defines the corresponding prefix complexity and shows that the inequalities obeyed by this prefix complexity are also obeyedBy von Neumann entropy.


Indeterminate-length quantum coding
Indeterminate-length quantum codes, in which code words may exist in superpositions of different lengths, provide an alternate approach to quantum data compression and are explored in this paper.
Quantum information is incompressible without errors.
This work generalizes this variable-length and faithful scenario to the general quantum source producing mixed states rho(i) with probability p(i), and finds the optimal compression rate in the limit of large block length differs from the one in the fixed- length and asymptotically faithful scenario.
Lossless quantum data compression and variable-length coding
A general framework for variable-length quantum messages is developed in close analogy to the classical case and it is shown that it is possible to reduce the number of qbits passing through a quantum channel even below the von Neumann entropy by adding a classical side channel.
A quantum analog of huffman coding
A Huffman-coding inspired scheme for quantum data compression that can be made to be polynomial in log N by a massively parallel implementation of a quantum gate array is constructed.
On lossless quantum data compression with a classical helper
It turned out that the optimal compression can always be achieved by a code obtained by this scheme and a von Neumann entropy lower bound to rates of these codes and a necessary and sufficient condition to achieve the bound are obtained.
Elements of Information Theory
The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.
Quantum Complexity Theory
  • SIAM J. Comput.
  • 1997
  • Cai, “On Lossless Quantum Data Compressi on with a Classical Helper”,IEEE Trans. Inf. Th.50 1208-1219
  • 2004
  • Felbinger, “Lossless quantum data compr ession and variable-length coding”,Phys. Rev. A. 65 032313
  • 2002