Towards a geometrical interpretation of quantum-information compression (6 pages)

@article{Mitchison2003TowardsAG,
  title={Towards a geometrical interpretation of quantum-information compression (6 pages)},
  author={Graeme J. Mitchison and Richard Jozsa},
  journal={Physical Review A},
  year={2003},
  volume={69},
  pages={32304}
}
Let S be the von Neumann entropy of a finite ensemble E of pure quantum states. We show that S may be naturally viewed as a function of a set of geometrical volumes in Hilbert space defined by the states and that S is monotonically increasing in each of these variables. Since S is the Schumacher compression limit of E, this monotonicity property suggests a geometrical interpretation of the quantum redundancy involved in the compression process. It provides clarification of previous work in… 
View PDF on arXiv
22 Citations
Highly Influential Citations
1
Background Citations
7
Methods Citations
1
Results Citations
1

22 Citations

Matrices of fidelities for ensembles of quantum states and the holevo quantity

This work combines geometrical and probabilistic aspects of the ensemble in order to obtain meaningful bounds for χ by constructing various correlation matrices involving fidelities between every pair of states from the ensemble.

Properties of subentropy

The notion of conditional subentropy is introduced and it is shown that it can be used to provide an upper bound for the guessing probability of any classical-quantum state of two qubits; it is conjecture that the bound applies also in higher dimensions.

Entropy of quantum channel in the theory of quantum information

    W. Roga
    Computer Science, Physics
  • 2011
The Holevo information associated with this ensemble is shown to be bounded by the entropy exchanged during the preparation process between the initial state and the environment, and the additivity of the entropy of a channel is proved.

On a Sufficient Condition for Additivity in Quantum Information Theory

In this paper we give a sufficient condition for additivity of the minimum output entropy for a pair of given channels and an analytic verification of this condition for specific quantum channels…

An entropy inequality

Let S(ρ) = -Tr(ρ log ρ) be the von Neumann entropy of an N-dimensional quantum state ρ and e2(ρ) the second elementary symmetric polynomial of the eigenvalues of ρ. We prove the inequality…

A quantum channel with additive minimum output entropy

We give a direct proof of the additivity of the minimum output entropy of a particular quantum channel which breaks the multiplicativity conjecture. This yields additivity of the classical capacity…

A Classical Interpretation of the Scrooge Distribution

A real-amplitude version of the Scrooge distribution is considered for which it is found that there is a non-trivial but natural classical interpretation and a step that may shed light on the relation between quantum mechanics and classical probability theory.

Characterizing the quantumness of mixed-state ensembles via the coherence of Gram matrix with generalized α-z-relative Rényi entropy

By utilizing the generalized α-z-relative Rényi entropy of coherence of Gram matrices, this work presents a new quantifier of quantumness of mixed-sate ensembles and reveals its peculiar properties.

Symmetric polynomials in information theory: entropy and subentropy

It is seen that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory.

Gram Matrices of Mixed-State Ensembles

Gram matrices arise naturally in the consideration of pair-wise overlap of a family of vectors or quantum pure states, and play an important role in synthesizing information of a family of pure…

11 References

Illustrating the concept of quantum information

This work discusses some illustrative examples, including a strengthening of the well-known no-cloning theorem leading to a property of permanence for quantum information, and considerations arising from information compression that reflect on fundamental issues.

Entropy as a function of geometric phase

We give a closed-form solution of von Neumann entropy as a function of geometric phase modulated by visibility and average distinguishability in Hilbert spaces of two and three dimensions. We show…

Distinguishability of states and von Neumann entropy

The result shows that the notion of distinguishability within an ensemble is a global property that cannot be reduced to considering distinguishability of each constituent pair of states.

Quantum coding.

    Schumacher
    Physics
    Physical review. A, Atomic, molecular, and optical physics
  • 1995
The von Neumann entropy S of the density operator describing an ensemble of pure quantum signal states is equal to the number of spin-1/2 systems (quantum bits or \qubits") necessary to represent the signal faithfully.

Coding theorems of quantum information theory

Coding theorems and (strong) converses for memoryless quantum communication channels and quantum sources are proved: for the quantum source the coding theorem is reviewed, and the strong converse proven, and a new proof of the famous Holevo bound is obtained.

Limits for compression of quantum information carried by ensembles of mixed states

It is proved that for arbitrary coding schemes the least number of quantum bits per message needed to convey the signal states asymptotically faithfully is bounded from below by the Holevo function.

General fidelity limit for quantum channels.

A general limit on the fidelity of a quantum channel conveying an ensemble of pure states is derived and this limit applies to arbitrary coding and decoding schemes, establishing in full generality the quantum noiseless coding theorem.

Between entropy and subentropy

The von Neumann entropy and the subentropy of a mixed quantum state are upper and lower bounds, respectively, on the accessible information of any ensemble consistent with the given mixed state. Here…

A new proof of the quantum noiseless coding theorem

An account of the quantum noiseless coding theorem, including a new proof based on a simplified block coding scheme, and an illustrative example of quantum coding.

Introduction to Numerical Analysis

1. The numerical evaluation of expressions 2. Linear systems of equations 3. Interpolation and numerical differentiation 4. Numerical integration 5. Univariate nonlinear equations 6. Systems of…