Private quantum codes: introduction and connection with higher rank numerical ranges

@article{Kribs2014PrivateQC,
  title={Private quantum codes: introduction and connection with higher rank numerical ranges},
  author={David W. Kribs and Sarah Plosker},
  journal={Linear and Multilinear Algebra},
  year={2014},
  volume={62},
  pages={639 - 647}
}
  • D. KribsS. Plosker
  • Published 31 March 2014
  • Computer Science, Mathematics
  • Linear and Multilinear Algebra
We give a brief introduction to private quantum codes, a basic notion in quantum cryptography and key distribution. Private code states are characterized by indistinguishability of their output states under the action of a quantum channel, and we show that higher rank numerical ranges can be used to describe them. We also show how this description arises naturally via conjugate channels and the bridge between quantum error correction and cryptography. 
3 Citations

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References

SHOWING 1-10 OF 22 REFERENCES

Private quantum channels

It is shown that in order to transmit n qubits privately, 2n bits of shared private key are necessary and sufficient and may be viewed as the quantum analogue of the classical one-time pad encryption scheme.

Complementarity of private and correctable subsystems in quantum cryptography and error correction

It is shown that a subsystem is private for a channel precisely when it is correctable for a complementary channel, even for approximate notions of private and correctable defined in terms of the diamond norm for superoperators.

Quantum computation and quantum information

  • T. Paul
  • Physics
    Mathematical Structures in Computer Science
  • 2007
This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal

Optimal encryption of quantum bits

It is shown that $2n$ random classical bits are both necessary and sufficient for encrypting any unknown state of n quantum bits in an informationally secure manner and a connection is made between quantum encryption and quantum teleportation that allows for a different proof of optimality of teleportation.

Private quatnum channels, conditional expectations, and trace vectors

It is shown that trace vectors completely describe the private states for quantum channels that are themselves conditional expectations, and a new geometric characterization of single qubit private quantum channels is given that relies on trace vectors.

GENERALIZED NUMERICAL RANGES AND QUANTUM ERROR CORRECTION

For a noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the joint rank-k numerical range associated with the error operators of the channel is non-empty. In

Theory of quantum error-correcting codes

A general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions is developed and necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction are obtained.