-
[show abstract]
[hide abstract]
ABSTRACT: In this paper we investigate the use of quantum information to share classical secrets. While every quantum secret sharing scheme is a quantum error-correcting code, the converse is not true. Motivated by this we sought to find quantum codes which can be converted to secret sharing schemes. If we are interested in sharing classical secrets using quantum information, then we show that a class of pure [[n,1,d]]q Calderbank-Shor-Steane (CSS) codes can be converted to perfect secret sharing schemes. These secret sharing schemes are perfect in the sense that the unauthorized parties do not learn anything about the secret. Gottesman had given conditions to test whether a given subset is an authorized or unauthorized set; they enable us to determine the access structure of quantum secret sharing schemes. For the secret sharing schemes proposed in this paper the access structure can be characterized in terms of minimal codewords of the classical code underlying the CSS code. This characterization of the access structure for quantum secret sharing schemes is thought to be notable.
Phys. Rev. A. 08/2009; 80(2).
-
[show abstract]
[hide abstract]
ABSTRACT: In this paper we investigate the use of quantum information to share
classical secrets. While every quantum secret sharing scheme is a quantum error
correcting code, the converse is not true. Motivated by this we sought to find
quantum codes which can be converted to secret sharing schemes. If we are
interested in sharing classical secrets using quantum information, then we show
that a class of pure $[[n,1,d]]_q$ CSS codes can be converted to perfect secret
sharing schemes. These secret sharing schemes are perfect in the sense the
unauthorized parties do not learn anything about the secret. Gottesman had
given conditions to test whether a given subset is an authorized or
unauthorized set; they enable us to determine the access structure of quantum
secret sharing schemes. For the secret sharing schemes proposed in this paper
the access structure can be characterized in terms of minimal codewords of the
classical code underlying the CSS code. This characterization of the access
structure for quantum secret sharing schemes is thought to be new.
05/2009;
-
[show abstract]
[hide abstract]
ABSTRACT: The parameters of a nondegenerate quantum code must obey the Hamming bound.
An important open problem in quantum coding theory is whether or not the
parameters of a degenerate quantum code can violate this bound for
nondegenerate quantum codes. In this paper we show that Calderbank-Shor-Steane
(CSS) codes with alphabet $q\geq 5$ cannot beat the quantum Hamming bound. We
prove a quantum version of the Griesmer bound for the CSS codes which allows us
to strengthen the Rains' bound that an $[[n,k,d]]_2$ code cannot correct more
than $\floor{(n+1)/6}$ errors to $\floor{(n-k+1)/6}$. Additionally, we also
show that the general quantum codes $[[n,k,d]]_q$ with $k+d\leq
{(1-2eq^{-2})n}$ cannot beat the quantum Hamming bound.
12/2008;
-
[show abstract]
[hide abstract]
ABSTRACT: Subsystem codes protect quantum information by encoding it in a tensor factor of a subspace of the physical state space. Subsystem codes generalize all major quantum error protection schemes, and therefore are especially versatile. This paper introduces numerous constructions of subsystem codes. It is shown how one can derive subsystem codes from classical cyclic codes. Methods to trade the dimensions of subsystem and co-subystem are introduced that maintain or improve the minimum distance. As a consequence, many optimal subsystem codes are obtained. Furthermore, it is shown how given subsystem codes can be extended, shortened, or combined to yield new subsystem codes. These subsystem code constructions are used to derive tables of upper and lower bounds on the subsystem code parameters.
12/2008;
-
[show abstract]
[hide abstract]
ABSTRACT: In this paper we investigate the encoding of operator quantum error correcting codes i.e. subsystem codes. We show that encoding of subsystem codes can be reduced to encoding of a related stabilizer code making it possible to use all the known results on encoding of stabilizer codes. Along the way we also show how Clifford codes can be encoded. We also show that gauge qubits can be exploited to reduce the encoding complexity.
07/2008;
-
[show abstract]
[hide abstract]
ABSTRACT: Previous work on network coding capacity for random wired and wireless networks have focused on the case where the capacities of links in the network are independent. In this paper, we consider a more realistic model, where wireless networks are modelled by random geometric graphs with interference and noise. In this model, the capacities of links are not independent. By employing coupling and martingale methods, we show that, under mild conditions, the network coding capacity for random wireless networks still exhibits a concentration behavior around the mean value of the minimum cut.
05/2008;
-
[show abstract]
[hide abstract]
ABSTRACT: Recently, quantum error-correcting codes were proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit flip and phase flip errors. An example for a channel which exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit flips and phase flips can be related to relaxation and dephasing time, respectively. We give systematic constructions of asymmetric quantum stabilizer codes that exploit this asymmetry. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.
05/2008;
-
[show abstract]
[hide abstract]
ABSTRACT: Subsystem codes are the most versatile class of quantum error-correcting codes known to date that combine the best features of all known passive and active error-control schemes. The subsystem code is a subspace of the quantum state space that is decomposed into a tensor product of two vector spaces: the subsystem and the co-subsystem. A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and co-subsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilizer codes. The existence of numerous families of MDS subsystem codes is established. Propagation rules are derived that allow one to obtain longer and shorter subsystem codes from given subsystem codes. Furthermore, propagation rules are derived that allow one to construct a new subsystem code by combining two given subsystem codes.
01/2008;
-
[show abstract]
[hide abstract]
ABSTRACT: Recently, it has been shown that the max flow capacity can be achieved in a multicast network using network coding. In this paper, we propose and analyze a more realistic model for wireless random networks. We prove that the capacity of network coding for this model is concentrated around the expected value of its minimum cut. Furthermore, we establish upper and lower bounds for wireless nodes using Chernoff bound. Our experiments show that our theoretical predictions are well matched by simulation results.
11/2007;
-
[show abstract]
[hide abstract]
ABSTRACT: Ashikhmin and Litsyn showed that all binary stabilizer codes - pure or impure - of sufficiently large length obey the quantum Hamming bound, ruling out the possibility that impure codes of large length can outperform pure codes with respect to sphere packing. In contrast we show that impure subsystem codes do not obey the quantum Hamming bound for pure subsystem codes, not even asymptotically. We show that there exist arbitrarily long Bacon-Shor codes that violate the quantum Hamming bound.
11/2007;
-
[show abstract]
[hide abstract]
ABSTRACT: In this paper, we study network coding capacity for random wireless networks. Previous work on network coding capacity for wired and wireless networks have focused on the case where the capacities of links in the network are independent. In this paper, we consider a more realistic model, where wireless networks are modeled by random geometric graphs with interference and noise. In this model, the capacities of links are not independent. We consider two scenarios, single source multiple destinations and multiple sources multiple destinations. In the first scenario, employing coupling and martingale methods, we show that the network coding capacity for random wireless networks still exhibits a concentration behavior around the mean value of the minimum cut under some mild conditions. Furthermore, we establish upper and lower bounds on the network coding capacity for dependent and independent nodes. In the second one, we also show that the network coding capacity still follows a concentration behavior. Our simulation results confirm our theoretical predictions.
09/2007;
-
[show abstract]
[hide abstract]
ABSTRACT: Duadic group algebra codes are a generalization of quadratic residue codes. This paper addresses an open problem raised by Zhu concerning the existence of duadic group algebra codes. These codes can be used to construct degenerate quantum stabilizer codes that have the nice feature that many errors of small weight do not need error correction.
Information Theory, 2007. ISIT 2007. IEEE International Symposium on; 07/2007
-
[show abstract]
[hide abstract]
ABSTRACT: Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. A quantum Singleton bound for pure convolutional stabilizer codes is given. A familiy of quantum convolutional codes is derived from generalized Reed-Solomon codes. These codes are shown to be optimal with respect to the (quantum) Singleton bound.
Information Theory, 2007. ISIT 2007. IEEE International Symposium on; 07/2007
-
[show abstract]
[hide abstract]
ABSTRACT: Classical Bose-Chaudhuri-Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of length n can contain its dual code only if its designed distance delta=O(radicn), and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and - consequently - the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters
IEEE Transactions on Information Theory 04/2007; · 3.01 Impact Factor
-
[show abstract]
[hide abstract]
ABSTRACT: Subsystem codes are a generalization of noiseless subsystems, decoherence free subspaces, and quantum error-correcting codes. We prove a Singleton bound for GF(q)-linear subsystem codes. It follows that no subsystem code over a prime field can beat the Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the Hamming bound. A number of open problems concern the comparison in performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal MDS stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding. Comment: 18 pages more densely packed than classically possible
03/2007;
-
[show abstract]
[hide abstract]
ABSTRACT: We show that any stabilizer code over a finite field is equivalent to a graphical quantum code. Furthermore we prove that a graphical quantum code over a finite field is a stabilizer code. The technique used in the proof establishes a new connection between quantum codes and quadratic forms. We provide some simple examples to illustrate our results. Comment: 5 pages, 2 figures, paper presented at the 2002 IEEE International Symposium on Information Theory
03/2007;
-
[show abstract]
[hide abstract]
ABSTRACT: Quantum convolutional codes can be used to protect a sequence of qubits of arbitrary length against decoherence. We introduce two new families of quantum convolutional codes. Our construction is based on an algebraic method which allows to construct classical convolutional codes from block codes, in particular BCH codes. These codes have the property that they contain their Euclidean, respectively Hermitian, dual codes. Hence, they can be used to define quantum convolutional codes by the stabilizer code construction. We compute BCH-like bounds on the free distances which can be controlled as in the case of block codes, and establish that the codes have non-catastrophic encoders. Comment: 4 pages, minor changes, accepted for publication at the 10th Canadian Workshop on Information Theory (CWIT'07)
03/2007;
-
[show abstract]
[hide abstract]
ABSTRACT: Duadic group algebra codes are a generalization of quadratic residue codes. This paper settles an open problem raised by Zhu concerning the existence of duadic group algebra codes. These codes can be used to construct degenerate quantum stabilizer codes that have the nice feature that many errors of small weight do not need error correction; this fact is illustrated by an example.
02/2007;
-
[show abstract]
[hide abstract]
ABSTRACT: Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. Two families of quantum convolutional codes are derived from generalized Reed-Solomon codes and from Reed- Muller codes. A Singleton bound for pure convolutional stabilizer codes is given.
02/2007;
-
[show abstract]
[hide abstract]
ABSTRACT: We investigate various aspects of operator quantum error-correcting codes or, as we prefer to call them, subsystem codes. We give various methods to derive subsystem codes from classical codes. We give a proof for the existence of subsystem codes using a counting argument similar to the quantum Gilbert-Varshamov bound. We derive linear programming bounds and other upper bounds. We answer the question whether or not there exist [[n,n-2d+2,r>0,d]]<sub>q</sub> subsystem codes. Finally, we compare stabilizer and subsystem codes with respect to the required number of syndrome qudits.
11/2006;
