Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBounds on entanglement assisted source-channel coding via the Lovasz theta number and its variants
76
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study zero-error entanglement assisted source-channel coding (communication in the presence of side information). Adapting a technique of Beigi, we show that such coding requires existence of a set of vectors satisfying orthogonality conditions related to suitably defined graphs $G$ and $H$. Such vectors exist if and only if $vartheta(overline{G}) le vartheta(overline{H})$ where $vartheta$ represents the Lovasz number. We also obtain similar inequalities for the related Schrijver $vartheta^-$ and Szegedy $vartheta^+$ numbers. These inequalities reproduce several known bounds and also lead to new results. We provide a lower bound on the entanglement assisted cost rate. We show that the entanglement assisted independence number is bounded by the Schrijver number: $alpha^*(G) le vartheta^-(G)$. Therefore, we are able to disprove the conjecture that the one-shot entanglement-assisted zero-error capacity is equal to the integer part of the Lovasz number. Beigi introduced a quantity $beta$ as an upper bound on $alpha^*$ and posed the question of whether $beta(G) = lfloor vartheta(G) rfloor$. We answer this in the affirmative and show that a related quantity is equal to $lceil vartheta(G) rceil$. We show that a quantity $chi_{textrm{vect}}(G)$ recently introduced in the context of Tsirelsons conjecture is equal to $lceil vartheta^+(overline{G}) rceil$. In an appendix we investigate multiplicativity properties of Schrijvers and Szegedys numbers, as well as projective rank.
rate research
Read More
In this dissertation, I present a general method for studying quantum error correction codes (QECCs). This method not only provides us an intuitive way of understanding QECCs, but also leads to several extensions of standard QECCs, including the operator quantum error correction (OQECC), the entanglement-assisted quantum error correction (EAQECC). Furthermore, we can combine both OQECC and EAQECC into a unified formalism, the entanglement-assisted operator formalism. This provides great flexibility of designing QECCs for different applications. Finally, I show that the performance of quantum low-density parity-check codes will be largely improved using entanglement-assisted formalism.
We show how to protect a stream of quantum information from decoherence induced by a noisy quantum communication channel. We exploit preshared entanglement and a convolutional coding structure to develop a theory of entanglement-assisted quantum convolutional coding. Our construction produces a Calderbank-Shor-Steane (CSS) entanglement-assisted quantum convolutional code from two arbitrary classical binary convolutional codes. The rate and error-correcting properties of the classical convolutional codes directly determine the corresponding properties of the resulting entanglement-assisted quantum convolutional code. We explain how to encode our CSS entanglement-assisted quantum convolutional codes starting from a stream of information qubits, ancilla qubits, and shared entangled bits.
We provide several formulas that determine the optimal number of entangled bits (ebits) that a general entanglement-assisted quantum code requires. Our first theorem gives a formula that applies to an arbitrary entanglement-assisted block code. Corollaries of this theorem give formulas that apply to a code imported from two classical binary block codes, to a code imported from a classical quaternary block code, and to a continuous-variable entanglement-assisted quantum block code. Finally, we conjecture two formulas that apply to entanglement-assisted quantum convolutional codes.
We introduce two generalizations of Kochen-Specker (KS) sets: projective KS sets and generalized KS sets. We then use projective KS sets to characterize all graphs for which the chromatic number is strictly larger than the quantum chromatic number. Here, the quantum chromatic number is defined via a nonlocal game based on graph coloring. We further show that from any graph with separation between these two quantities, one can construct a classical channel for which entanglement assistance increases the one-shot zero-error capacity. As an example, we exhibit a new family of classical channels with an exponential increase.
This paper provides upper and lower bounds on the optimal guessing moments of a random variable taking values on a finite set when side information may be available. These moments quantify the number of guesses required for correctly identifying the unknown object and, similarly to Arikans bounds, they are expressed in terms of the Arimoto-Renyi conditional entropy. Although Arikans bounds are asymptotically tight, the improvement of the bounds in this paper is significant in the non-asymptotic regime. Relationships between moments of the optimal guessing function and the MAP error probability are also established, characterizing the exact locus of their attainable values. The bounds on optimal guessing moments serve to improve non-asymptotic bounds on the cumulant generating function of the codeword lengths for fixed-to-variable optimal lossless source coding without prefix constraints. Non-asymptotic bounds on the reliability function of discrete memoryless sources are derived as well. Relying on these techniques, lower bounds on the cumulant generating function of the codeword lengths are derived, by means of the smooth Renyi entropy, for source codes that allow decoding errors.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
