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arXiv:quant-ph/0412039 · Physical Review A

Probabilistic superdense coding

Arun Kumar Pati, Preeti Parashar, Pankaj Agrawal

Published Jul 22, 2005• 97 citations
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Abstract

We explore the possibility of performing superdense coding with nonmaximally entangled states as a resource. Using this we find that one can send two classical bits in a probabilistic manner by sending a qubit. We generalize our scheme to higher dimensions and show that one can communicate $2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}d$ classical bits by sending a $d$-dimensional quantum state with a certain probability of success. The success probability in superdense coding is related to the success probability of distinguishing nonorthogonal states. The optimal average success probabilities are explicitly calculated. We consider the possibility of sending $2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}d$ classical bits with a shared resource of a higher dimensional entangled state $(D\ifmmode\times\else\texttimes\fi{}D,D>d)$. It is found that more entanglement does not necessarily lead to higher success probability. This also answers the question as to why we need ${\mathrm{log}}_{2}d$ ebits to send $2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}d$ classical bits in a deterministic fashion.

DOI
10.1103/physreva.72.012329

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We explore the possibility of performing superdense coding with nonmaximally entangled states as a resource. Using this we find that one can send two classical bits in a probabilistic manner by sending a qubit. We generalize our scheme to higher dimensions and show that one can communicate 2log2d2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}d2log2​d classical bits by sending a ddd-dimensional quantum state with a certain probability of success. The success probability in superdense coding is related to the success probability of distinguishing nonorthogonal states. The optimal average success probabilities are explicitly calculated. We consider the possibility of sending 2log2d2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}d2log2​d classical bits with a shared resource of a higher dimensional entangled state (D\ifmmode×\else\texttimes\fiD,D>d)(D\ifmmode\times\else\texttimes\fi{}D,D>d)(D\ifmmode×\else\texttimes\fiD,D>d). It is found that more entanglement does not necessarily lead to higher success probability. This also answers the question as to why we need log2d{\mathrm{log}}_{2}dlog2​d ebits to send 2log2d2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}d2log2​d classical bits in a deterministic fashion.
The famous superdense coding protocol of Bennett and Wiesner demonstrates that it is possible to communicate two bits of classical information by sending only one qubit and using a shared EPR pair. Our first result is that an arbitrary protocol for achieving this task (where there are no assumptions on the sender’s encoding operations or the dimension of the shared entangled state) is locally equivalent to the canonical Bennett-Wiesner protocol. In other words, the superdense coding task is rigid . In particular, we show that the sender and receiver only use additional entanglement (beyond the EPR pair) as a source of classical randomness. We also investigate several questions about higher-dimensional superdense coding, where the goal is to communicate one of d 2 possible messages by sending a d -dimensional quantum state, for general dimensions d . Unlike the d =2 case (i.e., sending a single qubit), there can be inequivalent superdense coding protocols for higher d . We present concrete constructions of inequivalent protocols, based on constructions of inequivalent orthogonal unitary bases for all d > 2. Finally, we analyze the performance of superdense coding protocols where the encoding operators are independently sampled from the Haar measure on the unitary group. Our analysis involves bounding the distinguishability of random maximally entangled states, which may be of independent interest.
In quantum superdense coding, two parties previously sharing entanglement can communicate a two bit message by sending a single qubit.We study this feature in the broader framework of general probabilistic theories.We consider a particular class of theories in which the local state space of the communicating parties corresponds to Euclidian hyperballs of dimension n (the case n = 3 corresponds to the Bloch ball of quantum theory).We show that a single n-ball can encode at most one bit of information, independently of n.We introduce a bipartite extension of such theories for which there exist dense coding protocols such that n log 1 2 + ( ) bits are communicated if entanglement is previously shared by the communicating parties.For n 3, > these protocols are more powerful than the quantum one, because more than two bits are communicated by transmission of a system that locally encodes at most one bit.We call this phenomenon hyperdense coding (HDC).Our HDC protocols imply superadditive classical capacities: two entangled systems can encode n log 1 2 2 + > ( ) bits, even though each system individually encodes at most one bit.In our examples, HDC and superadditivity of classical capacities come at the expense of violating tomographic locality or dynamical continuous reversibility.
A method of generating superdense coding based on quantum hyper-entanglement and facilitated by quantum networks is discussed. Superdense coding refers to the coding of more than one classical bit into each qubit. Quantum hyperentanglement refers to quantum entanglement in more than one degree of freedom, e.g. polarization, energy-time, and orbital angular momentum (OAM). The new superdense coding scheme permits 2L bits to be encoded into each qubit where L is the number of degrees of freedom used for quantum hyper-entanglement. The superdense coding procedure is based on a generalization of the Bell state for L degrees of freedom. Theory describing the structure, generation/transmission, and detection of the generalized Bell state is developed. Circuit models are provided describing the generation/transmission process and detection process. Detection processes are represented mathematically as projection operators. A mathematical proof that that the detection scheme permits the generalized Bell states to be distinguished with 100% probability is provided. Measures of effectiveness (MOEs) are derived for the superdense coding scheme based on open systems theory represented in terms of density operators. Noise and loss related to generation/transmission, detection and propagation are included. The MOEs include various probabilities, quantum Chernoff bound, a measure of the number of message photons that must be transmitted to successfully send and receive a message, SNR and the quantum Cramer Rao’ lower bound. Quantum networks with quantum memory are used to increase the efficiency of the superdense coding scheme.
The utilization of a ddd-level partially entangled state, shared by two parties wishing to communicate classical information without errors over a noiseless quantum channel, is discussed. We analytically construct deterministic dense coding schemes for certain classes of nonmaximally entangled states, and numerically obtain schemes in the general case. We study the dependency of the maximal alphabet size of such schemes on the partially entangled state shared by the two parties. Surprisingly, for d>2d>2d>2 it is possible to have deterministic dense coding with less than one ebit. In this case the number of alphabet letters that can be communicated by a single particle is between ddd and 2d2d2d. In general, we numerically find that the maximal alphabet size is any integer in the range [d,d2][d,{d}^{2}][d,d2] with the possible exception of ${d}^{2}\ensuremath{-}1$. We also find that states with less entanglement can have a greater deterministic communication capacity than other more entangled states.