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arXiv:2111.01563 · Physica Scripta

Absolutely secure distributed superdense coding: entanglement requirement for optimality

Sagnik Dutta, Asmita Banerjee, Prasanta Kumar Panigrahi

Published Jan 11, 2023• 12 citations
arXivHTMLPDF

Abstract

Abstract Superdense coding uses entanglement as a resource to communicate classical information efficiently through quantum channels. A superdense coding method is optimal when its capacity reaches Holevo bound. We show that for optimality, maximal entanglement is a necessity across the bipartition of Alice and Bob, but neither absolute nor genuine multipartite entanglement is required. Unlike the previous schemes, which can transmit either even or odd bits of information, we demonstrate a generalized dense coding protocol using the genuine multipartite entangled GHZ state to send arbitrary information bits. Expressed in the eigenbasis of different Pauli operators, GHZ state is characterized by a unique parity pattern which enables us to formulate a security checking technique to ensure absolute security of the protocol. We show this method is better applicable in a scenario, where the initial information is distributed among spatially separated parties. Finally, optimizing the number of qubit(s) sent to Bob, we construct a distributed dense coding method, which completely depicts absolutely secure quantum communication between many to one party.

DOI
10.1088/1402-4896/acb23d

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Abstract Superdense coding uses entanglement as a resource to communicate classical information efficiently through quantum channels. A superdense coding method is optimal when its capacity reaches Holevo bound. We show that for optimality, maximal entanglement is a necessity across the bipartition of Alice and Bob, but neither absolute nor genuine multipartite entanglement is required. Unlike the previous schemes, which can transmit either even or odd bits of information, we demonstrate a generalized dense coding protocol using the genuine multipartite entangled GHZ state to send arbitrary information bits. Expressed in the eigenbasis of different Pauli operators, GHZ state is characterized by a unique parity pattern which enables us to formulate a security checking technique to ensure absolute security of the protocol. We show this method is better applicable in a scenario, where the initial information is distributed among spatially separated parties. Finally, optimizing the number of qubit(s) sent to Bob, we construct a distributed dense coding method, which completely depicts absolutely secure quantum communication between many to one party.
Quantum communication, as a vital area of quantum information science, aims to leverage quantum phenomena—particularly entanglement—to achieve tasks that are impossible or inefficient in classical systems. It plays a key role in the development of secure communication protocols and advanced information processing. This paper focuses on one such protocol, dense coding, which enables the transmission of two classical bits using only one entangled quantum particle.”We introduce the mathematical and physical foundations of quantum communication, such as the entangled state and Kronecker product. Then we propose the quantum communication protocol, namely dense coding using Bell states. It offer unique advantages that are beyond the classical parts. For this purpose, we in detail analyze the quantum measurements using Pauli gates and identity matrix, as well as the transmission of particle from Alice to Bob. They are necessary steps for dense coding. Then we analyze how Bob finishes the protocol using measurement over two qubits.
This study presents a generalized nnn--bit superdense coding protocol that enables the transmission of n classical bits of information using an entangled nnn--qubit quantum system and the transmission of n−1n-1n−1 qubits. The protocol involves creating a maximally entangled nnn--qubit GHZ state, encoding the classical message with Pauli--Z and Pauli--X gates, and then transmitting and decoding the message via quantum communication, quantum operations, and measurements. The key novelty of this work is an explicit, scalable nnn--bit encoding routine, realized on the GHZ channel with a low--depth encoding layer using only single--qubit Pauli--X and Pauli--Z operations. The protocol was tested on real quantum hardware using Qiskit 2.0 and the IBM--Fez quantum computer for message lengths of 4, 6, 8, and 10 bits, as a circuit--level proof of concept on a single processor; the physical transmission of qubits between the parties was not implemented. Results show that success rates decrease as message length, transpiled circuit depth and gate count increase; the latter two are set mainly by transpilation rather than by the encoding itself. A mutual information analysis shows that the transmitted information exceeds the classical bound of n−1n-1n−1 bits only for short messages. Strategies to improve performance include sending messages in shorter segments and advances in qubit coherence and gate fidelity. This work offers a practical and easily scalable quantum communication protocol with potential applications in quantum networks and communication systems.
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.
The beauty of the quantum circuit lies in the fact that it can perform certain tasks which are impossible in the classical world. For example, consider teleportation and superdense coding. Teleportation is a quantum task in which an unknown quantum state is transmitted from a sender (Alice) to a spatially separated receiver (Bob) via an entangled quantum channel and with the help of some classical communications. Dense coding is another communication scheme in which if Alice and Bob share a prior entanglement then one of them can send n bits of classical information to the other by communicating m-qubits (m < n). Before we go into the technical details of these two schemes let us try to lucidly visualize the uniqueness of teleportation. Suppose Alice and Bob are spatially separated and Alice wants to send an unknown quantum state |ψ〉 = α|0〉 + β|1〉 to Bob. For some reason she is not allowed to send the state directly through a quantum channel. She cannot measure it and send the information to Bob because the moment she will try to measure the state, it will collapse either to |0〉 or to |1〉. But the state can be sent by using a simple quantum circuit (pre-shared entanglement) and some classical communications. In the process the unknown quantum state gets destroyed at Alice’s end and reappears at Bob’s end. The state never exists in the communication channel. This process is called quantum teleportation. The uniqueness of quantum teleportation will be clearer if we note that the Fax is not an example of teleportation because the original copy remains with the sender. Again a letter posted in a letter box is not an example of teleportation because the letter exists (travels) in its original form in the communication channel. In brief, there is no classical analogue of quantum teleportation. The same is true for dense coding. In this chapter we will describe these two unique quantum communication schemes and discuss their significance and applications.