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Physical Review A

Control power of high-dimensional controlled dense coding

Wen-Xin Duan, Tie-Jun Wang

Published May 12, 2022• 30 citations
Publisher

Abstract

We quantitatively analyze and evaluate the control power in a high-dimensional noisy quantum channel for transmitting classical information. We calculate the control power of the controlled dense coding scheme in a four-dimensional extended Greenberger-Horne-Zeilinger (GHZ) class state channel. Different from controlled teleportation, there is no tradeoff between the control power and the classical capacity of the optimal four-dimensional GHZ state quantum channel. Only when the channel is noisy and the sender is allowed to gain some control authority can the tradeoff between the control power and the classical capacity be activated, and the overall control power for transmitting classical information will be enhanced by reducing the classical capacity of the high-dimensional quantum channel. This noise induced characteristic is very different from that of transmitting quantum information, and it may provide new ideas and methods to develop the resource theory of quantum channels.

DOI
10.1103/physreva.105.052417

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We quantitatively analyze and evaluate the control power in a high-dimensional noisy quantum channel for transmitting classical information. We calculate the control power of the controlled dense coding scheme in a four-dimensional extended Greenberger-Horne-Zeilinger (GHZ) class state channel. Different from controlled teleportation, there is no tradeoff between the control power and the classical capacity of the optimal four-dimensional GHZ state quantum channel. Only when the channel is noisy and the sender is allowed to gain some control authority can the tradeoff between the control power and the classical capacity be activated, and the overall control power for transmitting classical information will be enhanced by reducing the classical capacity of the high-dimensional quantum channel. This noise induced characteristic is very different from that of transmitting quantum information, and it may provide new ideas and methods to develop the resource theory of quantum channels.
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To quantitatively analyze the control power of arbitrary ddd-dimensional controlled teleportation (CT), this paper investigates channels of entangled high-dimensional three-qudit standard Greenberger-Horne-Zeilinger (GHZ) and GHZ-type states. The upper bound on the controller's power is established in the ddd-dimensional CT scheme. We identify a trade-off between the control power and the capacity of the quantum channels. The control power can be enhanced by reducing the dimension of the teleported information in a suitably high-dimensional quantum channel, or by increasing the dimension of the qudit held by the controller. Under controlled teleporting of a single qubit in a perfect CT scheme, the control power is maximized in the three-dimensional GHZ-type state channel. This maximum control power exceeds the highest predicted value in previous works and the upper bounds of control power in a two-dimensional quantum channel. Our studies provide a feasible method for achieving higher control power, and it may provide approaches to define high-dimensional entanglement of multibody systems.
A highly entangled bipartite quantum state is more advantageous for the quantum dense coding protocol than states with low entanglement. Applications of quantum channels are most likely to be commercially important only in the multiparty regime, where such a correspondence does not exist even for pure quantum states. We establish a connection between the multiparty capacity of classical information transmission in quantum dense coding and several multipartite quantum correlation measures of the shared state, used in the dense coding protocol. In particular, we show that for the noiseless channel, if multipartite quantum correlations of an arbitrary multipartite state of arbitrary number of qubits are the same as those of the corresponding generalized Greenberger-Horne-Zeilinger state, then the multipartite dense coding capability of the former is the same as or better than that of the generalized Greenberger-Horne-Zeilinger state. Interestingly, in a noisy-channel scenario, where we consider both uncorrelated and correlated noise models, the relative abilities of the quantum channels to transfer classical information can get inverted by administration of a sufficient amount of noise. When the shared state is an arbitrary multipartite mixed state, we also establish a link between the classical capacity for the noiseless case and multipartite quantum correlation measures.
We investigate minimal control power (MCP) for controlled dense coding defined by the channel capacity. We obtain MCPs for extended three-qubit Greenberger-Horne-Zeilinger (GHZ) states and generalized three-qubit W states. Among those GHZ states, the standard GHZ state is found to maximize the MCP and so does the standard W state among the W-type states. We find the lower and upper bounds of the MCP and show for pure states that the lower bound, zero, is achieved if and only if the three-qubit state is biseparable or fully separable. The upper bound is achieved only for the standard GHZ state. Since the MCP is nonzero only when three-qubit entanglement exists, this quantity may be a good candidate to measure the degree of genuine tripartite entanglement.
Ideal Controlled quantum teleportation, discovered theoretically by Karlsson and Bourennane, is a weird paradigm of nonlocal transplantation of probability-amplitudes of an arbitrary unknown generic information-qubit possessed by a sender (Alice) on a qubit preserved by a distantly located receiver (Bob) under control or cooperation by a third party (Charlie) via maximally entangled Greenberger-Horne-Zeilinger (GHZ) state employed as the quantum channel with local operation and classical communication. The GHZ state is a fragile quantum resource and extremely sensitive to inevitable noisy environment causing decoherence. It fuels, therefore, an imperatively pressing demand to strategize a scheme for noise-resistant control quantum teleportation. We revisit the ideal control quantum teleportation but with a noisy Greenberger-Horne-Zeilinger State in which each intertwining qubit suffers with differing noises and inequivalent levels, namely, bit-flip, amplitude-flip and phase-flip, i.e., asymmetric noise ambience. The controlling power, introduced by Li and Ghose, is invoked to assess the performance of the afore-described noisy strategy. It is seen that differing levels of noises breaks the classical bound of controlling power and, thereby, demonstrating the loosening authority of controller. Our findings suggest that to which extent of asymmetric noise ambience noisy control quantum teleportation is noise-tolerant.