Bellman's optimality principle has been of enormous importance in the development of whole branches of applied mathematics, computer science, optimal control theory, economics, decision making, and classical physics. Examples are numerous: dynamic programming, Markov chains, stochastic dynamics, calculus of variations, and the brachistochrone problem. Here we show that Bellman's optimality principle is violated in a teleportation problem on a quantum network. This implies that finding the optimal fidelity route for teleporting a quantum state between two distant nodes on a quantum network with bipartite entanglement will be a tough problem and will require further investigation.

Optimal path for a quantum teleportation protocol in entangled networks

We study the activation of entanglement in teleportation protocols. To this end, we a present derivation of the average fidelity of teleportation process with noisy classical channel for qudits. In our work we do not make any assumptions about the entangled states shared by communicating parties. Our result allows us to specify the minimum amount of classical information required to beat the classical limit when the protocol is based on the Bell measurements. We also compare average fidelity of teleportation obtained using noisy and perfect classical channel with restricted capacity. The most important insight into the intricacies of quantum information theory that we gain is that though entanglement, obviously, is a necessary resource for efficient teleportation it requires a certain threshold amount of classical communication to be more useful than classical communication. Another interesting finding is that the amount of classical communication required to activate entanglement for teleportation purposes depends on the dimension d of the system being teleported but is not monotonic reaching maximum for d = 4.

Activation of entanglement in teleportation

We study the activation of entanglement in teleportation protocols. To this end, we present a derivation of the average fidelity of the teleportation process with a noisy classical channel for qudits. In our work, we do not make any assumptions about the entangled states shared by communicating parties. Our result allows us to specify the minimum amount of classical information required to beat the classical limit when the protocol is based on the Bell measurements. We also compare the average fidelity of teleportation obtained using a noisy and perfect classical channel with restricted capacity. The most important insight into the intricacies of quantum information theory that we gain is that, although entanglement is obviously a necessary resource for efficient teleportation, it requires a certain threshold amount of classical communication to be more useful than classical communication. Another interesting finding is that the amount of classical communication required to activate entanglement for teleportation purposes depends on the dimension d of the system being teleported but is not monotonic, reaching a maximum for d = 4.

https://iopscience.iop.org/article/10.1088/1751-8113/46/43/435301/pdf

Since decoherence caused by some environment is always present in practice, the quantum channels usually exist in the form of non-maximally entangled states instead of maximally entangled states. We present an approach to deterministically teleport an arbitrary two-qubit state with a probabilistic quantum channel composed of a one-dimensional six-qubit cluster state. To avoid damaging the states to be teleported, the channel is modulated in advance, and Hadamard transformation and Bell state measurement are applied instead of six qubit von Neumann measurement. The scheme is flexible because the channel can be modulated either by the sender or by the receiver, and we may decide which party modulates the channel according to the distribution of available particle and gate operation resources. Furthermore, we provide different ways of redistributing qubits between the sender and the receiver, as well as alternative solutions of constructing the scheme. Finally, for general cases, we extend this scheme to deterministic teleportation of an arbitrary n-qubit state.

Deterministic teleportation of an arbitrary two-qubit state via a six-qubit cluster state

We study dense coding under the condition that the sender’s encoding operations be imperfect. In order to formally describe the effect of the imperfect encoding operations, we use four kinds of quantum noise processes. In this way, the imperfect operation is the corresponding perfect operation followed by a quantum noise process. We show the relation among the average probability of decoding the correct information, the non-maximally entangled state, the imperfect encoding operations, and the receiver’s measurement basis.

Dense Coding Process with Imperfect Encoding Operations

Teleportation protocol with non-ideal conditional local operations

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Teleportation protocol with non-ideal conditional local operations