The necessary and sufficient condition for perfect teleportation and superdense coding and all the suitable states for teleportation and superdense coding

TL;DR

This work establishes a framework for assessing when three-qubit states can serve as resources for perfect teleportation (PTP), and two forms of superdense coding (PSDC-2 and PSDC-3), through the lens of LU-invariance. It proves PTP and PSDC-2 are LU invariant and that PTP/PSDC-2 require exactly $1$ $\ebit$ of shared entanglement, with many separable states also qualifying as resources. The analysis classifies suitable states by SLOCC class via Schmidt decompositions, showing GHZ-class states can support PTP and PSDC-2 under precise SD constraints, while W-class states typically do not support PSDC-3. PSDC-3, in contrast, is not LU invariant and is highly constrained, with the GHZ state $\tfrac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$ being the unique SD compatible under PSDC-3, and no subclass of the W or A/B/C-AB classes meeting PSDC-3 criteria. Together, these results provide a complete entanglement-based criterion for three-qubit resources across these protocols and extend to a general n-qubit teleportation condition, offering clear guidance for protocol design and resource evaluation.

Abstract

It is known that two local unitaries (LU) equivalent states possess the same amount of entanglement and can be used to perform the same tasks in quantum information theory (QIT). For a protocol for a task in QIT, we call a protocol LU invariant if two LU-equivalent states are either both suitable for the protocol or neither is. So far, no one has discussed whether a protocol for a task in QIT is LU invariant. In [Phys. Rev. A, 74, 062320 (2006)], Agrawal and Pati proposed the perfect teleportation protocol (PTP) and the protocol for superdense coding to transmit 2-bit classical information by sending one qubit (PSDC-2) and 3-bit classical information by sending two qubits (PSDC-3). In this paper, we show that PTP and PSDC-2 are LU invariant. That is, two LU equivalent states are suitable for PTP and PSDC-2 or neither of them is. We show that PSDC-3 is not LU invariant. We also indicate that the teleportation proposed in <cite>Nielsen</cite> is not LU invariant. We give a necessary and sufficient condition for a state to be suitable for PTP, PSDC-2, and PSDC-3, respectively. Via the LU invariance of PTP and PSDC-2, we prove that a state is suitable for PTP and PSDC-2 if and only if it has 1 ebit of shared entanglement, respectively and find all genuine entangled states and separable states which are suitable for PTP and PSDC-2, respectively. So far, no one has indicated that PTP and PSDC-2 do not require genuine entanglement. Agrawal and Pati suggested to study if there are subclasses of W SLOCC class which are suitable for PSDC-3. So far, it still remains an unsolved question. We show that any state of the SLOCC class W is not suitable for PSDC-3.