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arXiv:2602.11293 · Annals of Physics

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

Dafa Li

quant-phSubmitted Feb 11, 2026• Announced Feb 13, 2026• 1 citation
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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.

Journal reference
Annals of Physics, Vol. 488 (May 2026) 170377
DOI
10.1016/j.aop.2026.170377
Comments
35 pages, no figures

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As we know, the states of triqubit systems have two important classes: GHZ-class and W-class. In this paper, the states of W-class are considered for teleportation and superdense coding, and they are generalized to multi-particle systems. First we describe two transformations on the shared resources for teleportation and superdense coding. With these transformations, we obtain a sufficient and necessary condition for a state of W-class being suitable for perfect teleportation and superdense coding. For the state which was thought to be not suitable for sending three classical bits by sending two qubits by Agrawal and Pati (2006 Phys. Rev. A 74 062320), we show that it may be used to fulfil that task, if entangled unitary operations on two qubits are allowed. We generalize the states of W-class to multi-qubit systems and multi-particle systems with higher dimension. We propose two protocols for teleportation and superdense coding by using W-states of multi-qubit systems that generalize the protocols by using | W ⟩ 123 proposed by Agrawal and Pati. We obtain an optimal way to partition some W-states of multi-qubit systems into two subsystems, such that the entanglement between them achieves maximum value.
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We investigate the usefulness of a recently introduced five-qubit state by Brown et al. [I. D. K. Brown, S. Stepney, A. Sudbery, and S. L. Braunstein, J. Phys. A 38, 1119 (2005)] for quantum teleportation, quantum-state sharing, and superdense coding. It is shown that this state can be utilized for perfect teleportation of arbitrary single and two-qubit systems. We devise various schemes for quantum-state sharing of an arbitrary single- and two-particle state via cooperative teleportation. We later show that this state can be used for superdense coding as well. It is found that five classical bits can be sent by sending only three quantum bits.
Agrawal and Pati \cite{agrawal} had shown that there exist a class of three-qubit W-states, which act as a resource state for the perfect teleportation of a single qubit. They have used three-qubit measurement basis to execute their teleportation protocol. In this work, we modify Agrawal-Pati teleportation protocol. Our modified teleportation protocol is based on two-qubit measurement basis for the perfect teleportation of a single qubit. We deduce the condition for the successful execution of modified teleportation protocol and this leads us to new class of three-qubit W-states, which is required as a resource state to achieve the modified perfect teleportation protocol. We have constructed operators that can be used to verify the condition of teleportation in experiment. This verification is necessary for the detection of whether the given three-qubit state is useful in modified teleportation protocol or not. Then we have shown that the set containing three-qubit W-states used in modified teleportation scheme is larger than the set of shared three-qubit W-states used in Agrawal-Pati teleportation protocol. Further we quantify the amount of entanglement contained in the newly identified shared three-qubit W-class of states needed for the perfect teleportation of a single qubit. Moreover, we show that the three-qubit W-class of shared states used in the two teleportation protocols can be prepared using NMR set up.
We are now ready to look at our first protocols for quantum information. In this section, we examine two communication protocols which can be implemented using the tools we have developed in the preceding sections. These protocols are known as superdense coding and quantum teleportation. Both are inherently quantum: there are no classical protocols which behave in the same way. Both involve two parties who wish to perform some communication task between them. In descriptions of such communication protocols (especially in cryptography), it is very common to name the two parties ‘Alice’ and ‘Bob’, for convenience. We will follow this tradition. We will repeatedly refer to communication channels. A quantum communication channel refers to a communication line (e.g. a fiberoptic cable), which can carry qubits between two remote locations. A classical communication channel is one which can carry classical bits (but not qubits).1 The protocols (like many in quantum communication) require that Alice and Bob initially share an entangled pair of qubits in the Bell state The above Bell state is sometimes referred to as an EPR pair. Such a state would have to be created ahead of time, when the qubits are in a lab together and can be made to interact in a way which will give rise to the entanglement between them. After the state is created, Alice and Bob each take one of the two qubits away with them. Alternatively, a third party could create the EPR pair and give one particle to Alice and the other to Bob. If they are careful not to let them interact with the environment, or any other quantum system, Alice and Bob’s joint state will remain entangled. This entanglement becomes a resource which Alice and Bob can use to achieve protocols such as the following. Suppose Alice wishes to send Bob two classical bits of information. Superdense coding is a way of achieving this task over a quantum channel, requiring only that Alice send one qubit to Bob. Alice and Bob must initially share the Bell state Suppose Alice is in possession of the first qubit and Bob the second qubit.