Two quantum algorithms for communication between spacelike separated locations

The `no communication' theorem prohibits superluminal communication by showing that any measurement by Alice on an entangled system cannot change the reduced density matrix of Bob's state, and hence the expectation value of any measurement operator that Bob uses remains the same. We argue that the proof of the `no communication' theorem is incomplete and superluminal communication is possible through state discrimination in a higher-dimensional Hilbert space using ancilla qubits. We propose two quantum algorithms through state discrimantion for communication between two observers Alice and Bob, situated at spacelike separated locations. Alice and Bob share one qubit each of a Bell state $\frac{1}{\sqrt 2}(\ket{00}+\ket{11})$. While sending classical information, Alice measures her qubit and collapses the state of Bob's qubit in two different ways depending on whether she wants to send $0$ or $1$. Alice's first measurement is in the computational basis, and the second measurement is again in the computational basis after applying Hadamard transform to her qubit. Bob's first algorithm detects the classical bit with probability of error $<\frac{1}{2^k}$, but Alice and Bob need to share $k$ Bell states for communicating a single classical bit. Bob's second algorithm is more complex, but Bob can detect the classical bit deterministically using four ancilla qubits. We also discuss possible applications of our algorithms.