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

Two quantum algorithms for communication between spacelike separated locations