# Tag Info

There is a physical meaning to the complex amplitude. The phase of the state affects what happing to the state. For example, consider photon qubit, which might be located in a superposition of $2$ optic fibers (fiber $A$ is $|0\rangle$, fiber $B$ is $|1\rangle$) they might be in the same phase, they might be mirrors (phase $-1 / \pi / 180^\circ$) or phase $... 4 Following up on a comment in @gIS's answer, there is a particular sense in which the computational power of algorithms involving only real amplitudes is very much equivalent, up to a small overhead, as the power of algorithms involving both real and complex amplitudes. I say this because it is known that Toffoli gates and Hadamard gates are sufficient for ... 4 I'm assuming you are asking specifically about the need for complex numbers in the context of a quantum algorithm written as a decomposition in terms of quantum gates. If you are instead asking about the need for complex numbers more in general in quantum mechanics, the answer would be a bit different, depend on what precisely you mean with "need", ... 2 This is basically up to you: which elements are you transposing? If you're talking about transposing just the third system, then you'd be talking about $$|abc\rangle\langle xyz|\mapsto |abz\rangle\langle xyc|$$ but you could do this on any of the three individual subsystems, or any of the three pairs of subsystems. Of course, if you're talking about doing ... 1 Yes, the reasoning is correct. In fact, it can be generalized beyond pure states. By definition, every mixed quantum state$\rho$is a positive semidefinite operator with unit trace. Since every positive semidefinite operator is Hermitian, we may interpret$\rho$as an observable. In this case, the expectation of observable$\rho$in state$\sigma$$$\... 1 I'll consider vector spaces over \mathbb R or \mathbb C. In finite dimensions, an inner product space is automatically complete, and thus a Hilbert space, so no issues there. More generally, one important feature of a Hilbert space V is the isomorphism between V and its (continuous) dual. This duality is what allows to work seamlessly in bra-ket ... 1 Explicit indices The difficulty here arises from making indices implicit in tensor product expressions. For example, the unitary U corresponding to controlled-NOT gate on qubits 1 and 2 and identity on qubit 3 is often written down as$$ U=\text{CNOT}\otimes I\tag1$$but a similar unitary$U'$corresponding to controlled-NOT on qubits$1$and$3\$ ...