The Postulates of Quantum Mechanics

Let’s begin by summarizing what we know about quantum states so far. In previous lessons, we’ve seen that quantum states are vectors in 2-D complex vector space. These bras (row vectors) and kets (column vectors) depict the possible locations that we might find a qubit in. We form a multi-qubit system by taking the tensor product of the state vectors of single qubits. To know which state a qubit is in, we must measure it. We measure it using complex hardware in the computational basis and transfer the state information into classical bits that we can read. Which state we end up getting is entirely probabilistic until the first measurement. The complex amplitudes dictate this probability associated with the state vectors. And lastly, no measurement on the system is gentle enough to leave the qubit in an unperturbed state.

All these ideas take root from the postulates at the heart of quantum mechanics. We’ve already discussed a bunch of them, as you’ll notice. But going through them in a sequence revises the concepts we have learned so far and makes it easier for us to understand the ones we haven’t discussed.

Quantum states are described by vectors

A quantum state is represented by a vector in 2-D complex vector space. This state, a ket-vector $|\psi\rangle$, has complex amplitudes which control the probabilities of measuring the quantum particle in one of its possible states.

A key point to make here is that this state vector completely describes the state of an isolated system. In such a case, it is a unit vector in that vector space. The keyword here is isolated system. An isolated system does not interact with foreign objects, not even with other quantum particles outside the system, because interaction with foreign objects also localizes the quantum particles. So, systems that are not in isolation do not have a quantum state of their own.

This is the reason we do not see the eccentricities of quantum superposition play out in the real world as objects we see around us are localized because they’ve been implicitly “measured” in those interactions.

Quantum states evolve with time

The second postulate provides an important clarification. The previous might have made it seem that isolated quantum states do not change at all when in fact, they do. Quantum states evolve with time. This was formalized by Schrodinger’s Equation:

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