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Fundamentals Solutions

01 /Quantum computers with magical advantage? 02 /Bra-ket notation 03 /Born rule 04 /Bloch sphere 05 /Measurement operators 06 /Experimental and theoretical measurements 07 /How to measure in any basis 08 /State decomposition 09 /Why every maximally entangled state is secretly a Bell state 10 /Coherence 11 /Relaxation 12 /Quantum materials: the environment of a qubit 13 /Introduction to quantum error correction (QEC) 14 /Multiqubit systems 15 /Multiqubit gates 16 /Entanglement 17 /Measuring multiqubit systems 18 /Superdense coding 19 /Teleportation 20 /Entanglement swapping

How to measure in any basis

Let’s try this for ourselves. Which operator U do we need to apply to a qubit to measure it in the X basis? What about the Y basis?

So we try to find an operator U such that U† ZU equals X. It turns out that if we take U=H (the Hadamard gate), we obtain:

One way to get to this answer is to solve the following linear algebra problem:

Then solving for a, b, c and d will lead you to the Hadamard gate.

If we want to measure in the Y basis, we can solve a similar equation and end up with the following matrix:

One way to see this is that if we apply this gate to the Z basis states, we end up with the following:

The resulting states we can recognize as the eigenstates of the Y basis.

Second part of the question: What does the 0 state collapse to if we measure it in the X basis using our protocol?

If we measure the 0 state in the X basis it will collapse to one of the eigenstates of the X basis. So it will either collapse in the plus or minus state.