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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

State decomposition

How would you decompose the |0> state in the Y-basis? Note: the basis states of the Y-basis are:   

To find the state decomposition of the 0 state in the Y-basis, we have to determine the coefficients in front of the eigenstates. The notation for a general state in the Y-basis is:

where the coefficients alpha and beta are determined by taking the inner product between the basis states:

So the decomposition of the zero state in the Y-basis becomes: