Projective measurements of quantum processor

[EightBells]

Homework Statement

The quantum processor will provide information about the final state of
the system by performing projective measurement |1><1|. What are possible outcomes of
this measurement and what are the probabilities for this outcomes for each state listed
in Eq. (2). Explain, how you can perform projective measurements for states |±x> and
|±y> using the processor equipped with the standard single qubit gates and the projective
measurements |1><1|.

Relevant Equations

Eq. 2 gives the six states: |0>, |1>, |±x>=(1/sqrt(2))*(|0>±|1>), |±y>=(1/sqrt(2))*(|0>±i*|1>)

Am I correct in thinking that the system measures the probability |<f|1>|^2 for some state <f|? Then the probabilities for each of the six states would be:

|<0|1>|^2= 0
|<1|1>|^2= 1
|<+x|1>|^2= |(1/√2)|^2 = 1/2
|<-x|1>|^2= |(-1/√2)|^2 = 1/2
|<+y|1>|^2= |(-i/√2)|^2 = 1/2
|<-y|1>|^2= |(i/√2)|^2 = 1/2

However, what, specifically, is the question asking for when it asks for the possible outcomes of the measurements?

Performing projective measurements in|±x> and|±y>, am I looking for the probabilities |<f|±x>|^2 and |<f|±y>|^2, but in terms of state |1> which is done using the qubit gates? I know ZH|1> = |+x>, H|1> = |-x>, and SH|1> = |-y>, but I can't find a gate or series of gates that transforms |1> to |+y>, and I'm not sure how to use this to calculate the probabilities |<f|±x>|^2 and |<f|±y>|^2.

 

[member 553083]

The possible outcomes of the measurements are the four states |±x> and |±y>. The probabilities for each of these states can be calculated using the equation |<f|1>|^2, where |f> is one of the four states. For example, the probability of measuring the state |+x> is |<+x|1>|^2 = |(1/√2)|^2 = 1/2. Similarly, the probability of measuring the state |-x> is |<-x|1>|^2 = |(-1/√2)|^2 = 1/2, the probability of measuring the state |+y> is |<+y|1>|^2 = |(-i/√2)|^2 = 1/2, and the probability of measuring the state |-y> is |<-y|1>|^2 = |(i/√2)|^2 = 1/2.