added 38 characters in body; edited tags
Sanchayan Dutta
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Wikipedia saysNielsen and Chuang (on page 188 exercise 4.33) says that the circuit including CNOT and Hadamard is performing a measurement in the Bell basis. But I can't see how.

The matrix representing the circuit functionality is:

$$\begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & -1 \\ 0 & 1 &-1 & 0 \end{bmatrix}$$

Now, what I have found out is, that instead of measuring in the Bell state, it is actually simply converting $$|00\rangle$$ to $$\beta_{00}$$, $$|01\rangle$$ to $$\beta_{01}$$ and vice versa and so on. So it's actually transforming from one basis to another. But how is it a measurement in the Bell state?

Suppose you take the vector $$(1,0,0,0)$$ in the Bell state it's written $$(1,0,1,0)$$ then when measuring in Bell state I would expect you get either $$(1,0,0,0)$$ which is $$|00\rangle + |11\rangle$$ or $$(0,0,1,0)$$ which is $$|00\rangle-|11\rangle$$. But why isn't it doing the measurement but just converting from basis to basis and yet Wikipedia says it's a measurement?

Sanchayan Dutta
• 15.5k
• 5
• 39
• 91

Wikipedia says that the circuit including CNOT and Hadamard is performing a measurement in the Bell basis. But I can't see how.

The matrix representing the circuit functionality is:

$$\begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & -1 \\ 0 & 1 &-1 & 0 \end{bmatrix}$$

Now, what I have found out is, that instead of measuring in the Bell state, it is actually simply converting $$|00\rangle$$ to $$\beta_{00}$$, $$|01\rangle$$ to $$\beta_{01}$$ and vice versa and so on. So it's actually transforming from one basis to another. But how is it a measurement in the Bell state?

Suppose you take the vector $$(1,0,0,0)$$ in the Bell state it's written $$(1,0,1,0)$$ then when measuring in Bell state I would expect you get either $$(1,0,0,0)$$ which is $$|00\rangle + |11\rangle$$ or $$(0,0,1,0)$$ which is $$|00\rangle-|11\rangle$$. But why isn't it doing the measurement but just converting from basis to basis and yet Wikipedia says it's a measurement?

improved mathjax statements

Wikipedia says that the circuit including CNOT and Hadamard is performing a measurement in the Bell basis. But I can't see how.

The matrix representing the circuit functionality is:

$$\begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & -1 \\ 0 & 1 &-1 & 0 \end{bmatrix}$$

Now, what I have found out is, that instead of measuring in the Bell state, it is actually simply converting $$|00\rangle$$ to B00$$\beta_{00}$$, $$|01\rangle$$ to B01$$\beta_{01}$$ and vice versa and so on. So it's actually transforming from one basis to another. But how is it a measurement in the Bell state?

Suppose you take the vector $$(1,0,0,0)$$ in the Bell state it's written $$(1,0,1,0)$$ then when measuring in Bell state I would expect you get either $$(1,0,0,0)$$ which is $$|00\rangle + |11\rangle$$ or $$(0,0,1,0)$$ which is $$|00\rangle-|11\rangle$$. But why isn't it doing the measurement but just converting from basis to basis and yet Wikipedia says it's a measurement?

deleted 25 characters in body
Sanchayan Dutta
• 15.5k
• 5
• 39
• 91