2

I am reading Quantum Computing and Quantum infırmation. And it says:

Adjoints and Hermitian operators Suppose $A$ is any linear operator on a Hilbert space, $V$ . It turns out that there exists a unique linear operator $A^†$ on $V$ such that for all vectors $|v\rangle, |w\rangle ∈ V$ ,

$$(|v\rangle, A|w\rangle)=(A^†|v\rangle, |w\rangle)$$

This linear operator is known as the adjoint or Hermitian conjugate of the operator A. From the definition it is easy to see that $(AB)^† = B^†A^†$.

How come only having the definition of adjoint it is trivial to see $(AB)^† = B^†A^†$ ?

iRestMyCaseYourHonor
  • 145

1 Answers1

2

On the one hand, we have that $\langle ABv,w\rangle = \langle v,(AB)^{*}w\rangle$. On the other hand, we have that \begin{align*} \langle ABv,w\rangle = \langle Bv,A^{*}w\rangle = \langle v,B^{*}A^{*}w\rangle \end{align*}

Since the adjoint operator exists and is unique, we conclude that $(AB)^{*} = B^{*}A^{*}$, as desired.

Hopefully this helps.

user0102
  • 21,572
  • Sorry but I started suspecting everything now. How do we have ⟨,⟩=⟨,()∗⟩ ? Is it stemming from the definition of inner product? Is it because of the nonlinear w.r.t first element property? And also aren't conjugate and adjoint operations slightly different meaning former one lacks the transpose? Did you use conjugate notation as a notation for adjoint? – iRestMyCaseYourHonor May 03 '20 at 08:58
  • It stems from the definition of the adjoint, meaning he's using * for the adjoint. There is not just a single notation for adjoint or conjugation. – ms_ May 03 '20 at 09:16
  • Okay I guess now I understand. First we defined our inner product so that we have the conjugate transpose w.r.t to the first element. And then we defined the hermitian adjoint. Then, in the main part of this proof, user1337 used properties of inner product to arrive the conclusion that ()∗=∗∗ . I understand this $(|⟩,|⟩)=(^†|⟩,|⟩)$ equallity makes † equal to * operation which is defined within the innerproduct. I hope I am right.. – iRestMyCaseYourHonor May 03 '20 at 09:42