- #1

- 272

- 20

Here is a link to a lecture by Scott Aaronson.

http://www.scottaaronson.com/democritus/lec9.html

About half way in the lecture he talks about the "qubit". In that section he introduces a 2x2 unitary matrix which rotates a vector by 45°. He applies that transformation to the state |0>. When he does that he gets the mixed state 1.0/√2 ( |0>, |1> ). I understand where that mixed state comes form.

Next he applies the transformation another time to the mixed result and he ends up with the state |1>. I understand where that comes from also.

It is his next step that I do not understand. He displays a binary tree which is supposed to show all the possible "paths" when applying the transformation twice. Where does that tree come from? I see two "paths" but I do not see 4 paths. Where does the path that results in the state -|0> come from?

EDIT: I think I see where the state -|0> comes from. If you apply the transformation a third time you get the mixed state 1.0/√2 ( -|0>, |1> ) and if you apply the transformation a fourth time you get the state -|0>. I am still not sure how he constructs that tree.

Here is screen capture of the tree...

https://lh3.googleusercontent.com/KvO3ZBgTUQsLbxSEEUuF6Mo9pXYdV2LJHHAXOUzC7Qwjtio0WoxmiMipIYuLoBImj7Xp-G_UqAvKi6NYei4AqOLedhNPtKSnv1ebfMK1BcORFjvFhw-9_1ZwUsc0SXB5iPiC6q_qVi5bgVWF6WPlWREKBJPxdmSVZImHH1-TCnz2XyyRkLbPxBvhQSZqWONKLD0C-xh_dAgvqeBg112wLdSiGz3gT-0A3VGp4kdXGw_75OVpXSBbboUO3I_izoP0C7xWHwWOKKD1Y62qY2bhFWB-Qbvli0opxIUFDU_Oh83bfDNkNgZyA7pmjGTWfUJbA2-7o0UXLHNpkZjQxl-9isYmASDu0mHlsfc4qkc6DnNZ1MbCclcNljsR_9nUqsxzJB8uP1-UoHQGe8uzEo5z_VT-tM1n-ZvWG5uA-GNU4gm8FlhsFL-8tROFmEmO9EE0VOs2xEoIYsY2bj5XatWBJ59LL-9PQ-Fm_W_VUdM2Ry98DKgPTwQnFutHGiDSo-6ueXow0_SvqacctzhzpdhDU_0n9mPh_b6aMtgwqh1WrwFpv2aZQYQE1rc0KmMe-zMeLzmhDSzjrdAJBmQNk8HRGSaBVe2UnjEvjL4UNtvy8ayCyEM3XNiR=w1350-h741-no

http://www.scottaaronson.com/democritus/lec9.html

About half way in the lecture he talks about the "qubit". In that section he introduces a 2x2 unitary matrix which rotates a vector by 45°. He applies that transformation to the state |0>. When he does that he gets the mixed state 1.0/√2 ( |0>, |1> ). I understand where that mixed state comes form.

Next he applies the transformation another time to the mixed result and he ends up with the state |1>. I understand where that comes from also.

It is his next step that I do not understand. He displays a binary tree which is supposed to show all the possible "paths" when applying the transformation twice. Where does that tree come from? I see two "paths" but I do not see 4 paths. Where does the path that results in the state -|0> come from?

EDIT: I think I see where the state -|0> comes from. If you apply the transformation a third time you get the mixed state 1.0/√2 ( -|0>, |1> ) and if you apply the transformation a fourth time you get the state -|0>. I am still not sure how he constructs that tree.

Here is screen capture of the tree...

https://lh3.googleusercontent.com/KvO3ZBgTUQsLbxSEEUuF6Mo9pXYdV2LJHHAXOUzC7Qwjtio0WoxmiMipIYuLoBImj7Xp-G_UqAvKi6NYei4AqOLedhNPtKSnv1ebfMK1BcORFjvFhw-9_1ZwUsc0SXB5iPiC6q_qVi5bgVWF6WPlWREKBJPxdmSVZImHH1-TCnz2XyyRkLbPxBvhQSZqWONKLD0C-xh_dAgvqeBg112wLdSiGz3gT-0A3VGp4kdXGw_75OVpXSBbboUO3I_izoP0C7xWHwWOKKD1Y62qY2bhFWB-Qbvli0opxIUFDU_Oh83bfDNkNgZyA7pmjGTWfUJbA2-7o0UXLHNpkZjQxl-9isYmASDu0mHlsfc4qkc6DnNZ1MbCclcNljsR_9nUqsxzJB8uP1-UoHQGe8uzEo5z_VT-tM1n-ZvWG5uA-GNU4gm8FlhsFL-8tROFmEmO9EE0VOs2xEoIYsY2bj5XatWBJ59LL-9PQ-Fm_W_VUdM2Ry98DKgPTwQnFutHGiDSo-6ueXow0_SvqacctzhzpdhDU_0n9mPh_b6aMtgwqh1WrwFpv2aZQYQE1rc0KmMe-zMeLzmhDSzjrdAJBmQNk8HRGSaBVe2UnjEvjL4UNtvy8ayCyEM3XNiR=w1350-h741-no

Last edited: