CS378, MA375T, PHY341
Homework 1
Homework 1
Introduction to Quantum Information Science Due Sunday, January 29th at 11:59 PM
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1. Stochastic and Unitary Matrices.
a) Of the following matrices, which ones are stochastic? Which ones are unitary?
10 01 12 10 A= ,B= ,C= 3,D= ,
00 10 013 0i
21 111 34 3i4
E=2,F=√ ,G=55,H=55 −1 1 2 1 −1 4 −3 4 −3i
b) Show that any stochastic matrix that’s also unitary must be a permutation matrix.
c) Stochastic matrices preserve the 1-norms of nonnegative vectors, while unitary matrices preserve 2-norms. Give an example of a 2 × 2 matrix, other than the identity matrix, that
a44 the preserves 4-norm of real vectors b : that is, a + b .
d) [Extra credit] Give a characterization of all matrices that preserve the 4-norms of real vectors. Hopefully, your characterization will help explain why preserving the 2-norm, as quantum mechanics does, leads to a much richer set of transformations than preserving the 4-norm does.
2. Tensor Products.
a) Calculate the tensor product 31 ⊗ 54 .
b) Of the following length-4 vectors, decide which ones are factorizable as 2 × 2 tensor
products, and factorize them. (Here the vector entries should be thought of as labeled by
00, 01, 10, and 11 respectively.)
92 0 14 0 0
1 1 1 1 1 A=94 , B=0, C=41 , D=21 , E=02 .
9 4 2 29 0 14 0 12
3. Prove that there’s no 2 × 2 real matrix A such that 210
A = 0 −1 . Due Sunday, January 29th at 11:59 PM
CS378, MA375T, PHY341 Homework 1
This observation perhaps helps to explain why the complex numbers play such a central role in quantum mechanics.
4. Dirac notation. √√
a) Let |ψ⟩ = |0⟩+2|1⟩ and |φ⟩ = 2i|0⟩+3|1⟩ . What’s ⟨ψ|φ⟩? 5 13
b) Usually quantum states are normalized: ⟨ψ|ψ⟩ = 1. The state |φ⟩ = 2i |0⟩ − 3i |1⟩ is not normalized. What constant A makes |ψ⟩ = |φ⟩ a normalized state?
c) Define |i⟩ = |0⟩+i|1⟩ and |−i⟩ = |0⟩−i|1⟩. Show explicitly that the vectors |i⟩ and |−i⟩
22 form an orthonormal basis for C2.
d) Write the normalized vector |ψ⟩ from part b in the {|i⟩ , |−i⟩} basis.
Due Sunday, January 29th at 11:59 PM Page 2
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