Quantum Computing Find the Formula for the Density Matrix - Mathematics Assignment Help

Assignment Task:

Task:

Instructions: Each problem is worth 10 points.

1. Let ψ = P ij aijvi ⊗wj ∈ HA ⊗HB. Find the formula for the density matrix ρA if the basis {w1, . . . , wn} of HB is not orthonormal.

2. A source emits electrons in the same unknown spin state ψ = a |↑zi+b |↓zi. If we measure spin along Z-axis, we get |↑zi with probability 0.64 and |↓zi with probability 0.36. If we measure spin along X-axis, we get |↑xi with probability 0.82 and |↓xi with probability 0.18.

(a) What can be said about ψ (i.e., the values of a and b)?

(b) What are the probabilities of observing |↑yi and |↓yi if the spin is measured along Y-axis?

3. Show that local unitary transformations in W do not change the density matrix ρV .

4. Show that a state ψ ∈ V ⊗ W is not entangled if and only if ρ 2 V = ρV .

5. Let ψ ∈ V ⊗ B, where B is a q-bit space with basis {|0i, |1i}, and V is a higher-dimensional space. Show that the rank of ρV is at most 2.

6. Consider a 4 q-bit system with two 2 q-bit parts: H = V ⊗ W , where V = B ⊗ B and W = B ⊗ B. Consider the following state in H: ψ = 1 √ 12 (2 |0001i − i|0010i + i|0101i + i|0111i + |1110i − 2i|1111i). Find the density matrix ρV . Is ψ entangled? 1

7. In the notations of the previous problem, find the Schmidt decomposition for ψ. For problems involving Octave, please include your code and a screenshot/printout of an execution run.

8. In Octave/Matlab, implement a function, generating a random N-qbit (N is an argument).

9. In Octave/Matlab, implement a probabilistic funcion, measuring value of k-th bit of an N-qbit. (Input: N-qbit, N, k; Output: observed value, collapsed N-qbit.

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