T1: 2. Bipartite Systems

Question 1

Define a bipartite system

Answer 1

A quantum system with Hilbert space H, which can be partitioned into subspaces a and b, with Hilbert spaces H_a and H_b

Question 2

Define local operation

Answer 2

Any operation performed by Alice or Bob (or equiv) on their subsystem, i.e. any time evolution or measurement

Question 3

Define classical communication

Answer 3

Any way in which Alice and Bob (or equiv) may communicate by sending classical bits

Question 4

Define a separable (and hence, entangled) state

Answer 4

A separable state is one which can be written as the tensor product of two other states. An entangled state is a non-separable state

Question 5

How do we define a mixed ensemble of separable states?

Answer 5

Using the definition of the density matrix, replacing ϱ_i with the tensor of ϱ_1i and ϱ_2i where these denotes density matrices in the first and second Hilbert spaces.

Question 6

How do operators behave in tensored Hilbert spaces?

Answer 6

An operator which belongs to a Hilbert space can only act upon states in that Hilbert space.

Question 7

In the matrix representation, describe the tensor product of two vectors.

Answer 7

For each element in the first vector, insert the second vector and multiply out.

Question 8

In the matrix representation, describe the tensor product of two matrices.

Answer 8

For each element in the first matrix, insert the second matrix and multiply out.

Question 9

What is the dimension of an n-qubit system?

Answer 9

2^n

Question 10

What is the outcome of two consecutive projectors F_i F_j where i ≠ j?

Answer 10

Zero: all projectors are mutually orthogonal.

Question 11

Where does degeneracy come performing an operator on one system in a bipartite system?

Answer 11

The operator on the bipartite system will be of the form
F_A ⊗ I (for an operator acting on system A). Using the spectral decomposition of the identity, we see multiple states corresponding to the same overall operator F_i.