Lecture 1 - Sep 1, 2020

pure state quantum mechanics
|psi> in C^2,  or C^d,   or C^2 otimes C^2, ... or C^2^{otimes n}

mixed-states QM
|psi>  -->   |psi><psi|  in H(C^d)  = Hermitian d x d matrices
(p_i, |psi_i>)_{i=1...m}  --> rho = sum_{i=1}^m p_i |psi_i><psi_i|

general rule
density matrices = {rho in H(C^d) : tr rho = 1 and rho >=0}
"rho >= 0"   <---> rho is positive semidefinite <--> eig(rho) >= 0

quantum analogue of probability simplex
{p in R^d : p_1, ..., p_d >=0, sum_i p_i = 1}

measure rho in |1>, ..., |d> basis (or do POVM |1><1|, ..., |d><d|)
probability of outcomes are rho_11, rho_22, ..., rho_dd
diag(rho) is prob distribution
eig(rho) is too
exercise: What are the possible relations between these distributions?

* mixed states from entangled states *
|psi>_AB in C^{d_A} otimes C^{d_B}
psi = |psi><psi|
look only at system A
rho_A  = tr_B [ psi ] = psi_A = (id otimes tr)(psi)
psi_B = tr_A [psi]

tr : L(C^d) -> C
id: L(C^d) -> L(C^d)
L(V) = linear operators over V, i.e. matrices

tr_B = id otimes tr
tr_A = tr otimes id

|psi> = sum_ij c_ij |i> otimes |j>
psi = sum_ijkl c_ij c_kl* (|i> otimes |j>) (<k| otimes <l|)
 = sum_ijkl c_ij c_kl* |i><k| otimes |j><l|

id [ |i><k|] = |i><k|
tr [ |j><l|] = delta_jl

psi_A  = sum_ijkl c_ij c_kl* |i><k| delta_jl
= sum_ijk c_ij c_kj* |i><k|

C = (c_11 c_12  ...
        c_21 ...    )

c_kj* = (C^dag)_jk

psi_A = C C^dag

|gamma> = sum_ij d_ij |i> (otimes) |j>
|psi><gamma| = sum_ijkl c_ij d_kl* ...

example 1:
C is rank 1.  |alpha> in C^{d_A}, |beta> in C^{d_B}
C = |alpha><beta|  is d_A x d_B
C_{ij} = alpha_i \bar{beta}_j
|psi> = |alpha> otimes |bar{beta}> in C^{d_Ad_B} = C_{d_A} otimes C^{d_B}
i.e. |psi> is unentangled, a product state

psi_A  = |alpha><alpha| = alpha
psi_B = |bar(beta)><bar(beta)| = bar(beta)

|psi> product <---> psi_A pure (i.e. rank 1) <--> psi_B pure
(I've only shown --> direction.)

example 2:
d = d_A = d_B
C unitary?
C = U
psi_A = C C^dag =  U U^dag = I.  tr \neq 1

instead  C= U / sqrt(d)
psi_A = I / d, maximally mixed state
U = sum_{i=1...d} |u_i><i|,  where |u_1>,...,|u_d> are orthonormal
|psi>_AB = 1/sqrt(d) sum_i |u_i>_A |i>_B
measure B, get each i with probability 1/d, residual state is |u_i>_A
psi_A = 1/d sum_i |u_i><u_i| = I/d for any orthonormal basis {|u_i>}
exercise: check sum_i |u_i><u_i| = I

singular value decomposition for C
C = U D V^dag,  where U,V are unitary, D is diagonal, nonnegative entries
diag(D) = svals(C), # nonzeros = rank(C)
psi_A = C C^dag = U D V^dag (U D V^dag)^dag = U D^2 U^dag
implications:
1. eig(psi_A) = svals(C)^2
svals(C) are called "Schmidt coefficients",  SVD is called "Schmidt decomposition"
exercise: show eig(psi_B) = eig(psi_A), up to adding zeros.
|psi> = |1> otimes |1>
d_A =1, d_B =3
p_A = [1]
psi_B is 3x3 matrix
[1 0 0]
[0 0 0]
[0 0 0]

2. psi_A doesn't depend on V
exercise: Show that local unitaries on A affect only U, local unitaries on B affect only V.

purifications
given rho_A, can we find |psi>_AB such that rho_A = psi_A ?
|psi>_AB is a purification of rho_A, or of psi_A

need to find factorization rho_A = C C^dag

next time: always possible.
factorization is not unique.  will explore non-uniqueness
then make robust

application: bit commitment
no-go thm for bit commitment, based on the robustness of purification