lecture 1

lecture 1

1. Classical K D

   • Uses Cipher, has conditions
     • 1. Needs to work!
     • 2. Can not give Eve any info, so m should have no depependence on e
     • Example: One Time Pad

2. Density matrix

   • Difficult to write some mixed states (such as entangled (non-seperable) states, so use density matrix
   • Aver
   • Mixed state
     • Formally given with density matrix, rho = |psi><psi|
     • If there is a single probability (Rho has a single element) then it is a PURE state
       • "Rho has rank 1"
     • If there multiple states with different probabilities then we can think of rho as mixing the probabilities and this is a MIXED state
       • "Rho has rank bigger than 1"
   • Useful way to describe states
     • DOES NOT GIVE INFORMATION ABOUT HOW THE STATE WAS PREPARED, THERE IS OFTEN NOT A SINGLE UNIQUE WAY
   • Bloch Sphere representation
   • To combine two density products, it just the tensor product save as states
   • REQUIRES CONDITIONS: TR(Density_matrix)=1, Density matrix is hermitian & should have no negative eigenvalires
   • Probability of measuring outcome b is also given as

3. Quantum One time pad

   • Can encrypt qubits similar to cbits in the classical one time pad

4. Classical (quantum) states

   • Classical State (c-state) Definition: An Isolated system, X, is in a classical state if the combined density matrix is diagonal in the standard basis
   • Classical-Quantum state (cq-state) Definition:
     • Generally they look a little like:

5. Partial Trace

   • Reduced states
     • A state which has been partially measured and so the wave state has collapsed partially leaving a reduced state,
       • Comparison of the two methods
   • Can use partial trace to solve these reduced states:
     • Examlpe
     • Summery
       • There is a second method that looks like this but I don't like it as much:
         • My preferred method only works for when rho(ab) = rho(a) tensor rho(b). Need to use the other method if not. Might need to review notes before exams

6. To calculate the post measurement of state of a reduced state: Right side (left side is the probability of an outcome but it derives the post measurement state on the right. (rearrange the equation with knowledge that the trace is cyclic so you can form the rhs)

   • Can think of the second take on finding a reduced states as measuring qubit B and forgetting the outcome (so factor out each ourcome of qubit B and sum for each probability related to that outcome. Could also think of it as measuring an infiinte amount of times and then asking what ratio of post measurement states you would expect or could think of it as "think about measuring it and then consider what probabilities of post measurement state you get , This gives the reduced state. of A

7. POVM: Positive operator-valued measure.

   • E.g. Using POVM: Partial trace of a pure state is sometimes a mixed state

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