Q1L2b.txt

#
# File:   content-mit-8371x-subtitles/Q1L2b.txt
#
# Captions for 8.421x module
#
# This file has 66 caption lines.
#
# Do not add or delete any lines.  If there is text missing at the end, please add it to the last line.
#
#----------------------------------------

For quantum error correction, we need
to describe what an error is, in a quantum mechanical context.
More generally, let us try to describe
all possible ways in which a input density matrix may evolve
to an output density matrix.
We will describe this via a map called E.
E should be able to capture any physical process happening
to this input state, not just unitary evolution,
but non-unitary, as well.
The model which we will employ will
be one which has two parts.
We will suppose that we have a system, which
may start at a pure state, say, psi for now.
And this system evolves in a unitary manner,
but not just by itself, rather, it
is coupled to another state, which
we will call the environment.
Let us say the initial state of the environment
is a ket e, a pure state, but that need not
be a pure state, in general.
The environment, let us say, is measured
after this unitary transform.
And we will consider the measurement
as happening in an orthonormal basis, e0, e1,
and so forth, up through the dimension of the environment.
The question is, what is the output state, rho?
We may write the output state before the measurement,
by understanding that the unitary transform gives
us U times e times psi.
It is convenient to invent some notation
to describe this output state.
We will use the circle plus symbol to represent "or."
Using this, then we may write the output state rho
as a stochastic combination, where,
when the measurement result is 0, we get e0 u e,
and this is an operator which acts on psi.
Or, if the measurement result is 1,
we get e1 U e acting on psi, and so forth.
This may be compactly written as a sum
over all possible measurement outcomes,
where ek U e, we emphasize, is a matrix, an operator, which
acts on psi.
We call this operator an operation element.
Historically, it is also called, sometimes, a Krauss operator.
And this operator, E sub k, will play a principal role
in our study of quantum error correction.
We may also sometimes call them error operators.
Concisely, therefore, we may write the output density
matrix in this form, as the sum over k Ek rho Ek dagger.
This is the map, E, acting on rho.
The main constraint is that the sum
of the square of these operators, Ek dagger Ek
must equal identity.
We can check that this is true, explicitly,
by substituting in the definition of E sub k,
first using Ek dagger and then Ek.
And then, noting that this term in the middle, the sum over ek
ek, is identity, because this is a complete set of states
for the basis of the environment.
Thus, this sum, overall, gives the identity operator,
as desired.
This setup, with the system environment model giving us
an equation which maps the input density matrix to the output
density matrix, is an important idea for quantum error
correction, and we will return to this model several times.