Quantum Channels are generalizations of Quantum Operations. Since, in general our operations maybe noisy (inherently) and the system we are dealing with maybe open so Quantum Channels us the study of noisy open Quantum Operations.
The above only holds for closed quantum systems. Thus, in case of noisy open quantum systems we have the following.
Here, is a combination of:
subtracting systems or partial tracing
Review of Density Matrices
Moreover, to deal with noisy open systems we also need to review density matrices which are the general way of representing noisy quantum states.
Now since, is positive semi-definite thus, .
This results in the Bloch Ball representation for general mixed state qubits, which serves as a generalization of the Bloch Sphere that serves as representation for pure state qubits. Pure state qubits have eigenvalues of or , .
The eigenvalue of the maximally mixed state is where the state is actually given by the following.
However, in both these above cases, we are still dealing with closed (albeit noisy) for the random case.
Generalized Quantum Operations
We can refer to Generalized Quantum Operations as Quantum Channels (noisy and open) whose definitions we shall rediscover and formalize here.
Now, it should be noted that there are three major equivalent formalisms for Quantum Channels:
Mathemtically Simplified (Kraus)
Axiomatic (TPCP maps)
Introduction to Steinspring Representation
For closed systems we have a probabilistic combinations of the above (noisy closed channels).
This can be generalized for noisy open channels as combinations of unitaries , adding systems together and subtracting systems .
But why is partial tracing allowed? Doesn't this "delete" information?
The information has basically flown out of your lab to the heavens (but still exists in nature) and in most cases, you simply can't access it anymore, kinda like your ex-girlfriend.
An operator is called an isometry if and only if for all . In other words, an isometry is a genralization of norm preserving operators.
Now, properties of linear isometries:
Linear isometries are not always unitary operators, though, as those require additionally that and .
By the Mazur–Ulam theorem, any isometry of normed vector spaces over is affine. Affine transformations are those that preserve lines and parallelism (but not necessarily distances and angles).
Isometries allow us to restate the requirements of a noisy open quantum channel. Now, it can be formed by of and . Moreover, the purpose of the partial trace is to trash the environment shit out of my lab.
Thus, we have the following formalism.
Partial tracing can be delayed (), an idea similar to deferred measurements. Also, from a philosophical standpoint (Church of the Larger Hilbert Space) you can defer it forever.
Now, the above circuit is equivalent to the below circuit. As a result, inductively, combinations of isometries and partial trace is equivalent to the operator.
Kraus Operator Picture
We can fix a orthonormal bases for , and have such that
where and furthermore
such that in general and thus, we have the Kraus operator condition (stated below).
Thus, we can have a equivalent and simplified formalism of the Steinspring Operator , where are Kraus operators.
Conversely, given satisfying Kraus operator conditions, is a quantum operator with as the isometry involved in the transformation.
Examples of Kraus Operators for Quantum Channels
single Kraus operator
where thus we also have the following holding true, .
, regardless of the initial state .:
simple kind of white noise effect:
where we have the following:
By God, properties of should be as follows:
Linear (assume this or die, basically) and Hermitian preserving
Trace preserving (TP) such that
Completely positive (CP) since
But why completely positive? Well, because we are dealing with isometries of form .
Here, completely positive means that if is applied on a subsystem, then the complete system and the subsystems must remain positive.
Equivalence of TPCP maps and Kraus operators
then is TPCP and converse holds true as well.: If
This implies that the map is injective. In fact it is known as the Choi-Jamiołkowski isomorphism — a correspondence between quantum channels and quantum states (described by density matrices).
operators and with operator elements and respectively, if then: Given
Measurements as Quantum Operations
Measurement can be thought of as a quantum operation where the input is any quantum state and the output is classical, (diagonal density matrix).
The probabilities, , should depend on the state. Further, should be a linear function of the density matrix . From this we can work out the properties that should obey:
thus over thus, we have are positive semi-definite :
Thus, we have the following:
These conditions still leave room for noisy measurements, etc.
We can also talk about non-demolition measurements, which do not discard the quantum systems aftermeasurements. Consider the following quantum channel.
This channel has the following interpretation: with probability as
the state of the system after the application of this channel is . This is similar to having a measurement that outputs the post measured state with probability .
Quantum Norms and Distance Metrics
: Now that we have defined channels and states of information, how do you differentiate between two items of information? What does it mean to say that information is preserved by some process?
Well for these questions it is necessary to develop distance measures. There is a certain arbitrariness in the way distance measures are defined, both classically and quantum mechanically, and the community of people studying quantum computation and quantum information has found it convenient to use a variety of distance measures over the years. Two of those measures, the trace distance and the fidelity, have particularly wide currency today.
Norm is a distance metric such that
Examples of norms are as follows
norm corresponds to probability distributions
norm corresponds to pure states
Similarly, schatten -norm where is a vector of singular values of a matrix, say .
In case of density matrices (in general to positive semi-definite Hermitian operators), we have thus, .
If , then then, .
Thus, is a density matrix and .
Let us consider a simple system with as measurement operators.
Then, we have
But why is it that two objects with different norms can co-exist in the same operator framework? The answer lies in the notion of duality.
Given a norm there exists a dual norm such that, .
is dual to
and are dual to each other
is dual to
and are dual to each other
Introduction to Trace Norm
Then, where denotes measurement,
In fact, we can tighten this inequality even further to obtain
and finally we get the required inequality
Thus, I canof two states, no matter what. So, basically when you wanna protect against noise all we are doing is slowing down the rate of noise (indistinguishable nonsense).