Classical channel vs quantum channel

In summary, a classical channel in quantum information theory is a map between states that is defined by a conditional probability distribution. This is represented by the formula ∑x,y (Tr lx><xlX)P(ylx)ly><yl, where P(ylx) is the probability distribution and the trace appears as a way to calculate the mean value of X in the state. In contrast, a quantum channel is a map between states in general, and can send definite states given by some probability distribution. The formula for a classical channel of two bits is similar to the first formula, but with the addition of the trace of lij><ijlX. X refers to an operator in this context. The image of a projector |z
  • #1
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I have a hard time understanding what a channel that transmits classical information is in the framework of quantum information theory. My textbook says the following:

Let P(ylx) be a classical channel defined by a conditional probability distribution. We define the corresponding quantum channel by:

∑x,y (Tr lx><xlX)P(ylx)ly><yl

How should I interpret this formula and what is the role of the trace appearing? And in general what is a classical channel as opposed to a quantum channel? I guess the answer is that a classical channel can send definite states given by some probability distribution, while quantum channels are maps between states in general.

Another place in the book it states that a classical channel of two bits is of the form:
C(X) = ∑i,j lij><ijl Tr lij><ijlX
Is this equivalent with the first formula?
 
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  • #2
Could you explain what is X in your question?
Have you links to this subject?
 
  • #3
The first thing which comes to my mind is that the trace of ##\rho X## is the mean value of X in the state ##\rho##
Now if X is an operator it sends X to another operator. How can we describe it? What is the image of a projector |z><z|?
 

1. What is the difference between a classical channel and a quantum channel?

A classical channel is a communication channel that transmits classical bits of information (0s and 1s) between two parties. A quantum channel, on the other hand, can transmit quantum states, which are combinations of 0s and 1s, between two parties. This means that a quantum channel has the ability to transmit information that is in a superposition of both 0 and 1 states, while a classical channel can only transmit information in one of these two states.

2. How do classical and quantum channels differ in terms of security?

Classical channels are vulnerable to eavesdropping, where an unauthorized third party can intercept and read the transmitted information. However, quantum channels offer a higher level of security due to the principles of quantum mechanics, such as the no-cloning theorem, which make it impossible for an eavesdropper to intercept and copy the transmitted quantum information without being detected.

3. Can classical and quantum channels be used interchangeably?

No, classical and quantum channels cannot be used interchangeably. The type of channel needed depends on the type of information being transmitted. Classical channels are suitable for transmitting classical bits of information, such as text or images, while quantum channels are necessary for transmitting quantum information, such as quantum keys for secure communication.

4. What are some real-world applications of classical and quantum channels?

Classical channels are commonly used in everyday communication systems, such as the internet and telephone networks. Quantum channels, on the other hand, are used in quantum communication systems, such as quantum key distribution, which enables secure communication between two parties. Quantum channels are also used in quantum computing and quantum teleportation.

5. Are there any limitations to quantum channels?

While quantum channels offer a higher level of security compared to classical channels, they are not immune to all types of attacks. For example, quantum channels can be vulnerable to side-channel attacks, where an eavesdropper can gather information about the quantum state being transmitted without directly intercepting the information. Additionally, quantum channels have limitations in terms of distance, as the quantum state can become degraded over long transmission distances, impacting the accuracy of the transmitted information.

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