# Difference between Quantum Superposition and Mixed States

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• Trixie Mattel
In summary, the conversation discusses the differences between mixed states and quantum superposition. Mixed states occur when there is a lack of information about the state of a particle, while quantum superposition always exists in quantum systems. The difference lies in the probabilities used to describe the states - classical probabilities for mixed states and quantum probability amplitudes for superposition. It is also mentioned that for a particle to be in quantum superposition, the superimposed states must be eigenstates of a hermitian operator.
Trixie Mattel
Hello, sorry, I do realize that this question has been asked before but there are just a few things I would like to figure out.

So, in my mind the differences lie in knowing the states that the system could end up in, and also the difference in the probabilities.

Is this thinking correct:

For a mixed state, there exists classical randomness. So for example some machine prepares a state
|ψ1> and that is the desired state I would like to get. However the machine is faulty and also prepares some state |ψ2>. The probability of getting |ψ1> is P1 and the probability of state |ψ2> being prepare is 1-P1.
The probabilities here are classical. Also |ψ2> is some state, made by the faulty machine, that I am not really aware of. I am not sure of its characteristics. Its prepared due to classical randomness. And so I get a mixed state of P1|ψ1> + P2|ψ2>Then there's for example a particle. Which can be in spin up or spin down, |0> and |1> respectively.
Here I am aware of the possible state the particle may end up in and the characteristics of those states.
However I can not be sure of which state the particle is in until i make a measurement (the wavefunction collapses) and then it will be in one state or the other. Here in QM we can think of the particle, before measurement as existing in a quantum superposition of both states. α|0> + β|1>
where α and β are not classical probabilities, they are quantum probability amplitudes. And also we can think of this quantum superposition of pure states as just another pure state before measurement.Is that a correct way of thinking of the difference between mixed states and quantum superposition?

Also, to be in quantum superposition, is it a necessity that the superimposed states are eigenstates of some hermitian operator??Thank you very much

A mixed state occurs because of a lack of information. For instance, if a particle is entailed entangled (darn autocorrect) with another but the state of that particle is unknown (or it is not even known that the particle has an entanglement partner), then its state is essentially a mixed state.

Quantum systems are always in superpositions. If they are in an eigenstate of a given observable, then they are in a superposition of states of any observable that does not not commute with the first one.

Last edited:
vanhees71, Jilang and Trixie Mattel

## 1. What is the main difference between quantum superposition and mixed states?

Quantum superposition is a state in which a quantum system exists in multiple states simultaneously, while mixed states are a statistical mixture of different quantum states. In superposition, the system has a definite probability of being in each state, whereas in mixed states, the system has a definite probability of being in a specific state.

## 2. How are quantum superposition and mixed states represented mathematically?

Quantum superposition is represented by a linear combination of different quantum states, while mixed states are represented by a density matrix. The density matrix contains information about the probabilities of the system being in different states.

## 3. What are some real-life applications of quantum superposition and mixed states?

Quantum superposition is a key concept in quantum computing, allowing for the creation of qubits that can store and manipulate multiple pieces of information simultaneously. Mixed states are utilized in quantum cryptography, where the probabilistic nature of mixed states can be used to securely transmit information.

## 4. Can quantum superposition and mixed states be observed in everyday objects?

No, quantum superposition and mixed states can only be observed in the microscopic world of quantum mechanics. In everyday objects, the principles of classical mechanics apply, and objects can only exist in one state at a time.

## 5. How do quantum superposition and mixed states relate to the concept of wave-particle duality?

Quantum superposition and mixed states are closely related to wave-particle duality, as they both demonstrate the probabilistic nature of quantum particles. Just as particles can exist in multiple states simultaneously, they also exhibit wave-like behavior, such as interference and diffraction. Mixed states also show how particles can have both particle-like and wave-like properties at the same time.

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