Understanding interference and decoherence

In summary, the concept of inner product and transition probability helps to explain how decoherence causes interference to disappear in quantum systems, and how it relates to the double slit experiment.
  • #1
Talisman
95
6
I'm trying to make sense of various explanations of why decoherence causes interference to disappear, but I'm afraid I don't quite grok it.

There's a bit of explanation in this thread.

And some more on wikipedia.

The first link starts with [tex]|\psi{\rangle} = |a{\rangle} + |b{\rangle}[/tex] and considers [tex]{\langle}\psi | \psi{\rangle} = {\langle}a|a{\rangle} + {\langle}b|b{\rangle} + 2 Re({\langle}a|b{\rangle})[/tex]. Clearly a and b are not meant to be basis vectors here; in that case, why did we write [tex]\psi[/tex] as a sum of them to begin with?

Wikipedia shows the following:
[tex]prob(\psi \Rightarrow \phi) = |{\langle} \psi |\phi {\rangle}|^2 = |\sum_i\psi^*_i \phi_i |^2 = \sum_{ij} \psi^*_i \psi_j \phi^*_j\phi_i= \sum_{i} |\psi_i|^2|\phi_i|^2 + \sum_{ij;i \ne j} \psi^*_i \psi_j \phi^*_j\phi_i[/tex]
In both cases, when we work out the algebra in the inner product, we find that the expansion contains terms involving the product of components of each state, which are called "cross terms."

What I don't understand is the significance of taking the inner product of a state with itself in the first case, or what the "transition probability" refers to in the second. Also, how does this relate to the double slit experiment? Aren't the position states represented by the electron going through each slit orthogonal?
 
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  • #2
The inner product of a state with itself in the first case is a measure of the probability that the system will remain in its original state, while the transition probability in the second case is the probability that the system will transition from its initial state to its final state, given the two states. This can be seen as relating to the double slit experiment in that the initial state is the electron going through one of the two slits, and the final state is the electron going through both slits at the same time. As the two states are orthogonal, the transition probability between them is zero, and so interference is not observed. This is because the cross terms between the two states cancel each other out, meaning that the probability of the electron being detected at any given point on the screen is not affected by the electron going through both slits (i.e., there is no interference). Decoherence prevents interference from occurring by introducing random noise into the system which causes the two states to become entangled with their environment, making them no longer orthogonal. This means that the cross terms between the two states no longer cancel each other out, and so the interference pattern can be observed.
 
  • #3


First of all, let's clarify what interference and decoherence are. Interference is a phenomenon observed in quantum mechanics where particles can exhibit wave-like behavior and produce a pattern of constructive and destructive interference. This is commonly demonstrated in the double-slit experiment where a single particle can interfere with itself and create an interference pattern on a screen.

Decoherence, on the other hand, is the process by which a quantum system interacts with its environment, causing the system to lose its quantum coherence and behave more like a classical system. In other words, the system's quantum properties, such as superposition and interference, become "entangled" with the environment and are no longer observable.

Now, let's address your specific questions. In the first link, the author is discussing the inner product of a state with itself, which is also known as the norm or magnitude of the state. This is a common mathematical operation in quantum mechanics and is used to calculate the probability of finding a particle in a particular state. In the case of the double slit experiment, the state of the particle is represented by a superposition of two states, one going through each slit. The inner product of this state with itself gives us the probability of finding the particle in either of the two states, which explains the presence of the cross terms in the expansion.

In the second link, the author is discussing the transition probability, which is the probability of a particle transitioning from one state to another. This is also calculated using the inner product of the two states involved. In the case of the double slit experiment, this is related to the probability of the particle transitioning from one slit to the other and causing interference.

To address your last question, the position states of the electron going through each slit are indeed orthogonal. However, in the quantum world, particles can exist in a superposition of states, which means they can simultaneously exist in multiple states until they are observed. In the double slit experiment, the particle is in a superposition of going through both slits, which allows for the interference pattern to be observed. But as soon as the particle interacts with its environment, decoherence occurs, and the particle is forced to "choose" one path, resulting in the disappearance of interference.

In summary, interference and decoherence are two fundamental concepts in quantum mechanics that are crucial to understanding the behavior of particles at the quantum level. The inner product and transition probability play a key role in calculating the probability of finding a particle in a
 

1. What is interference and decoherence?

Interference and decoherence are two related concepts in quantum mechanics that describe how particles interact and behave in a quantum system. Interference refers to the phenomenon where particles can overlap and interact with each other, resulting in observable patterns. Decoherence, on the other hand, is the process by which a quantum system loses its coherence due to interactions with the environment, ultimately leading to the collapse of the quantum state into a classical state.

2. How does interference and decoherence affect quantum computing?

In quantum computing, interference and decoherence can greatly impact the accuracy and stability of quantum operations. Interference can be harnessed to perform calculations and store information in a quantum system, but decoherence can disrupt these processes and introduce errors. Therefore, minimizing decoherence is a major challenge in quantum computing.

3. What are some examples of interference and decoherence in everyday life?

Interference and decoherence are not just theoretical concepts in quantum mechanics, but they can also be observed in everyday life. For example, the colors on a soap bubble are a result of interference between light waves. Decoherence can also be seen in the way a hot cup of coffee cools down as it interacts with the cooler air in the room.

4. How do scientists study interference and decoherence?

Scientists use a variety of tools and techniques to study interference and decoherence, including quantum simulators and interferometers. These experiments help researchers understand the behavior of quantum particles and how they interact with the environment.

5. Can interference and decoherence be controlled or manipulated?

While decoherence is often seen as a hindrance to quantum computing, scientists are actively researching ways to control and manipulate it. This includes using error-correcting codes and quantum error correction techniques to minimize the effects of decoherence. Additionally, new materials and technologies are being developed to reduce the impact of the environment on quantum systems.

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