Nigel:
Take a very simple physical system exhibiting decoherence. Perhaps involving the beta decay of an unstable nucleus in vacuum and its subsequent detection at a screen. I'm assuming that decoherence occurs as the electron interacts with the atoms (electron shells) of the screen.
Could you describe the situation, showing and explaining all logical steps?
How did the electron screen interaction vs non-interaction induce decoherence? How, in physics terms, was the superposition lost?
Gemini:
You might want to start by watching Sabine Hossenfelder's excellent video tutorial about decoherence. And then resume reading.
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Let's break down in words how the electron-screen interaction, versus non-interaction, leads to decoherence and the apparent "loss" of superposition in physical terms.
The Essence of Decoherence in This Scenario
Imagine the electron is initially in a quantum superposition of "arriving at the screen" (∣A⟩) and "not arriving at the screen" (∣N⟩). Think of this like being in a quantum "both/and" situation regarding its path to the detector.
Before Interaction (Coherent Superposition)
Quantum Coherence: In the initial superposition (1/√2)(∣A⟩+∣N⟩)the electron is in a coherent superposition. This means the ∣A⟩ and ∣N⟩ components are not just alternatives, but they are quantum mechanically interfering with each other. If we were to perform a measurement that's sensitive to this superposition (not just measuring if it's in ∣A⟩ or ∣N⟩ individually, but something that probes the phase relationship between them), we would observe interference effects.
Environment is Ignorant: At this stage, the environment (the screen atoms) is in its initial state ∣E0⟩. It's essentially "ignorant" about whether the electron is in state ∣A⟩ or ∣N⟩. There is no correlation or entanglement between the electron and the environment yet.
During Interaction with the Screen (Entanglement and "Measurement")
Interaction as a "Measurement-like" Process: The interaction of the "arriving" electron (∣A⟩) with the screen atoms is crucial. Think of the screen as performing a kind of "measurement" on the electron. It's not a conscious measurement, but physically, the screen atoms are sensitive to the electron's arrival.
Environment Gets "Informed" (Entanglement):
If the electron is in state ∣A⟩ (arriving), the interaction changes the state of the environment from ∣E0⟩ to ∣E1⟩. This change could be excitation of atoms, ionization, momentum transfer, etc. Importantly, ∣E1⟩ is a different physical state of the environment compared to ∣E0⟩. The environment has now become "aware" that an electron in state ∣A⟩ interacted with it.
If the electron is in state ∣N⟩ (not arriving), we assumed for simplicity that the environment remains in ∣E0⟩. The environment stays "unaffected" or in its initial state.
Correlation is Created (Entanglement): Because of this state-dependent interaction, the system and environment become entangled. The final total state (1/√2)(∣A,E1⟩+∣N,E0⟩) shows that the electron's state is now correlated with the environment's state. If you know the environment is in state ∣E1⟩, you know the electron must have been in state ∣A⟩. If the environment is in ∣E0⟩, the electron must have been in state ∣N⟩.
After Interaction (Decoherence and Loss of Observable Superposition)
"Which Path" Information Leaks to the Environment: The crucial point is that the environment now contains "which path" information about the electron. The environment's state (∣E1⟩ vs. ∣E0⟩) effectively tells you whether the electron was in the "arriving" or "not arriving" path.
Observable Superposition for the Electron is Lost (Relative to Us Ignoring the Environment): When we perform the partial trace over the environment, we are mathematically averaging over or ignoring the environment's state. From the perspective of an observer who only looks at the electron and does not measure the environment, the electron's state appears to have changed from a pure superposition to a mixed state.
No More Interference (in the Reduced System): The off-diagonal elements in the reduced density matrix, which represented quantum coherence and the potential for interference, vanish. If we try to perform interference experiments on the electron alone (without measuring the environment), we will no longer observe the interference patterns we would have seen with the initial pure superposition.
Classical Mixture Emerges: The reduced density matrix
ρS=(½)(∣A⟩⟨A∣+∣N⟩⟨N∣)=(½)I
represents a classical statistical mixture. It's as if the electron is either in state ∣A⟩ with 50% probability or in state ∣N⟩ with 50% probability, but not in a quantum superposition of both for any observation we make that only involves the electron.
Analogy: "Quantum Spies"
Think of it like this:
Initial Superposition: The electron is like a secret agent who is in a superposition of two possible missions (state ∣A⟩ and ∣N⟩). As long as no one "observes" the agent's mission, it's truly in both possibilities simultaneously (quantum coherence).
The Screen as "Quantum Spies": The screen atoms are like "quantum spies" that are designed to interact differently depending on which mission the agent is on.
Entanglement = Spies Learn the Mission: If the agent is on mission ∣A⟩, the spies get "activated" and their state changes (to ∣E1⟩). If the agent is on mission ∣N⟩, the spies remain in their "dormant" state (∣E0⟩). Now, the state of the spies (environment) is correlated with the agent's mission.
Decoherence = We Ignore the Spies: If we, as observers, only look at the agent and completely ignore the "spy network" (the environment), the agent now appears to be in a mixed state. It's as if the agent has randomly chosen one mission or the other. We've lost the ability to observe the quantum interference that was present in the initial superposition, because the "which mission" information has leaked out into the environment ("spy network"), which we are ignoring.
In summary, the superposition is "lost" not in the sense that it disappears from the universe, but in the sense that the quantum coherence, and the associated interference effects, become unobservable when we focus solely on the system (the electron) and ignore the now-entangled environment (the screen). The "quantumness" of the superposition effectively bleeds out into the vast, uncontrolled degrees of freedom of the environment.
Nigel:
In this example of exactly one electron we could not measure an interference effect which, surely, would only show up in a pattern of repeated experiments? Does that suggest your response should be slightly nuanced or am I misunderstanding something?
Gemini:
You are absolutely right to point that out! My explanation, as it stands, is indeed slightly nuanced and needs further clarification regarding the role of single versus repeated experiments and how interference manifests.
Let me refine and nuance my response to address this correctly:
The Nuance: Interference and Repeated Experiments
Interference is inherently a statistical phenomenon, observed in patterns that build up over repeated experiments (or in ensembles). You are correct that a single electron, in isolation, will not produce an interference pattern. Interference patterns emerge from the collective behavior of many identically prepared quantum systems or from repeated measurements on systems prepared in the same initial state.
However, the quantum state (including superposition and coherence) exists for each individual electron. Even for a single electron in a superposition, say (1/√2)(∣A⟩+∣N⟩), that electron is in a superposition state. The density matrix:
ρS(0)=(½)(matrix 1 1 1 1)
| 1/2 | 1/2 |
| 1/2 | 1/2 |
describes the state of this single electron. The off-diagonal elements indicating coherence are properties of this single electron's quantum state.
Decoherence acts on the quantum state of each individual electron. When we talk about decoherence due to the screen interaction, we are saying that for each electron that interacts with the screen in this way, its quantum state evolves from a coherent superposition towards a mixed state.
The loss of interference we discuss with decoherence manifests in the statistical pattern that would emerge over many repeated experiments. If we were to perform an experiment designed to reveal interference between the ∣A⟩ and ∣N⟩ paths (even in our simplified model), we would need to repeat the experiment many times with electrons prepared in the initial superposition.
Clarifying the Role of Repeated Experiments and Ensembles
Let's imagine we could design an experiment to test for interference between the "Arriving" (∣A⟩) and "Not Arriving" (∣N⟩) states. Think of it analogously to a double-slit experiment, but in our abstract "arrival state" space.
Without Decoherence (Ideal Scenario): If we could prepare many electrons in the initial superposition state ∣ψS (0)⟩=(1/√2) (∣A⟩+∣N⟩) and then perform a measurement designed to detect interference between ∣A⟩ and ∣N⟩, we would observe an interference pattern over many repeated trials. This pattern would be a consequence of the off-diagonal elements in ρS(0). The probabilities of different outcomes in our interference-sensitive measurement would be influenced by the coherence in the initial superposition.
With Decoherence (Screen Interaction): Now, let's introduce the screen interaction that causes decoherence. After each electron interacts with the screen and we trace out the environment, the state of each electron is described by the decohered density matrix
ρS =(1/2)(matrix 1 0 0 1):
| 1/2 | 0 |
| 0 | 1/2 |
If we now perform the same interference-sensitive measurement as in step 1, but now on electrons that have undergone decoherence, we would not observe the same interference pattern. The pattern would be washed out or disappear. This is because the off-diagonal elements, responsible for interference, are gone in ρS.
Analogy with Classical Coin Flips vs. Quantum Superposition
Think about flipping a coin many times:
Classical Mixture (like Decohered State): If we have a box of coins where half are pre-set to heads and half to tails, and we randomly pick a coin and flip it (without knowing beforehand if it's a "heads-set" or "tails-set" coin), and repeat this many times, we'll get roughly 50% heads and 50% tails. This is analogous to a mixed state. There's no interference, just statistical probabilities.
Quantum Superposition (like Coherent State): Imagine something more quantum. We prepare each electron in a true superposition state. Now, in our hypothetical interference experiment, we observe a pattern over many trials that is not just 50/50 for some measurement outcomes. It deviates from classical probabilities due to quantum interference. This is analogous to the initial coherent superposition.
Key Nuance Summary:
Decoherence affects the quantum state of each individual system, shifting it from a coherent superposition towards a mixed state.
Interference is observed in patterns that emerge from repeated experiments (or ensembles) because these patterns reveal the underlying probability distributions dictated by the quantum state.
Decoherence leads to a change in these statistical patterns. Interference patterns that would be present for coherent states are lost or diminished when decoherence creates mixed states.
Therefore, my initial response wasn't wrong in saying superposition is "lost," but it lacked the nuance that this loss is observed in the statistical patterns of repeated experiments, and that decoherence is a process acting on the quantum state of each individual system within such an ensemble. The density matrix formalism precisely captures this shift from a state capable of producing interference (coherent, pure) to one that is not (mixed, decohered) within an ensemble description, while being relevant for the quantum state of each individual system as well.
Nigel:
Thanks, but I'm still a little puzzled. If we kept track of all the environmental changes (or no change) - we might suppose the detector or screen was just a single atom - what changes? I'm guessing you're going to say that the electron and screen would together remain in a combined superposition (as they, in fact, do anyway).
I suppose my confusion is that keeping track of is not well-defined here. We don't want to add some third element, some macroscopic observer looking at both the electron and the screen and therefore themselves entering a superposition like an endless chain of Wigner's Friends!
I guess the core of my question is: what does it mean in this example, in physical terms, to disregard the possible changes in the environment? But I'm confused, so you are invited to straighten me out!
Gemini:
You've correctly identified a potential for confusion and are asking for a deeper physical interpretation of "disregarding the environment." Let's unpack this carefully:
1. System + Environment Remain in Superposition (Jointly)
You are absolutely right. If we consider the entire system – electron and screen (even if it's just a single atom) – they remain in a joint, entangled superposition after the interaction. In our example, this is the state:
∣Ψ(t)⟩=(1/√2)(∣A,E1⟩+∣N,E0⟩)
The total wavefunction is still a coherent superposition. No quantum superposition has been fundamentally destroyed in the universe. If we could, in principle, perform measurements on the combined system (electron and screen atom), we could still see interference effects that reflect this joint superposition.
2. "Keeping Track" = In Principle Measurable Correlation
"Keeping track" of the environmental changes means, in principle, having the ability to distinguish between the environmental states ∣E0⟩ and ∣E1⟩ and to correlate these environmental states with the system states ∣A⟩ and ∣N⟩.
If we could measure the environment perfectly: We could perform a measurement on the screen atom to determine if it's in state ∣E0⟩ or ∣E1⟩. If we found it in ∣E1⟩, we would know with certainty that the electron is in state ∣A⟩. If we found it in ∣E0⟩, we'd know the electron is in state ∣N⟩. In this ideal scenario, we could restore the "pure state" description, but it would be a correlated pure state of the combined system.
3. "Disregarding" = Choosing Not to Measure/Consider the Environment, Leading to Partial Trace
"Disregarding" the environment means we are consciously choosing not to perform measurements on the screen atom and are only interested in describing the state of the electron itself. This is a very common situation in physics. We are often interested in a particular system (like an electron, a molecule, a qubit) and its properties, without needing to track every single detail of its surroundings.
Mathematical Tool: Partial Trace: The mathematical operation of the partial trace (TrE) is the precise tool to describe this situation. It's designed to give us the quantum state of a subsystem when we are ignoring or averaging over the degrees of freedom of the rest of the system (the environment).
4. Physical Interpretation of Partial Trace: Loss of Local Information and Emergence of Mixed State
When we perform the partial trace, we are essentially:
Averaging over possible environmental states: We are considering all possible states of the environment that are consistent with our lack of measurement on it.
We don't know if the environment is in ∣E0⟩ or ∣E1⟩ (or some more complex state in a more realistic scenario). So we are averaging over these possibilities.
Losing the Off-Diagonal Coherences for the System Alone: The partial trace operation mathematically eliminates the off-diagonal terms in the system's reduced density matrix that were responsible for quantum coherence and interference relative to measurements on the system alone.
Analogy to Classical Probability and Ignorance
Think of classical probability. If you have two coins, and they are entangled in the sense that if coin A is heads, coin B is tails, and vice versa (perfect anti-correlation). The joint state is a pure, correlated state.
If you know the outcome of coin A, you know the outcome of coin B.
However, if you only look at coin A and ignore coin B, coin A will appear to be in a mixed state of 50% heads and 50% tails. You've lost the perfect correlation information by disregarding coin B. The "mixedness" reflects your ignorance about the state of coin B and the correlations.
Decoherence is analogous, but in the quantum realm. The electron becomes entangled with the environment. When we "ignore" the environment, the electron appears to be in a mixed state. The "mixedness" of the reduced density matrix for the electron reflects our ignorance of the environment's state and the electron-environment correlations.
5. No Wigner's Friend Paradox
The process of taking the partial trace and "disregarding the environment" does not lead to an infinite regress of Wigner's Friends. It's a well-defined mathematical operation that provides a consistent description of a subsystem of an entangled system.
Objectivity (but relative): The reduced density matrix ρS is an objective description of the state of the electron given that we are not measuring or considering the environment. It's not dependent on a "macroscopic observer" or consciousness. It's about the physical situation of entanglement and the choice of which degrees of freedom we are focusing on for our description.
Consistent Quantum Mechanics: Partial trace is a standard operation within quantum mechanics. It's not introducing any new interpretations or paradoxes. It's a tool to describe subsystems within the standard quantum formalism.
6. Physical Meaning of Disregarding – Information Leakage
Physically, "disregarding the environmental changes" means:
Information has leaked from the system to the environment: The interaction has caused "which path" information about the electron (was it ∣A⟩ or ∣N⟩?) to become encoded in the state of the environment (∣E1⟩ vs. ∣E0⟩).
We are choosing not to access this information: By tracing out the environment, we are effectively erasing our access to this "which path" information. For any measurements we perform solely on the electron, the outcomes will be consistent with a mixed state (no interference).
Classicality Emerges (Apparent): For the system alone, classical probabilities emerge. The quantum superposition, from the viewpoint of the system alone, has effectively "decohered" into a classical mixture.
Decoherence is not about destroying superposition in the universe. It's about how entanglement with an environment and our choice to focus on a subsystem leads to a relative loss of coherence for that subsystem. The "disregarding" of the environment, mathematically implemented by the partial trace, reflects our limited perspective and the leakage of information into unobserved degrees of freedom, resulting in the system appearing to behave more classically.
Your confusion touches on the core interpretation of decoherence. It's not about wavefunction collapse in the measurement sense, but rather about the practical disappearance of quantum interference for a system when it becomes entangled with a large, complex, and typically unmeasured environment.
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