Is Entanglement Commutative in Quantum Computing?

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  • Thread starter Thread starter Kenneth Adam Miller
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Discussion Overview

The discussion revolves around the nature of entanglement in quantum computing, specifically addressing whether entanglement is commutative when involving multiple qubits and entangled photons. Participants explore the implications of entanglement during interactions between photons and qubits, and how these interactions affect the overall entanglement within the system.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions if striking a supercooled qubit with entangled light results in the qubit being suspended in entanglement, seeking to understand the commutativity of entanglement.
  • Another participant explains that the total amount of entanglement between a pair of entangled photons and a qubit remains constant, despite potential changes in the entanglement between individual components due to interactions.
  • A follow-up comment suggests that the entanglement structure changes, with one participant proposing that the entanglement between the photon and the qubit creates a subspace that is influenced by their interaction metrics.
  • A participant introduces the concept of monogamy of entanglement, stating that the entanglement relationships must adhere to specific inequalities, indicating a trade-off in entanglement distribution among the components.

Areas of Agreement / Disagreement

Participants appear to agree on the constancy of total entanglement in the system, but there is ongoing exploration of how individual entanglement relationships change, indicating that the discussion remains unresolved regarding the implications of these interactions.

Contextual Notes

Participants reference concepts such as the monogamy of entanglement and the interaction metrics, which may require further clarification or definition for a complete understanding. The discussion does not resolve the implications of entanglement commutativity in all scenarios.

Kenneth Adam Miller
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I've seen diagrams of quantum computer components at a high level that discusses multiplexing laser reflections over many qubits, and I have to believe that entanglement as a hardware operation has to be scaled to the many qubits by means of some operation that is applied to each of them simultaneously. That being said, if I remember correctly, there were examples of the slit experiment at a microscopic level to give readers at a introductory level an impression of what the hardware was doing. But I don't think that that was strictly what was actually at that level. Perhaps I working with a very vague understanding, but what I want to know is, if you have light that is entangled, and you strike a super cooled qubit of any kind, does that mean that that qubit is also suspended in entanglement? In other words, is entanglement commutative?
 
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If you have a pair of entangled photons A and B, and one of those photons B interacts with a qubit C, the amount of entanglement between A and the joint system BC remains the same (assuming no additional environmental interaction).

The amount of entanglement between A and B may change due to B interacting with C, but if A has no further interaction with B or C, the total entanglement between A and BC must remain constant.
 
Ok, so it's as though A now shares a total entanglement with all three, but BC sort of share a subspace determined by the metrics of their interaction, is that correct?
 
If I understand you correctly, yes.

A shares entanglement with BC, and the amount of entanglement between A and BC is the same before and after B and C interact.
What's different is how much entanglement A shares with just B, or with just C.

There's a useful concept called the monogamy of entanglement that says the amount of entanglement B shares with AC cannot be less than the sum of the entanglement between A and B, and between B and C.
E(A:BC)\geq E(A:B) + E(A:C)
So, as A becomes less entangled with B, A must be more entangled with C, (or at least, the maximum possible entanglement between A and C increases.
 
Fascinating. Thank you.
 

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