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The concept of measurement in quantum mechanics is usually discussed without specifying how much time such measurement might take. In principle, one can imagine that the time needed to perform measurement of some properties of a physical system can be longer when a particular system is more complex in a certain sense. For example, measurement time may depend on the "extent" of entanglement. This might mean that measurement time may increase exponentially with the number of qubits as their state becomes more entangled. Is there anything in quantum mechanics theory that would prevent that from happening? If not, how can a quantum algorithm designer be sure that measurement will not be the primary limiting factor influencing the time-complexity of an algorithm? Thank you.

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  • $\begingroup$ See comments to this post quantumcomputing.stackexchange.com/questions/18001/…. Number of qubits to be measured is minimized as possible just to avoid effects you mentioned. $\endgroup$ Apr 22 at 15:22
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    $\begingroup$ But why would measurement take exponentially long, depending on the amount of entanglement? Why would measurement know about entanglement? $\endgroup$ Apr 22 at 16:03
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    $\begingroup$ Agreeing with above: more measurements will take longer. But if the actual measurement took more time depending on the state of the system (and therefore depending on the measurement result), then you would already "know the result" just because your measurement is taking longer $\endgroup$ Apr 22 at 17:45

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Initially I think the idea that:

measurement time may depend on the "extent" of entanglement...

is ruled out by the no-signaling theorem. For example, let's have Alice on Earth and Bob on Mars share an entangled Bell pair. If measurement "knows about" the entanglement present in the Bell pair, and takes longer if the qubits are entangled than if they are not, then Alice on Earth could send a signal to Bob on Mars by deciding whether or not to measure her qubit (and, destroy the entanglement therebetween). If she decides not to pre-measure her qubit then when Bob measures his qubit, by hypothesis it would take longer for Bob to measure his qubit than it would have if Alice had decided to pre-measure hers, and Bob could get a signal from Alice based on her pre-measurement decision, in contradiction of the no-signaling theorem.

Alternatively perhaps the OP is asking about wall-clock time. Indeed, measurement time is critical for any real-world quantum processor, and does vary based on processor-type and modality, but it should scale linearly with the number of qubits (and not with the amount of entangled).

Lastly perhaps as @Martin suggests the question is about quantum state tomography - that is, learning all about the state $|\psi\rangle$ and all of its coefficients. I do not know much about tomography, but the OP might want to review various topics around shadow tomography for example.

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  • $\begingroup$ Thank you for your comments. I am still quite confused by the issue. If I understand the comments correctly, it seems that the main argument about possible dependence of the measurement time on degree of entanglement is that this would somehow imply that the measurement system knows about entanglement via knowledge about measurement time. However, time is not a physical observable in quantum mechanics. In this sense it can't be "known". Sufficient measurement time is a fixed property of any given measurement apparatus. The question is: how much time is sufficient in general? $\endgroup$ Apr 24 at 14:40
  • $\begingroup$ The comment that "In this sense it can't be 'known'" doesn't make any sense to me. I sure know what time it is. Alice can look at her clock and see how long it takes her to measure her qubits; Bob can do the same. If Bob's measurement takes a long time depending on whether Alice had measured her qubits then Bob would know whether Alice had measured her qubits, and Alice could be sending a superluminal signal. Otherwise I don't understand your question. Good luck! $\endgroup$ Apr 24 at 15:25

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