- #1

mbond

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But how do I do that? I would like to use Born's rule, but what projector would project on ##|x\rangle##?

I appreciate your help.

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- Thread starter mbond
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In summary, the conversation discusses the possibility of using Born's rule to project a measurement onto a specific state with a high probability, particularly in the context of quantum computing algorithms such as the quantum Fourier transform. However, it is noted that a single measurement may not be sufficient to determine the wave function, and that multiple measurements may be needed to identify the desired state.

- #1

mbond

- 41

- 7

But how do I do that? I would like to use Born's rule, but what projector would project on ##|x\rangle##?

I appreciate your help.

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- #2

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$$\hat{J}=\sum_j j |j \rangle \langle j|,$$

is

$$P(x)=|\langle x|\psi \rangle|^2=|a_x|^2.$$

The projector to the corresponding eigenstate is

$$\hat{P}_x=|x \rangle \langle x|.$$

I hope I understood the question right. It's a bit vague ;-).

- #3

mbond

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I am looking for a projector that would return ##x## because it has the highest amplitude ##|a_x|^2##.

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Then I don't understand the question.

- #5

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There is no such projector. The only eigenvalues of any projector are 0 and 1, so a projector cannot return the number ##x##.mbond said:I am looking for a projector that would return ##x## because it has the highest amplitude ##|a_x|^2##.

But you can measure the observable ##J## defined by @vanhees71. If you do that, it is almost certain that the result of measurement will be ##x##.

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$$\hat{P}_x |\psi \rangle =|x \rangle \langle x|\psi \rangle.$$

- #7

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An operator is applied to a Hilbert-space vector, giving another vector, not a number. An operator has eigenvalues (numbers) and eigenvectors. A projection operator ##\hat{P}## has eigenvalues 0 and 1, because it obeys ##\hat{P}^2=\hat{P}##. From this it also follows that there's one eigenvector (moduloa an arbitrary phase factor) with eigenvalue 1, i.e., ##\hat{P}=|\psi \rangle \langle \psi|## with some normalized ##|\psi \rangle## (which is the one eigenvector with eigenvalue 1).

- #8

mbond

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I have a wave function resulting from a quantum Fourier transform. This wave function is a superposition with one of the amplitudes very high (because it corresponds to the period). How do I recover the period?

- #9

Morbert

Gold Member

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Measure ##\sum_j a_j|j\rangle\langle j|## a large number of times and see which outcome occurs the most. That is likely your ##a_x##.mbond said:

But how do I do that? I would like to use Born's rule, but what projector would project on ##|x\rangle##?

I appreciate your help.

Or if you want to keep it theoretical, ##p(a_j) = |\langle j|\psi\rangle|^2## so compute for each ##j## and see which ##p(a_j)## is largest.

- #10

mbond

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A wave function can't be measured more than once; it's collapsed after the 1st measurement.

- #11

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In general, a single measurement cannot determine the wave function, unless you you have some knowledge about the wave function before measurement.mbond said:A wave function can't be measured more than once; it's collapsed after the 1st measurement.

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This period refers to periodicity either in time or in space. If it is in time, then your wave function is approximately an energy eigenstate. If it is in space, then your wave function is approximately a momentum eigenstate. Hence you need to measure either the energy or the momentum, the value that you obtain is probably the value that you need. There is a probability that you will get a wrong value, but this probability is small. Since your wave function is not an exact eigenstate, there is no way to get the needed value with 100% certainty.mbond said:

I have a wave function resulting from a quantum Fourier transform. This wave function is a superposition with one of the amplitudes very high (because it corresponds to the period). How do I recover the period?

Last edited:

- #13

mbond

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- #14

DrClaude

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Slightly off-topic suggestion: when discussing QM, stop using ##x## for anything else than position.

- #15

martinbn

Science Advisor

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- #16

Nugatory

Mentor

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.mbond said:

That does not exist. The best we can do, and what is done in quantum computing, is to measure ##x##. It is probable, and approaches certainly as one of the ##a_i## approaches unity and the others approach zero, that the result will be the eigenvalue corresponding to that ##a_i##.

- #17

Morbert

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Sorry, I mean measure the observable ##\sum_j \lambda_j |j\rangle\langle j|## a large number of times, each on a different member of your ensemble. If you know the preparation has the form ##\sum_{j} a_j|j\rangle## with one amplitude approximately 1 and the others 0, then measuring ##\sum_j \lambda_j |j\rangle\langle j|## a large number of times will identify it.mbond said:A wave function can't be measured more than once; it's collapsed after the 1st measurement.

- #18

mbond

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##\langle\psi|\hat{O}|\psi\rangle=r|a_r|^2+\displaystyle\sum_{j\neq r}j|a_j|^2\approx r##

and the period I am looking for in the quantum Fourier transform ##|\psi\rangle## is ##P\approx\displaystyle \frac{2^n}{r}##.

Sorry for not being clear, and many thanks for the help.

- #19

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You cannot figure out the state by just measuring the observable. All you get are the probabilities, i.e., the ##|a_j|^2##. You get no information about the phases, i.e., the complex numbers ##a_j##.Morbert said:Sorry, I mean measure the observable ##\sum_j \lambda_j |j\rangle\langle j|## a large number of times, each on a different member of your ensemble. If you know the preparation has the form ##\sum_{j} a_j|j\rangle## with one amplitude approximately 1 and the others 0, then measuring ##\sum_j \lambda_j |j\rangle\langle j|## a large number of times will identify it.

There's a thorough discussion about, how to empirically find out the quantum state ("state tomography") in Ballentine, Quantum Mechanics.

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You cannot get the average value with a single measurement.mbond said:I make the measurement

##\langle\psi|\hat{O}|\psi\rangle##

- #21

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Actually, in the quantum computing community it is quite common to use ##x## for other observables or states.DrClaude said:Slightly off-topic suggestion: when discussing QM, stop using ##x## for anything else than position.

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But you said that you want to obtain the result with a single measurement. A quantum algorithm is not a single measurement. Moreover, most quantum algorithms still cannot be realized with actually existing quantum computer hardware.mbond said:In quantum computing, there are algorithms such as the quantum Fourier transform or Grover's algorithm

- #23

Morbert

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I interpreted his question to be asking to identify which ##j## had the largest amplitude: " If you know the preparation has the form ##\sum_j a_j|j\rangle##with one amplitude approximately 1 and the others 0"vanhees71 said:You cannot figure out the state by just measuring the observable. All you get are the probabilities, i.e., the ##|a_j|^2##. You get no information about the phases, i.e., the complex numbers ##a_j##.

There's a thorough discussion about, how to empirically find out the quantum state ("state tomography") in Ballentine, Quantum Mechanics.

A superposition in quantum mechanics refers to the combination of all possible states that a quantum system can be in simultaneously. For example, an electron in a superposition of spin states is both spin-up and spin-down at the same time until measured.

Measurement in quantum mechanics collapses the superposition into one of the possible eigenstates. The act of measuring forces the system to 'choose' a specific state, which is then observed. This collapse is instantaneous and non-deterministic, meaning the outcome cannot be predicted with certainty but follows probabilistic rules.

Born's rule is a fundamental principle in quantum mechanics that provides the probability of finding a system in a particular state after measurement. It states that the probability is equal to the square of the amplitude of the wave function corresponding to that state.

In practice, Born's rule is applied by taking the wave function of a quantum system, identifying the amplitude associated with the state of interest, and then squaring this amplitude to find the probability of measuring that state. This rule is essential for making predictions about the outcomes of quantum experiments.

Born's rule is crucial because it bridges the gap between the abstract mathematical formalism of quantum mechanics and empirical observations. It allows physicists to make quantitative predictions about the results of measurements, thus connecting theory with experimental data.

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