# What is a non-orthogonal measurement?

• JK423
In summary, a non-orthogonal measurement is a measurement that projects a state onto a set of non-orthogonal state.
JK423
Gold Member
As the title sais... Can someone explain to me what is a non-orthogonal measurement?
I searched the web by i can't find any definition

Hi JK423!

What is the context?

JK423 said:
what is a non-orthogonal measurement?

Do you have a specific experiment in mind?

Orthogonal means perpendicular and uncoupled, like x and y axes in Cartesian space. But you can also use other axes, for example you could obtain x' and y' by rotating everything a few degrees clockwise.

Say you measure the length of the projection along the x-axis of some typical vector v, and record the result. Now, if you were to measure next the y-axis projection of that vector then you would get another result completely independent from your first one, but if instead you had next measured the x' axis projection then you would expect to get a result very closely (though not quite 100%) interrelated with your first one.

Measurements in QM are often mathematically analogous to projecting a vector onto an axis, the angle between axes corresponding with degrees of correlation.

A good experiment right off hand would be the Aharonov-Bohm effect.

Thanks for the answers so far..
I'll explain to you my problem a little bit more analytically.
For example, i know that an orthogonal measurement projects a state |Ψ> onto an eigenstate of the observable we are measuring.
Now I am not sure what a non-orthogonal measurement does...
Of what I've understood (from the notes of Preskill I am studying) when we perform a non orthogonal measurement, |Ψ> is projected onto a state |Ψa> from a set of non-orthogonal state {|Ψa>}. [Like in an orthogonal measurement, |Ψ> is projected onto an (eigen)state from a set of orthogonal states (the rest of eigenstates)]
And there i got a problem. How can the set {|Ψa>} be of non-orthogonal states?
Since we are measuring an observable, then |Ψ> should be projected onto an eigenstate of that observable! So i conclude that |Ψa> is an eigenstate! But how can {|Ψa>} be of non-orthogonal eigenstates??

im confused..

Hi JK423!

Does this help … http://en.wikipedia.org/wiki/POVM" ?

Last edited by a moderator:
Thanks tiny-tim, but i had looked at it but didnt have the info i needed!
My professor helped me and figured it out..! :)

Guys: Would it be proper to say that 'non-orthogonal' is synonymous with 'isotropic', and in any context?

## 1. What is the definition of a non-orthogonal measurement?

A non-orthogonal measurement refers to a type of measurement in which the measurement outcomes are not mutually exclusive or independent. This means that the different outcomes of the measurement are not orthogonal or perpendicular to each other, and there is a possibility of obtaining multiple outcomes simultaneously.

## 2. How does a non-orthogonal measurement differ from an orthogonal measurement?

In an orthogonal measurement, the different outcomes are mutually exclusive and independent, meaning that only one outcome can occur at a time. However, in a non-orthogonal measurement, multiple outcomes can occur simultaneously, and they are not independent of each other.

## 3. What are some examples of non-orthogonal measurements?

Some examples of non-orthogonal measurements include measuring the position and momentum of a particle simultaneously, measuring the polarization of a photon in different directions, and measuring the spin of an electron in different axes.

## 4. Why are non-orthogonal measurements important in quantum mechanics?

Non-orthogonal measurements are important in quantum mechanics because they allow us to obtain information about different properties of a quantum system simultaneously. This is crucial in understanding the behavior of quantum systems and making accurate predictions.

## 5. What are some challenges associated with non-orthogonal measurements?

One of the main challenges of non-orthogonal measurements is that they can introduce uncertainty and imprecision in the measurement outcomes. This is due to the fact that the outcomes are not mutually exclusive, and there is a possibility of obtaining multiple outcomes simultaneously. Additionally, non-orthogonal measurements can be more complex to analyze and interpret compared to orthogonal measurements.

• Quantum Physics
Replies
2
Views
675
• Quantum Physics
Replies
4
Views
1K
• Linear and Abstract Algebra
Replies
9
Views
498
• Quantum Physics
Replies
9
Views
408
• Quantum Physics
Replies
13
Views
1K
• Quantum Physics
Replies
11
Views
1K
• Quantum Physics
Replies
22
Views
1K
• Quantum Physics
Replies
7
Views
1K
• Quantum Physics
Replies
4
Views
745
• Quantum Physics
Replies
12
Views
698