# States of entangled particles after no/partial/full measurement

I'm getting contradictory information from the internet concerning entangled particles, measurement, and state knowledge that I'm hoping can be cleared up with a simple setup.

Say we have a source of two entangled particles that emits the state $$|\psi\rangle = \frac{1}{\sqrt{2}}(|H_a, H_b\rangle +|V_a, V_b\rangle)$$ where $$H$$ and $$V$$ are the complete orthonormal basis states for a particular observable (e.g. polarization for photons). Also, particles $$a$$ and $$b$$ are spatially separated and are sent to Alice and Bob, respectively.

In the first scenario, Alice does not measure her particle's state. What state is Bob's particle in, also as of yet unmeasured? $$\rho_b = Tr_a[|\psi\rangle \langle \psi|] = \frac{1}{2}(|H_b\rangle \langle H_b| + |V_b\rangle \langle V_b|)$$ which is a maximally mixed state. So far so good.

Now a second scenario, where instead of not measuring her particle's state she does a state resolving measurement that will either result in $$H$$ or $$V$$ (secondary question: if her measurement is not state resolving is this the same as the first scenario?). For argument's sake let's say Alice's measurement returns $$H$$. Going back to the definition of $$|\psi\rangle$$, it seems to be Bob's particle must now, after Alice's measurement, be in state $$|\psi\rangle_b = | H\rangle$$, that is, a pure state, even if Bob does no measurement.

This is where there may be some confusion. As Bob has not done a measurement, I can see how the state of his particle according to him is the mixed state, but I don't see why his ignorance needs to correspond to reality. He has only partial information about the state of the particles, where Alice has full information. But according to some material I find the "true" state of particle $$b$$ is the mixed state after Alice's measurement, since Bob has not done a measurement.

This is more than just a problem of interpretation. Imagine a scenario where instead of one particle being sent to Alice and Bob, many entangled particle pairs are sent from this entangled source. For argument's sake let's say Alice measures all of them and they all happen to return $$V$$. If Bob now does measurements after Alice on his particles it seems to me all must return $$V$$. If they were truly in mixed states after Alice's measurement it is unlikely they will all return $$V$$, although possible. If they were all in pure state $$V$$ then all of Bob's measurements will return $$V$$. The "interpretation" where they will all definitely return $$V$$ is a better description of the physical phenomenon.

So, which is it? After Alice's measurements, are Bob's particles before he does any measurements in a mixed or pure state in "reality"? I may get criticized saying reality, and some might say all that matters is what the observers knows (which would lead to the mixed state answer), but as the above paragraph tries to illustrate, the stance taken lead to physically measurable differences.

• Flip a coin. Observe the outcome. Don't tell me what it is. You say the coin is heads up. I say there's a 50/50 chance it's heads up. Which corresponds to reality? [Your answer to this question will help to clarify what you mean by "reality", which in turn will help clarify what you're asking.] Commented May 15 at 15:30

The answer to this question involves what interpretation of quantum theory you're using so I'll describe the options.

The Copenhagen and statistical interpretations say that quantum theory doesn't describe how the world works but you should use quantum equations of motion and the Born rule anyway. You're not supposed to ask what's happening in reality. I think this makes it impossible to set up experiments properly since there is no standard of what counts as setting particular initial conditions if you have no account of what's happening in the experiment. Those interpretations would say you should stop asking your question.

Spontaneous collapse theories and pilot wave theories are other options:

https://arxiv.org/abs/2310.14969

https://arxiv.org/abs/1906.10761

They both produce experimental predictions that disagree with quantum theory and I haven't seen a clear account of what's happening in Bell correlation experiments from either of them.

The final option is to take quantum equations of motion seriously as a description of what's happening in reality and work out what they say about this experiment. In the Heisenberg picture each system is described by observables that evolve according to the relevant equation of motion. Each observable corresponds to some measurable quantity and the eigenvalues of the observables correspond to possible measurement outcomes. There is a relative state $$\rho$$ that tells you about what observable was measured last and what value you saw. The expectation values of the observable $$\hat{X}$$ are given by $$tr(\rho\hat{X})$$. When information about an observable is copied out of a system that suppresses interference between different values of that observable: this process is called decoherence. As a result reality as described by unmodified quantum theory looks a bit like a collection of parallel universes on the scales of space and time of everyday life and so it is called the many worlds interpretation:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

In entanglement experiments, the observables of each system carry quantum information about the system it is entangled with, but you can't get it by measuring that system alone so it is called locally inaccessible information (LIA). The observables of the entangled systems are unsharp so they exist in multiple versions but have LIA that can be carried in decoherent systems because it can't be copied out by interactions. The LIA produces correlations when results of measurements on the entangled systems are compared:

https://arxiv.org/abs/quant-ph/9906007

https://arxiv.org/abs/1109.6223

So how do you describe an entangled experiment when you're half way through the measurement on both halves? The observables are part of the description and the other part involves making the relevant adjustment to the relative state, which changes your relative state for the system you measured as described in this paper about relative states in the Heisenberg picture:

https://arxiv.org/abs/2008.02328

• I appreciate the reply with the many good references. While interpretations are involved, when it comes to a result, they must all agree. "Shut up and calculate" : ). In scenarios #2, where all $V$s are measured by Alice, if you agree that Bob will also measure all $V$s then I DO NOT see how one can say Bob's particles, before his measurement, are NOT pure states. If that is NOT the case, what are the states of the particles that will all "happen" to point to $V$ once measured? Commented May 16 at 14:24

Reality is in the eye of the beholder, and is something of a matter of which quantum interpretation you consult. But I can add a few useful comments.

A) You have fallen into a historical trap, going back to the famous EPR paper of 1935. Yes, it looks as if a measurement by Alice casts Bob into a certain outcome. But this misleading conclusion was demonstrated to disagree with the predictions of QM by Bell in 1964. Having Alice and Bob measure on the same basis (angle) is a special case. The problem becomes more clear when considering measurements on different bases (such as 120 degrees apart). I won’t try to explain the proof here, as it is outside the scope of an answer. You should take the time to check it out if you aren’t already familiar.

But the upshot is… no, a measurement by Alice does not determine Bob’s outcome although it provides information useful for a statistical prediction at any angle chosen for measurement. It just happens to be a 100% certain prediction at the same basis, but that idea falls apart otherwise.

B) The order of measurement is irrelevant, unless you first assume there is action at a distance. (Of course, that is possible.) There are numerous experiments that demonstrate this, again outside the scope of the answer. One such demonstrated that if that were the case, the effect would be at least 10,000 times the speed of light.

• Thanks for the reply. A) Theory can predict in original basis just fine. Question is simple, what is the state of particle $b$ after Alice's measurement if she measures $H$? What are the states of the Bob's particles if Alice measures all $V$s in scenario #2? Do you agree these particles have states? Maybe Bob says mixture but complete information yields a different result. You say the prediction will be 100% correct (pure state prediction I assume). Being 100% correct works for me! B) Standard QM says wavefunction collapse could be "instantaneous", so for this phenomenon FTL is fine. Commented May 16 at 14:06
• @user401228 I would say that after Alice finds a as H>, b acts as if it is in a V> state. Does that mean Alice changed b by her measurement? Existing experiments would question that. an and b need not ever co-exist to be entangled, and they can be entangled after they are observed (delayed choice entanglement swapping). If there is such a thing as collapse, spatiotemporal entanglement makes the word “instantaneous” a poor description. So I am not sure there is a single satisfactory answer. See for example this paper, and try to determine when collapse occurs: arxiv.org/abs/1203.4834 Commented May 16 at 15:29