# Are entanglement correlations truly random?

• I
Gold Member
Suppose we have two truly random sources A and B that generate bits ('0' or '1') synchronously. If we measure the correlation between the respective bits generated, we find a random, ie no, correlation.

Now suppose A and B are two detectors that register polarization-entangled photons passing respective polarization filters. We can define bits as 'detection'='1' and 'no detection'='0'. A and B individually yield random results. However, there is in almost every case a non-zero correlation, depending on the angle of the filters.

So my question would then be: since the detections of the entangled particles often exhibit a different correlation than truly random sources, are the detections purely random in case of entanglement? (or do they only seem random?)

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Demystifier
Gold Member
Define "truly random"!

Zafa Pi and bhobba
Gold Member
Define "truly random"!
So you are saying that two truly random sources could as well (anti-)correlate then?

In other words: two truly random (binary) sources don't correlate (are independent). I answered my own question.

So, the detections of the OP are not truly random then (and hence not random). They are dependent of each other.

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This section (relevant text bolded) especially so:

A lot of “quantum” woo-woo New Age nonsense invokes entanglement between particles as an excuse, and it’s true that, in principle, everything is entangled with everything else, since every particle now in existence was initially packed into a tiny volume just after the Big Bang. But unless you know all of the ways the state of an electron in your brain was rotated in the intervening 13.7 billion years, you’ll never be able to detect the entanglement between it an an electron in somebody else’s brain,...
Which, to me, seems to be saying that the correlation exhibited between any given particles is not random, but only undetectable. Perhaps in the way strong ciphertext, which also passes pretty much any test for randomness, cannot be decrypted unless you have the algorithm(s) and the key(s).

DrChinese
Gold Member
From the reference: "everything is entangled with everything else..."

Which, to me, seems to be saying that the correlation exhibited between any given particles is not random, but only undetectable. Perhaps in the way strong ciphertext, which also passes pretty much any test for randomness, cannot be decrypted unless you have the algorithm(s) and the key(s).
Everything is not really entangled with everything else. As far as anyone knows anyway, as there is something called entanglement monogamy. If 2 particles are maximally entangled, they cannot also be entangled with a third.

Of course, the quantum world is a good source for a stream of random bits. Paraphrasing Demystifier: how would you know if that stream is "truly" random? There really is no *final* test that would prove that to all. So I assume it to be the case.

bhobba and Simon Phoenix
Gold Member
Of course, the quantum world is a good source for a stream of random bits. Paraphrasing Demystifier: how would you know if that stream is "truly" random? There really is no *final* test that would prove that to all. So I assume it to be the case.
Isn't it true that if the subsequent bits generated by 'truly random' (binary) source A (P=.5) are independent (independent 'throws'), and likewise for source B, that then the bits of A and B don't exhibit a correlation (over large numbers of bits)?

Isn't it true that if we view A and B as random variables that correlate, that A an B may be random, but are not independent?

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Everything is not really entangled with everything else. As far as anyone knows anyway, as there is something called entanglement monogamy. If 2 particles are maximally entangled, they cannot also be entangled with a third.

Of course, the quantum world is a good source for a stream of random bits. Paraphrasing Demystifier: how would you know if that stream is "truly" random? There really is no *final* test that would prove that to all. So I assume it to be the case.
Maybe it's true that two monogamously entangled particles cannot be monogamous with a third, but they might be able to fool around a little as long as they aren't caught? Here half a million rubidium atoms are entangled: https://www.scientificamerican.com/article/quantum-entanglement-creates-new-state-of-matter1/

DrChinese
Gold Member
Maybe it's true that two monogamously entangled particles cannot be monogamous with a third, but they might be able to fool around a little as long as they aren't caught? Here half a million rubidium atoms are entangled: https://www.scientificamerican.com/article/quantum-entanglement-creates-new-state-of-matter1/
You can certainly entangle many particles, as per your reference. But they are not maximally entangled.

And yes, there could be regions of varying sizes within the universe in which there "less-than-maximum" entanglement between member particles. I would guess them as being small, although I guess there is no specific way to check that. If the member particles within the hypothetical region follow some conservation rule (such as total spin = 0) AND the constituent components cannot be identified as to which has which value, there will be entanglement. So essentially the components must all be in a superposition for the specific observable. For them to be entangled, that is.

Simon Phoenix
Define "truly random"!
The Maximal Lyapunov exponent?

Simon Phoenix
Gold Member
You can certainly entangle many particles, as per your reference. But they are not maximally entangled.
I would probably amend this statement to read "but they are not pairwise maximally entangled".

It certainly makes sense to talk about a maximally entangled state of 3 qubits, for example. In such a maximally entangled state it is clear that any 2 qubits out of the 3 are not in a maximally entangled state of 2 qubits.

For ##n## qubits the maximum possible information content of the correlation between them is ##n \rm {ln} 2##, whereas for classical bits this is ##(n-1) \rm {ln} 2##. In terms of correlation strength alone the difference between quantum and classical amounts to just one bit - which ties in with the fact that for ##n## classical bits the maximally correlated state (expressed quantum mechanically) can be 'purified' to the maximally entangled pure quantum state of ##n## qubits by the addition of a single extra qubit.

For qubits, the maximally entangled state is one in which all of the pairwise correlations between the qubits are simultaneously optimized.

Simon Phoenix
Gold Member
Perhaps in the way strong ciphertext, which also passes pretty much any test for randomness, cannot be decrypted unless you have the algorithm(s) and the key(s).
Yes and it's nice that you've brought up crypto here. There is an operational definition of pseudorandomness used in crypto which essentially states that a string* is pseudorandom if it cannot be distinguished from a truly random string in polynomial time with better than a probability that is a negligible function of the length of the string.

I don't know whether quantum computers change this perspective. Theoretically anyway the measurements on a qubit from an entangled pair will be truly random in that the strings produced cannot be distinguished from true randomness by anyone with infinite time and infinite computational resources.

* 'string' here is just a shorthand - really we're talking about distributions of strings - or, asymptotically, sequences of distributions.

Gold Member
Theoretically anyway the measurements on a qubit from an entangled pair will be truly random in that the strings produced cannot be distinguished from true randomness by anyone with infinite time and infinite computational resources.
Consider two practically perfectly fair dice. One dice in New York, the other in London. Now both dice get thrown simultaneously (by two experimentators) repeatedly in a series and each result is registered and each respective throw is compared. After examination it is established that if six is thrown, it is always thrown simultaneously by both dice. Both dice show (perfectly) random and uncorrelated patterns otherwise; only when six faces up, it faces up with both dice simultaneously always.

Now, since each dice individually exhibits a (practically) random pattern of results, would it follow according to you that they are truly random patterns?

Allow me to consider to doubt that, because the correlated sixes could have been correlated two's instead. We would have a different situation and still consider the results on both ends random.

Demystifier
Gold Member
The Maximal Lyapunov exponent?
How would you define Lyapunov exponent for a quantum system? And even for a classical deterministic system, how exactly would you associate maximal Lyapunov exponent with "true randomness"?

Simon Phoenix
Gold Member
Now, since each dice individually exhibits a (practically) random pattern of results, would it follow according to you that they are truly random patterns?
I'm not at all sure I know what your issue is here.

Let's suppose I have two coins attached by some rigid rod. For the sake of argument let's suppose this device is thrown and it can land heads or tails uniformly at random. Now suppose it's been arranged so that I can see only one of the coins and a colleague can only see the other (maybe by some video arrangement).

What's the issue here? The strings observed by me and my colleague are perfectly correlated - but whether we both see a tail or a head is perfectly random. There's one bit of entropy here.

Basically we have two events A and B - but they are not independent events, and in the case of the coins joined by a rigid rod it's exactly like throwing just one coin.

We could make it even simpler - throw a single coin on a glass table. A camera records the coin face from above the table and another camera records the coin face from below the table. The feeds from the respective cameras are fed to two different people. Each person sees a random string - but the strings are perfectly (anti) correlated. Does the fact that each observed string is perfectly correlated mean there's less randomness here? Is this what you're asking?

Is there something here I'm missing?

Zafa Pi
Gold Member
In your examples there is a structure involved: a rod in case one and two camera's in case two. This has to be taken into account in addition to the two random sources, for it is logical that a correlation shows up if it is inherent to the experiment. And that is what I mean: if two sources of random values show a correlation, they are, as you say, not independent. So maybe the issue is whether there are two random sources, or there is only one, or one-and-a-half for that matter. It seems to me the difference between one or two bits. One might say there is a limitation placed on a source if it is not a full bit. For instance, in the case of the rod, if coin A is random, B isn't, and vice-versa. One coin is, in this view, just copying the pattern of the other (and vice-versa).

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Yes and it's nice that you've brought up crypto here. There is an operational definition of pseudorandomness used in crypto which essentially states that a string* is pseudorandom if it cannot be distinguished from a truly random string in polynomial time with better than a probability that is a negligible function of the length of the string.

I don't know whether quantum computers change this perspective. Theoretically anyway the measurements on a qubit from an entangled pair will be truly random in that the strings produced cannot be distinguished from true randomness by anyone with infinite time and infinite computational resources.

* 'string' here is just a shorthand - really we're talking about distributions of strings - or, asymptotically, sequences of distributions.
Thanks, Simon. I imagine randomness is pretty tough to prove. I just use byte distribution, probability of bit change and lack of pattern (string) repetitions as indicators. I wonder if the ciphertext output from say (double/triple/etc?) )block-chained Rijndael on all-zeros plaintext would be distinguishable from "true randomness" by any known tests or measures? Perhaps unrelated, I still wonder how two entangled particles, say a billion lightyears apart (with ~c relative velocities), are not communicating faster than light. I mean, I know we can't (yet, maybe ever) use them to transmit data, but they themselves seem to me to be communicating (in touch) with each other.

Gold Member
Suppose you have two independent random sources of strings of bits (0's and 1's), for instance, take one such string of random 0's and 1's and cut it in two, so you obtain two independent strings. Each random string of 0's and 1's is totally non-correlative with any other such string. Also, if you shift (rotate) one string relatively to another, there still will be no correlation (because of the randomness).

Pairs of strings of bits generated by detectors of entangled particles are not random in that respect: there is one exact alignment of the pair of strings where there is a correlation. Every other alignment is random. It is the existence of this one exception of alignment, a sort of key-in-lock situation, that sets entanglement randomness apart from truly independent randomness.

* I am not entirely sure for I am not able to check with code. I'll try to back this up with code.

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How would you define Lyapunov exponent for a quantum system? And even for a classical deterministic system, how exactly would you associate maximal Lyapunov exponent with "true randomness"?
I was suggesting the association to the maximal Lyapunov exponent through its notion of predictability, entropy. The math level is well above me, but it looks like the foundation might be presented here:

On quantum lyapunov exponents (Majewski & Marciniak)
It was shown that quantum analysis constitutes the proper analytic basis for quantization of Lyapunov exponents in the Heisenberg picture. Differences among various quantizations of Lyapunov exponents are clarified.

and here:

Lyapunov exponent in quantum mechanics. A phase-space approach (Man'ko & Mendes)
Using the symplectic tomography map, both for the probability distributions in classical phase-space and for the Wigner functions of its quantum counterpart, we discuss a notion of Lyapunov exponent for quantum dynamics. Because the marginal distributions, obtained by the tomography map, are always well-defined probabilities, the correspondence between classical and quantum notions is very clear. Then we also obtain the corresponding expressions in Hilbert space. Some examples are worked out. Classical and quantum exponents are seen to coincide for local and non-local time-dependent quadratic potentials.For non-quadratic potentials classical and quantum exponents are different and some insight is obtained on the taming effect of quantum mechanics on classical chaos. A detailed analysis is made for the standard map. Providing an unambiguous extension of the notion of Lyapunov exponent to quantum mechanics, the method that is developed is also computationally efficient in obtaining analytical results for the Lyapunov exponent, both classical and quantum.

Looks like a methodology for application to experimental data is here:

Determining lyapunov exponents from a time series (Wolf, Swift, Swinney, & Vastano)
We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii reaction and Couette-Taylor flow.

Isn't the randomness introduced by the orientation of the detector with respect to the pair being measured?

Simon Phoenix
Gold Member
Pairs of strings of bits generated by detectors of entangled particles are not random in that respect: there is one exact alignment of the pair of strings where there is a correlation. Every other alignment is random. It is the existence of this one exception of alignment, a sort of key-in-lock situation, that sets entanglement randomness apart from truly independent randomness.
I'm sorry, entropy, but I'm really struggling to fully grasp what you're saying here.

Take a pair of (maximally) entangled qubits - and for the sake of argument we'll think of two spin-1/2 particles. We'll assume the usual set-up where independent measurements on particles from the entangled pairs are performed at two remote locations ##A## and ##B##.

Irrespective of the relative detector settings ##A## obtains a binary random string as output from the measurements, so does ##B##.

If both ##A## and ##B## are measuring spin-z, say, then their results are perfectly correlated. If ##A## measures spin-z and ##B## measures spin-x, then there is no correlation between their results. The degree of correlation between the data sets varies continuously from perfect to zero as the relative angle between the detectors is changed from zero to ## \pi /2##.

When the detectors are aligned (both spin-z, for example) then it's like the situation we have with the coins joined by a rigid rod; when the detectors are 'orthogonal' (spin-z and spin-x, for example) then it's exactly like performing two independent coin flips at two remote locations.

This latter fact is interesting in that it tells that any hidden variable model must have, or be capable of generating, at least 2 bits of randomness for each measured entangled pair if it is to be able to fully reproduce all of the quantum mechanical predictions. It's also possible to use this 'orthogonal' configuration to demonstrate that for a deterministic hidden variable model measurement can't be a passive process but must induce a 'change of state' analogous to that of QM.

Chris Miller
Gold Member
I'm sorry for the confusion: with the term 'aligned' I was talking about strings of (generated) random bits that are aligned or not. More about that later.
When the detectors are aligned (both spin-z, for example) then it's like the situation we have with the coins joined by a rigid rod; when the detectors are 'orthogonal' (spin-z and spin-x, for example) then it's exactly like performing two independent coin flips at two remote locations.

This latter fact is interesting in that it tells that any hidden variable model must have, or be capable of generating, at least 2 bits of randomness for each measured entangled pair if it is to be able to fully reproduce all of the quantum mechanical predictions.
I doubt that. In case of two entangled coins/bits/particles one could say, for instance, that coin A generates a random bit, and coin B takes on a value that is dependent of coin A; in case of aligned detectors there is clearly no randomness for B (rigid rod)(single bit), but in case of misaligned detectors, B is jumping between an aligned and unaligned bit (detector) randomly. B is always dependent of A (or vice-versa)(for this, non-locality is a requirement). The strange thing is that one cannot identify which side is dependent on which!
* I am not entirely sure for I am not able to check with code. I'll try to back this up with code.
I have written code that supports my claim.

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Simon Phoenix
Gold Member
I'm sorry for the confusion
It's not necessarily your fault I am quite often rather slow on the uptake.

However, I think you do have to be careful in talking about splitting up binary strings (not quite sure what you mean by 'aligning' here). For example, suppose we had a source of 2 spin-1/2 particles producing either the state ##|00 \rangle## or the state ##|11 \rangle## uniformly at random. And now we send one of these particles off to A and its partner to B for each of the pairs that are produced. The standard Bell-type measurements are done. It is then possible to split the (joint) data into two groups - each of which show a violation of a Bell inequality, even though there's absolutely nothing 'quantum' in terms of entanglement going on.

Post-selection can do some interesting things

The Maximal Lyapunov exponent?
This was your definition of "truly random", referring to a string of bits.

Could you please define what the Lyapunov exponent is for a bit string. I only know what it means for a dynamical system.

I know what a random variable is and I've seen different "tests for randomness" , but I've never seen a coherent definition of random. Could you help me out?
Would you say Champernowne's recipe for a bit string is random?

Gold Member
It's not necessarily your fault I am quite often rather slow on the uptake.

However, I think you do have to be careful in talking about splitting up binary strings (not quite sure what you mean by 'aligning' here). For example, suppose we had a source of 2 spin-1/2 particles producing either the state ##|00 \rangle## or the state ##|11 \rangle## uniformly at random. And now we send one of these particles off to A and its partner to B for each of the pairs that are produced. The standard Bell-type measurements are done. It is then possible to split the (joint) data into two groups - each of which show a violation of a Bell inequality, even though there's absolutely nothing 'quantum' in terms of entanglement going on.

Post-selection can do some interesting things
Now I'm lost!

I mean that if you have two strings of bits, you can lay them beside each other in different positions: bit (A)0 and (B)0 aligned, or bit (A)0 and (B)1 aligned, or bit (A)0 and (B)2 aligned etc. So, in a way, rotating string B with respect to string A (or shifting).

My definition of 'random' is reflexive: If string (of bits) A is random, and string of bits B is random, then the pairwise comparison of the bits has no correlation (is random).

It turns out, following my code, that if we have an entanglement experiment setup in which there is correlation, the random strings produced by entanglement are in every alignment non-correlative, except for the (0,0)-alignment, where, of course, there is a correlation.

So entanglement bitstrings are (of course) different form purely random bitstrings in my definition, the latter exhibiting no correlation in every alignment.

Nugatory
Mentor
My definition of 'random' is reflexive: If string (of bits) A is random, and string of bits B is random, then the pairwise comparison of the bits has no correlation (is random).
That definition doesn't work. A trivial counterexample would be the string S' that I get by flipping every bit in a random string S; S' is no less random than S itself (all I'm doing is changing the symbols that represent the string), yet there is a 100% anticorrelation between the two.