Some authors exclusively use "Correlation" to classify composite quantum states, whereas most only speak of "Entanglement".

Correlation basically means that measurements on the subsystems are stochastically dependent and entanglement means non-separability of the composite state.

I am wondering, are those classifications equivalent, or is there any hierarchy (e.g. if a composite state is non entangled, it is always uncorrelated). Does the entropy of entanglement (in some cases) predict whether a state is (un)correlated?

References to a proof would be much appreciated!

Feel free to criticize me on the casual definitions given above as well.

  • 1
    $\begingroup$ DId you mean to write "stochastically dependent" instead of "stochastically independent"? States of the form $A\otimes B$ can give correlated results for many measurements, but are unentangled. $\endgroup$ – WillO Dec 30 '19 at 18:25
  • $\begingroup$ @WillO Not for connected correlation functions. $\endgroup$ – Norbert Schuch Jan 4 '20 at 23:04
  • $\begingroup$ @NorbertSchuch: The phrase "connected correlation function" is new to me. I'm off to learn about it. $\endgroup$ – WillO Jan 4 '20 at 23:13
  • $\begingroup$ @WillO <XY>-<X><Y>. $\endgroup$ – Norbert Schuch Jan 5 '20 at 22:11

Entanglement is the source of non-classical correlations, but of course you can have correlation without entanglement.

For instance, for a two-particle system of spinful particles with the constraint that the total spin is zero, you always have the correlation that if you measure the spin of one particle to be up (in one direction), the spin of the other particle will be down (in that direction). This is true regardless of whether the system is in an entangled state.


Correlations are a property of a set of measurements. Two measurement outcomes are correlated if the associated probability distribution cannot be factorised, that is, when the outcome of one measurement gives information about the outcome of the other measurement. The concept of "correlations", in this sense, is not inherently quantum, although quantum mechanics can make for correlations stronger than those allowed by classical probability theory.

On the other hand, entanglement is a property of a state, with respect to some partition on the underlying space. A bipartite state $\rho$ is said to be entangled if it cannot be written as a convex combination of product states, that is, if it cannot be written in the form $\rho=\sum_k p_k \rho_k^A\otimes\rho_k^B$ for some $p_k\ge0, \sum_k p_k=1$ and states $\rho_k^A,\rho_k^B$. The bipartite structure is usually, although not necessarily, taken to refer to degrees of freedom of spatially separated particles. It can however refer to any pair of degrees of freedom of a quantum system.

Entangled states can produce nonclassical correlations, but this is not necessarily the case. For example, not all entangled states can produce Bell violations. On the other hand, entangled states always display some form of correlation: given a pure entangled state $|\psi\rangle$, write it in its Schmidt decomposition as $|\psi\rangle=\sum_k \sqrt{p_k} |u_k\rangle\otimes|v_k\rangle$. Then, measuring in the $\{|u_k\rangle\}_k$ basis on the first space and in the $\{|v_k\rangle\}_k$ basis in the second will give correlated outcomes (the $k$-th outcome for the first party implies that the second party also will have measured its $k$-th outcome).


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