Consider a composite quantum system which consists of two subsystems and , respectively with basis states and . All accessible states of the sytem lie in the tensor product of the subsystems’ Hilbert spaces and :
A subset of these states can be written as the tensor product (i.e. Kronecker product in a basis) of a state in and a state in , often abbreviated as :
The states that can be written in this way are called separable, and states that cannot are called entangled. Therefore, we are dealing with quantum entanglement if the state of subsystem cannot be fully described independently of the state of subsystem , and vice versa.
To detect and quantify entanglement, we can use the density operator . For a pure ensemble in a given (possibly entangled) state , is given by:
From this, we would like to extract the corresponding state of subsystem . For that purpose, we define the reduced density operator of subsystem as follows:
Where is called the partial trace of , which basically eliminates subsystem from . For a pure composite state , the resulting describes a pure state in if is separable, else, if is entangled, it describes a mixed state in . In the former case we simply find:
We call maximally entangled if its reduced density operators are maximally mixed, where is the dimension of and is the identity matrix:
Suppose that we are given an entangled pure state . Then the partial traces and of are mixed states with the same probabilities (assuming and have the same dimensions, which is usually the case):
There exists an orthonormal choice of the subsystem basis states and , such that can be written as follows, where are the probabilities in the reduced density operators:
This is the Schmidt decomposition, and the Schmidt number is the number of nonzero terms in the summation, which can be used to determine if the state is entangled (greater than one) or separable (equal to one).
By looking at the Schmidt decomposition, we can notice that, if and are the subsystem observables with basis eigenstates and , then measurement results of these operators will be perfectly correlated across and . This is a general property of entangled systems, but beware: correlation does not imply entanglement!
But what if the composite system is in a mixed state ? The state is separable if and only if:
Where are probabilities, and and can be any subsystem states. In reality, it is very hard to determine, using this criterium, whether an arbitrary given is separable or not.
As a final side note, the expectation value of an obervable acting only on is given by:
- J.B. Brask, Quantum information: lecture notes, 2021, unpublished.