In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two inner product spaces.
It has numerous applications in quantum information theory, for example in entanglement characterization and in state purification, and plasticity.
be Hilbert spaces of dimensions n and m respectively.
, there exist orthonormal sets
are real, non-negative, and unique up to re-ordering.
The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context.
Fix orthonormal bases
A general element of the tensor product can then be viewed as the n × m matrix By the singular value decomposition, there exist an n × n unitary U, m × m unitary V, and a positive semidefinite diagonal m × m matrix Σ such that Write
the diagonal elements of Σ.
The previous expression is then Then which proves the claim.
Some properties of the Schmidt decomposition are of physical interest.
of the tensor product in the form of Schmidt decomposition Form the rank 1 matrix
In other words, the Schmidt decomposition shows that the reduced states of
The strictly positive values
The total number of Schmidt coefficients of
, counted with multiplicity, is called its Schmidt rank.
is called a separable state.
has Schmidt rank strictly greater than 1.
A consequence of the above comments is that, for pure states, the von Neumann entropy of the reduced states is a well-defined measure of entanglement.
For the von Neumann entropy of both reduced states of
is a product state (not entangled).
The Schmidt rank is defined for bipartite systems, namely quantum states
The concept of Schmidt rank can be extended to quantum systems made up of more than two subsystems.
[1] Consider the tripartite quantum system:
There are three ways to reduce this to a bipartite system by performing the partial trace with respect to
Each of the systems obtained is a bipartite system and therefore can be characterized by one number (its Schmidt rank), respectively
These numbers capture the "amount of entanglement" in the bipartite system when respectively A, B or C are discarded.
The concept of Schmidt-rank vector can be likewise extended to systems made up of more than three subsystems through the use of tensors.
This kind of system is made possible by encoding the value of a qudit into the orbital angular momentum (OAM) of a photon rather than its spin, since the latter can only take two values.
The Schmidt-rank vector for this quantum state is