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.
Let and be Hilbert spaces of dimensions n and m respectively. Assume . For any vector in the tensor product , there exist orthonormal sets and such that , where the scalars 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 and . We can identify an elementary tensor with the matrix , where is the transpose of . A general element of the tensor product
can then be viewed as the n × m matrix
Write where is n × m and we have
Let be the m column vectors of , the column vectors of , and the diagonal elements of Σ. The previous expression is then
which proves the claim.
Some properties of the Schmidt decomposition are of physical interest.
Spectrum of reduced states
Consider a vector of the tensor product
in the form of Schmidt decomposition
Form the rank 1 matrix . Then the partial trace of , with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are . In other words, the Schmidt decomposition shows that the reduced states of on either subsystem have the same spectrum.
Schmidt rank and entanglement
The strictly positive values in the Schmidt decomposition of are its Schmidt coefficients, or Schmidt numbers. The total number of Schmidt coefficients of , counted with multiplicity, is called its Schmidt rank.
If can be expressed as a product
then is called a separable state. Otherwise, is said to be an entangled state. From the Schmidt decomposition, we can see that is entangled if and only if has Schmidt rank strictly greater than 1. Therefore, two subsystems that partition a pure state are entangled if and only if their reduced states are mixed states.
Von Neumann entropy
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 , and this is zero if and only if 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.
Consider the tripartite quantum system:
There are three ways to reduce this to a bipartite system by performing the partial trace with respect to or
Each of the systems obtained is a bipartite system and therefore can be characterized by one number (its Schmidt rank), respectively and . These numbers capture the "amount of entanglement" in the bipartite system when respectively A, B or C are discarded. For these reasons the tripartite system can be described by a vector, namely the Schmidt-rank vector
The concept of Schmidt-rank vector can be likewise extended to systems made up of more than three subsystems through the use of tensors.
Take the tripartite quantum state
The Schmidt-rank vector for this quantum state is .
- Huber, Marcus; de Vicente, Julio I. (January 14, 2013). "Structure of Multidimensional Entanglement in Multipartite Systems". Physical Review Letters. 110 (3): 030501. arXiv:1210.6876. Bibcode:2013PhRvL.110c0501H. doi:10.1103/PhysRevLett.110.030501. ISSN 0031-9007. PMID 23373906. S2CID 44848143.
- Krenn, Mario; Malik, Mehul; Fickler, Robert; Lapkiewicz, Radek; Zeilinger, Anton (March 4, 2016). "Automated Search for new Quantum Experiments". Physical Review Letters. 116 (9): 090405. arXiv:1509.02749. Bibcode:2016PhRvL.116i0405K. doi:10.1103/PhysRevLett.116.090405. ISSN 0031-9007. PMID 26991161. S2CID 20182586.
- Pathak, Anirban (2013). Elements of Quantum Computation and Quantum Communication. London: Taylor & Francis. pp. 92–98. ISBN 978-1-4665-1791-2.