Schmidt decomposition
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.
Theorem[]
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.
Proof[]
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
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 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
Then
which proves the claim.
Some observations[]
Some properties of the Schmidt decomposition are of physical interest.
Spectrum of reduced states[]
Consider a vector w of the tensor product
in the form of Schmidt decomposition
Form the rank 1 matrix ρ = w w*. Then the partial trace of ρ, with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are |αi|2. In other words, the Schmidt decomposition shows that the reduced states of ρ on either subsystem have the same spectrum.
Schmidt rank and entanglement[]
The strictly positive values in the Schmidt decomposition of w are its Schmidt coefficients. The number of Schmidt coefficients of , counted with multiplicity, is called its Schmidt rank, or Schmidt number.
If w can be expressed as a product
then w is called a separable state. Otherwise, w is said to be an entangled state. From the Schmidt decomposition, we can see that w is entangled if and only if w 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 ρ is , and this is zero if and only if ρ is a product state (not entangled).
Schmidt-rank vector[]
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 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 reasos the tripartite system can be described by a vector, namely the Schmidt-rank vector
Multipartite systems[]
The concept of Schmidt-rank vector can be likewise extended to systems made up of more than three subsystems through the use of tensors.
Example [2][]
Take the tripartite quantum state
this kind of systems is made possible by encoding the value of a qudit into the orbital angular momentum (OAM) of a photon rather than its SPIN.
The Schmidt-rank vector for this quantum state is .
See also[]
- Singular value decomposition
- Purification of quantum state
References[]
- ^ Huber, Marcus; de Vicente, Julio I. (January 14, 2013). "Structure of Multidimensional Entanglement in Multipartite Systems". Physical Review Letters. 110 (3): 030501. doi:10.1103/PhysRevLett.110.030501. ISSN 0031-9007.
- ^ 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. doi:10.1103/PhysRevLett.116.090405. ISSN 0031-9007.
Further reading[]
- Pathak, Anirban (2013). Elements of Quantum Computation and Quantum Communication. London: Taylor & Francis. pp. 92–98. ISBN 978-1-4665-1791-2.
- Linear algebra
- Singular value decomposition
- Quantum information theory