Schmidt decomposition

Short description: Process in linear algebra

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.

## Contents

* 1 Theorem
* 1.1 Proof
* 2 Some observations
* 2.1 Spectrum of reduced states
* 2.2 Schmidt rank and entanglement
* 2.3 Von Neumann entropy
* 3 Schmidt-rank vector
* 3.1 Multipartite systems
* 3.2 Example
* 4 See also
* 5 References
* 6 Further reading

## Theorem

Let H1 and H2 be Hilbert spaces of dimensions n and m respectively. Assume n≥m. For any vector w in the tensor product H1⊗H2, there exist orthonormal sets {u1,…,um}⊂H1 and {v1,…,vm}⊂H2 such that w=∑i=1mαiui⊗vi, where the scalars αi 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 {e1,…,en}⊂H1 and {f1,…,fm}⊂H2. We can identify an elementary tensor ei⊗fj with the matrix eifjT, where fjT is the transpose of fj. A general element of the tensor product

w=∑1≤i≤n,1≤j≤mβijei⊗fj

can then be viewed as the n × m matrix

Mw=(βij).

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

Mw=U[Σ0]V*.

Write U=[U1U2] where U1 is n × m and we have

Mw=U1ΣV*.

Let {u1,…,um} be the m column vectors of U1, {v1,…,vm} the column vectors of V‾, and α1,…,αm the diagonal elements of Σ. The previous expression is then

Mw=∑k=1mαkukvkT,

Then

w=∑k=1mαkuk⊗vk,

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

H1⊗H2

in the form of Schmidt decomposition

w=∑i=1mαiui⊗vi.

Form the rank 1 matrix ρ=ww*. 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 αi in the Schmidt decomposition of w are its Schmidt coefficients, or Schmidt numbers. The total number of Schmidt coefficients of w, counted with multiplicity, is called its Schmidt rank.

If w can be expressed as a product

u⊗v

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 −∑i|αi|2log⁡(|αi|2), 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

|ψ⟩∈HA⊗HB

The concept of Schmidt rank can be extended to quantum systems made up of more than two subsystems.[1]

Consider the tripartite quantum system:

|ψ⟩∈HA⊗HB⊗HC

There are three ways to reduce this to a bipartite system by performing the partial trace with respect to HA,HB or HC

{ρ^A=TrA(|ψ⟩⟨ψ|)ρ^B=TrB(|ψ⟩⟨ψ|)ρ^C=TrC(|ψ⟩⟨ψ|)

Each of the systems obtained is a bipartite system and therefore can be characterized by one number (its Schmidt rank), respectively rA,rB and rC. 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

r→=(rA,rB,rC)

### 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

Source:[2]

Take the tripartite quantum state |ψ4,2,2⟩=12(|0,0,0⟩+|1,0,1⟩+|2,1,0⟩+|3,1,1⟩)

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 (4,2,2).

## See also

* Singular value decomposition
* Purification of quantum state

## References

1. ↑ Huber, Marcus; de Vicente, Julio I. (2013-01-14). "Structure of Multidimensional Entanglement in Multipartite Systems" (in en). Physical Review Letters 110 (3). doi:10.1103/PhysRevLett.110.030501. ISSN 0031-9007. PMID 23373906. Bibcode: 2013PhRvL.110c0501H. https://link.aps.org/doi/10.1103/PhysRevLett.110.030501.
2. ↑ Krenn, Mario; Malik, Mehul; Fickler, Robert; Lapkiewicz, Radek; Zeilinger, Anton (2016-03-04). "Automated Search for new Quantum Experiments" (in en). Physical Review Letters 116 (9). doi:10.1103/PhysRevLett.116.090405. ISSN 0031-9007. PMID 26991161. Bibcode: 2016PhRvL.116i0405K. https://link.aps.org/doi/10.1103/PhysRevLett.116.090405.

## Further reading

* Pathak, Anirban (2013). Elements of Quantum Computation and Quantum Communication. London: Taylor & Francis. pp. 92–98. ISBN 978-1-4665-1791-2. https://books.google.com/books?id=cEPSBQAAQBAJ&pg=PA92.

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