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 [2]
* 4 See also
* 5 References
* 6 Further reading

## Theorem

Let [math]\displaystyle{ H_1 }[/math] and [math]\displaystyle{ H_2 }[/math] be Hilbert spaces of dimensions n and m respectively. Assume [math]\displaystyle{ n \geq m }[/math]. For any vector [math]\displaystyle{ w }[/math] in the tensor product [math]\displaystyle{ H_1 \otimes H_2 }[/math], there exist orthonormal sets [math]\displaystyle{ \\{ u_1, \ldots, u_m \\} \subset H_1 }[/math] and [math]\displaystyle{ \\{ v_1, \ldots, v_m \\} \subset H_2 }[/math] such that [math]\displaystyle{ w= \sum_{i =1} ^m \alpha _i u_i \otimes v_i }[/math], where the scalars [math]\displaystyle{ \alpha_i }[/math] 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 [math]\displaystyle{ \\{ e_1, \ldots, e_n \\} \subset H_1 }[/math] and [math]\displaystyle{ \\{ f_1, \ldots, f_m \\} \subset H_2 }[/math]. We can identify an elementary tensor [math]\displaystyle{ e_i \otimes f_j }[/math] with the matrix [math]\displaystyle{ e_i f_j ^\mathsf{T} }[/math], where [math]\displaystyle{ f_j ^\mathsf{T} }[/math] is the transpose of [math]\displaystyle{ f_j }[/math]. A general element of the tensor product

[math]\displaystyle{ w = \sum _{1 \leq i \leq n, 1 \leq j \leq m} \beta _{ij} e_i \otimes f_j }[/math]

can then be viewed as the n × m matrix

[math]\displaystyle{ \; M_w = (\beta_{ij}) . }[/math]

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

[math]\displaystyle{ M_w = U \begin{bmatrix} \Sigma \\\ 0 \end{bmatrix} V^* . }[/math]

Write [math]\displaystyle{ U =\begin{bmatrix} U_1 & U_2 \end{bmatrix} }[/math] where [math]\displaystyle{ U_1 }[/math] is n × m and we have

[math]\displaystyle{ \; M_w = U_1 \Sigma V^* . }[/math]

Let [math]\displaystyle{ \\{ u_1, \ldots, u_m \\} }[/math] be the m column vectors of [math]\displaystyle{ U_1 }[/math], [math]\displaystyle{ \\{ v_1, \ldots, v_m \\} }[/math] the column vectors of [math]\displaystyle{ \overline{V} }[/math], and [math]\displaystyle{ \alpha_1, \ldots, \alpha_m }[/math] the diagonal elements of Σ. The previous expression is then

[math]\displaystyle{ M_w = \sum _{k=1} ^m \alpha_k u_k v_k ^\mathsf{T} , }[/math]

Then

[math]\displaystyle{ w = \sum _{k=1} ^m \alpha_k u_k \otimes v_k , }[/math]

which proves the claim.

## Some observations

Some properties of the Schmidt decomposition are of physical interest.

### Spectrum of reduced states

Consider a vector [math]\displaystyle{ w }[/math] of the tensor product

[math]\displaystyle{ H_1 \otimes H_2 }[/math]

in the form of Schmidt decomposition

[math]\displaystyle{ w = \sum_{i =1} ^m \alpha _i u_i \otimes v_i. }[/math]

Form the rank 1 matrix [math]\displaystyle{ \rho = w w^* }[/math]. Then the partial trace of [math]\displaystyle{ \rho }[/math], with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are [math]\displaystyle{ | \alpha_i|^2 }[/math]. In other words, the Schmidt decomposition shows that the reduced states of [math]\displaystyle{ \rho }[/math] on either subsystem have the same spectrum.

### Schmidt rank and entanglement

The strictly positive values [math]\displaystyle{ \alpha_i }[/math] in the Schmidt decomposition of [math]\displaystyle{ w }[/math] are its Schmidt coefficients, or Schmidt numbers. The total number of Schmidt coefficients of [math]\displaystyle{ w }[/math], counted with multiplicity, is called its Schmidt rank.

If [math]\displaystyle{ w }[/math] can be expressed as a product

[math]\displaystyle{ u \otimes v }[/math]

then [math]\displaystyle{ w }[/math] is called a separable state. Otherwise, [math]\displaystyle{ w }[/math] is said to be an entangled state. From the Schmidt decomposition, we can see that [math]\displaystyle{ w }[/math] is entangled if and only if [math]\displaystyle{ w }[/math] 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 [math]\displaystyle{ \rho }[/math] is [math]\displaystyle{ -\sum_i |\alpha_i|^2 \log\left(|\alpha_i|^2\right) }[/math], and this is zero if and only if [math]\displaystyle{ \rho }[/math] is a product state (not entangled).

## Schmidt-rank vector

The Schmidt rank is defined for bipartite systems, namely quantum states

[math]\displaystyle{ |\psi\rangle \in H_A \otimes H_B }[/math]

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

Consider the tripartite quantum system:

[math]\displaystyle{ |\psi\rangle \in H_A \otimes H_B \otimes H_C }[/math]

There are three ways to reduce this to a bipartite system by performing the partial trace with respect to [math]\displaystyle{ H_A, H_B }[/math] or [math]\displaystyle{ H_C }[/math]

[math]\displaystyle{ \begin{cases} \hat{\rho}_A = Tr_A(|\psi\rangle\langle\psi|)\\\ \hat{\rho}_B = Tr_B(|\psi\rangle\langle\psi|)\\\ \hat{\rho}_C = Tr_C(|\psi\rangle\langle\psi|) \end{cases} }[/math]

Each of the systems obtained is a bipartite system and therefore can be characterized by one number (its Schmidt rank), respectively [math]\displaystyle{ r_A, r_B }[/math] and [math]\displaystyle{ r_C }[/math]. 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

[math]\displaystyle{ \vec{r} = (r_A, r_B, r_C) }[/math]

### 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 [math]\displaystyle{ |\psi_{4, 2, 2}\rangle = \frac{1}{2}\big(|0, 0, 0\rangle + |1, 0, 1\rangle + |2, 1, 0\rangle + |3, 1, 1\rangle \big) }[/math]

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 [math]\displaystyle{ (4, 2, 2) }[/math].

## 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): 030501. 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): 090405. 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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