Schmidt decomposition

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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 applications in quantum information theory and plasticity.

Theorem[edit]

Let H_1 and H_2 be Hilbert spaces of dimensions n and m respectively. Assume n \geq m. For any vector v in the tensor product H_1 \otimes H_2, there exist orthonormal sets \{ u_1, \ldots, u_m \} \subset H_1 and \{ v_1, \ldots, v_m \} \subset H_2 such that v = \sum_{i =1} ^m \alpha _i u_i \otimes v_i, where the scalars \alpha_i are non-negative and, as a set, uniquely determined by v.

Proof[edit]

The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal bases \{ e_1, \ldots, e_n \} \subset H_1 and \{ f_1, \ldots, f_m \} \subset H_2. We can identify an elementary tensor e_i \otimes f_j with the matrix e_i f_j ^T, where f_j ^T is the transpose of f_j. A general element of the tensor product

v = \sum _{1 \leq i \leq n, 1 \leq j \leq m} \beta _{ij} e_i \otimes f_j

can then be viewed as the n × m matrix

\; M_v = (\beta_{ij})_{ij} .

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

M_v = U \begin{bmatrix} \Sigma \\ 0 \end{bmatrix} V^T .

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

\; M_v = U_1 \Sigma V^T .

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

M_v = \sum _{k=1} ^m \alpha_k u_k v_k ^T ,

Then

v = \sum _{k=1} ^m \alpha_k u_k \otimes v_k ,

which proves the claim.

Some observations[edit]

Some properties of the Schmidt decomposition are of physical interest.

Spectrum of reduced states[edit]

Consider a vector in the form of Schmidt decomposition

v = \sum_{i =1} ^m \alpha _i u_i \otimes v_i.

Form the rank 1 matrix ρ = v v*. 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 state of ρ on either subsystem have the same spectrum.

In the language of quantum mechanics, a rank 1 projection ρ is called a pure state. A consequence of the above comments is that, for bipartite pure states, the von Neumann entropy of either reduced state is a well defined measure of entanglement.

Schmidt rank and entanglement[edit]

For an element w of the tensor product

H_1 \otimes H_2

the strictly positive values \alpha_i in its Schmidt decomposition are its Schmidt coefficients. The number of Schmidt coefficients of w is called its Schmidt rank.

If w cannot be expressed as

u \otimes v

then 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.

Crystal plasticity[edit]

In the field of plasticity, crystalline solids such as metals deform plastically primarily along crystal planes. Each plane, defined by its normal vector ν can "slip" in one of several directions, defined by a vector μ. Together a slip plane and direction form a slip system which is described by the Schmidt tensor P=\mu\otimes \nu. The velocity gradient is a linear combination of these across all slip systems where the scaling factor is the rate of slip along the system.