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 $H_1$ 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 real, non-negative, and, as a set, uniquely determined by $v$.

Proof

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

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

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

$\; M_v = U_1 \Sigma V^\star .$

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

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

Then

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

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

$H_1 \otimes H_2$

in the form of Schmidt decomposition

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

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 state of ρ on either subsystem have the same spectrum.

Schmidt rank and entanglement

The strictly positive values $\alpha_i$ in the Schmidt decomposition of w are its Schmidt coefficients. The number of Schmidt coefficients of $w$, counted with multiplicity, is called its Schmidt rank, or Schmidt number.

If w can be expressed as a product

$u \otimes v$

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 bipartite 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 $-\sum_i |\alpha_i|^2 \log|\alpha_i|^2$, and this is zero if and only if only if ρ is a product state (not entangled).

Crystal plasticity

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.