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 $$w$$ 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 $w= \sum_{i =1} ^m \alpha _i u_i \otimes v_i$, where the scalars $$\alpha_i$$ 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 $$\{ e_1, \ldots, e_n \} \subset H_1$$ and $$\{ f_1, \ldots, f_m \} \subset H_2$$. We can identify an elementary tensor $$e_i \otimes f_j$$ with the matrix $$e_i f_j ^\mathsf{T}$$, where $$f_j ^\mathsf{T}$$ is the transpose of $$f_j$$. A general element of the tensor product


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

can then be viewed as the n × m matrix


 * $$\; M_w = (\beta_{ij}) .$$

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


 * $$M_w = U \begin{bmatrix} \Sigma \\ 0 \end{bmatrix} V^* .$$

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


 * $$\; M_w = U_1 \Sigma V^* .$$

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


 * $$M_w = \sum _{k=1} ^m \alpha_k u_k v_k ^\mathsf{T} ,$$

Then


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

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 $$ \rho = w w^* $$. Then the partial trace of $$ \rho $$, with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are $$ | \alpha_i|^2 $$. In other words, the Schmidt decomposition shows that the reduced states of $$ \rho $$ 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, 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
 * $$u \otimes v$$

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

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

$$|\psi\rangle \in H_A \otimes H_B$$

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

Consider the tripartite quantum system:

$$|\psi\rangle \in H_A \otimes H_B \otimes H_C$$

There are three ways to reduce this to a bipartite system by performing the partial trace with respect to $$H_A, H_B$$ or $$H_C$$

$$\begin{cases} \hat{\rho}_A = Tr_A(|\psi\rangle\langle\psi|)\\ \hat{\rho}_B = Tr_B(|\psi\rangle\langle\psi|)\\ \hat{\rho}_C = Tr_C(|\psi\rangle\langle\psi|) \end{cases}$$

Each of the systems obtained is a bipartite system and therefore can be characterized by one number (its Schmidt rank), respectively $$r_A, r_B$$ and $$r_C$$. 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

$$\vec{r} = (r_A, r_B, r_C)$$

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 === Take the tripartite quantum state $$|\psi_{4, 2, 2}\rangle = \frac{1}{2}\big(|0, 0, 0\rangle + |1, 0, 1\rangle + |2, 1, 0\rangle + |3, 1, 1\rangle \big)$$

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 $$(4, 2, 2)$$.