Schmidt decomposition

In linear algebra, the Schmidt decomposition 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

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.

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} eta _{ij} e_i otimes f_j

can then be viewed as the "n" × "m" matrix

:$;$M_v = (eta_{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 egin{bmatrix} Sigma \ 0 end{bmatrix} V^T .

Write $$U =egin{bmatrix} U_1 & U_2 end{bmatrix} where $$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 _{i=1} ^m alpha_i u_i v_i ^T = sum _{i=1} ^m alpha_i u_i otimes v_i ,

which proves the claim.

Some observations

Some properties of the Schmidt decomposition are of physical interest.

Spectrum of reduced states

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 |"α_i" |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

For an element "w" of the tensor product :$$H_1 otimes H_2the strictly positive values "σ_i" in its Schmidt decomposition are its Schmidt coefficients. The number of Schmidt coefficients of $$w is called its Schmidt rank.

If "w" can not be expressed as :$$u otimes vthen "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, a bipartite pure state is entangled if and only if its reduced states are mixed states.

Crystal plasticity

In the field of plasticity, crystaline solids such as metals deform plasticly 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=muotimes u. 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.