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 numerous applications in quantum information theory, for example in entanglement characterization and in state purification, and plasticity.

Theorem[edit]

Let and be Hilbert spaces of dimensions n and m respectively. Assume . For any vector in the tensor product , there exist orthonormal sets and such that , where the scalars are real, non-negative, and unique up to re-ordering.

Proof[edit]

The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal bases and . We can identify an elementary tensor with the matrix , where is the transpose of . A general element of the tensor product

can then be viewed as the n × m matrix

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

Write where is n × m and we have

Let be the m column vectors of , the column vectors of , and the diagonal elements of Σ. The previous expression is then

Then

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 w of the tensor product

in the form of Schmidt decomposition

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 states of ρ on either subsystem have the same spectrum.

Schmidt rank and entanglement[edit]

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

If w can be expressed as a product

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[edit]

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 ρ is , and this is zero if and only if ρ is a product state (not entangled).

Schmidt-rank vector[edit]

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

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

Consider the tripartite quantum system:

There are three ways to reduce this to a bipartite system by performing the partial trace with respect to or

Each of the systems obtained is a bipartite system and therefore can be characterized by one number (its Schmidt rank), respectively and . These numbers capture the "amount of entanglement" in the bipartite system when respectively A, B or C are discarded. For these reasos the tripartite system can be described by a vector, namely the Schmidt-rank vector

Multipartite systems[edit]

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][edit]

Take the tripartite quantum state

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 .

See also[edit]

References[edit]

  1. ^ Huber, Marcus; de Vicente, Julio I. (January 14, 2013). "Structure of Multidimensional Entanglement in Multipartite Systems". Physical Review Letters. 110 (3): 030501. doi:10.1103/PhysRevLett.110.030501. ISSN 0031-9007.
  2. ^ Krenn, Mario; Malik, Mehul; Fickler, Robert; Lapkiewicz, Radek; Zeilinger, Anton (March 4, 2016). "Automated Search for new Quantum Experiments". Physical Review Letters. 116 (9): 090405. doi:10.1103/PhysRevLett.116.090405. ISSN 0031-9007.

Further reading[edit]