This resource function is obsolete. Use the paclet Wolfram/QuantumFramework instead.

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QuantumSchmidtDecomposition

Perform a Schmidt decomposition on a pure discrete quantum state

Contributed by: Jonathan Gorard and Ruhi Shah

ResourceFunction["QuantumSchmidtDecomposition"][QuantumDiscreteState[…]]

performs a Schmidt decomposition on the specified QuantumDiscreteState.

Details and Options

The Schmidt decomposition in linear algebra represents a vector as the tensor product of two Hilbert spaces. In the context of quantum information theory, it represents a state vector as the tensor product of two quantum bases.

ResourceFunction["QuantumSchmidtDecomposition"] is only defined for pure QuantumDiscreteState objects containing at least two distinct subsystems (qudits).

By default, ResourceFunction["QuantumSchmidtDecomposition"][QuantumDiscreteState[…]] returns a second QuantumDiscreteState object corresponding to the Schmidt decomposition of the first, with a new (Schmidt-decomposed) tensor product basis.

With the option "GiveFullDecomposition"→True, ResourceFunction["QuantumSchmidtDecomposition"] returns a list containing the Schmidt-decomposed QuantumDiscreteState, as well as the two QuantumBasis objects involved in the tensor product.

The two bases involved in the tensor product consist of basis elements named u₁,u₂ and v₁,v₂, respectively.

Examples

Basic Examples (3)

Create a two-qubit pure discrete quantum state in the computational basis (default):

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Perform a Schmidt decomposition of the state, resulting in a new (Schmidt-decomposed) tensor product basis:

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Show the basis elements of the new (Schmidt-decomposed) tensor product basis:

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Perform a full Schmidt decomposition (returning both the decomposed state, and the two individual bases involved in the tensor product):

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Perform a Schmidt decomposition of a pure discrete quantum state with more than 2 subsystems by automatically partitioning the state into exactly two subsystems of (approximately) equal size:

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state = ResourceFunction["QuantumDiscreteState"][{"RandomPure", 4}];
{decomposition, basis1, basis2} = ResourceFunction["QuantumSchmidtDecomposition"][state, "GiveFullDecomposition" -> True]

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Perform a Schmidt decomposition of a state consisting of two three-dimensional qudits:

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state = ResourceFunction["QuantumDiscreteState"][{"RandomPure", 2}, 3];
{decomposition, basis1, basis2} = ResourceFunction["QuantumSchmidtDecomposition"][state, "GiveFullDecomposition" -> True]

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Options (2)

If "GiveFullDecomposition"→False (default), then QuantumSchmidtDecomposition returns only the decomposed state in the (Schmidt-decomposed) tensor product basis:

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On the other hand, if "GiveFullDecomposition"→True, then QuantumSchmidtDecomposition returns a list consisting of the decomposed state, as well as the two individual bases involved in the tensor product:

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