quantum state preparation (changes) in nLab

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Contents

Idea

In quantum physics and especially in quantum information theory, by state preparation one refers to a process (either in actual experiments or theoretically as a kind of quantum gate-operation) which, possibly conditioned on a classical parameter bb, “produces” a predefined quantum state |b⟩∈ℋ\vert b \rangle \,\in\, \mathscr{H} in a given space of quantum states ℋ\mathscr{H}.

For example, for a qbit data type there are, by definition of qbits, two canonical basis states |0⟩,|1⟩∈ℂ 2\vert 0 \rangle, \vert 1 \rangle \,\in\, \mathbb{C}^2 to be prepared, depending on a classical parameter which may be understood as of boolean type (where we set Bool≔{0,1}Bool \,\coloneqq\, \{0, 1\}).

The various fundamental quantum physics phenomena at play here may be identified with four modal-units of quantum modal logic (as discussed at quantum circuits via dependent linear types):

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quantum measurement	quantum state preparation
quantum superposition	quantum parallelism

Explicitly, in the language¹ of modal quantum logic on dependent linear types (see at quantum circuits via dependent linear types), the conditional preparation of a qbit-state reads as follows:

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and the unconditional state preparation is of this form:

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Other states may be prepared by first preparing a qbit-state and then sending it through some quantum gate.

For instance, the Bell state on two qbits may be prepared by first preparing a tensor product of two |0⟩\vert 0 \rangle-states and then sending one through a Hadamard gate and then the resulting two qbits through a CNOT gate:

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This is used in many common quantum circuits, such as for instance in the quantum teleportation-protocol.

References

Standard textbook accounts:

• Michael A. Nielsen, Isaac L. Chuang, §7.2 in: Quantum computation and quantum information, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), pdf, pdf]