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Quantum theory Definition

Quantum information

Builds on Hilbert spaceTensor product of Hilbert spacesDensity matrixQuantum channelPOVM

Definition

Quantum information theory uses the following framework.

  1. A quantum system is described by a finite-dimensional complex Hilbert space HH.
  2. A state of the system is a density matrix on HH, that is, a positive semidefinite operator of trace 11.
  3. A process from a system HH to a system KK is a quantum channel. It is a completely positive trace-preserving linear map from operators on HH to operators on KK.
  4. A measurement with finitely many outcomes is a POVM (Ej)j(E_j)_j on HH, and outcome jj occurs with probability tr⁡(Ejρ)\tr(E_j \rho).
  5. Two systems HAH_A and HBH_B together form the system HA⊗HBH_A \otimes H_B, their tensor product.

Quantum information is the information carried by states in this framework and transformed by channels and measurements.

Remarks

Example: a qubit is the system H=ℂ2H = \C^2, and ρ=12I\rho = \tfrac12 I is a state, since it is positive semidefinite with trace 11.

Non-example: the matrix diag(32,−12)\mathrm{diag}(\tfrac32, -\tfrac12) has trace 11 but a negative eigenvalue, so clause 2 fails and it is not a state.

Classical information is the special case where all states and measurement operators are diagonal in one fixed basis. Then states are probability vectors and channels are stochastic matrices. The unit of quantum information is the qubit, and its measures include the von Neumann entropy.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Density matrix, concept, no evidence levelDensity matrixconceptHilbert space, concept, no evidence levelHilbert spaceconceptPOVM, concept, no evidence levelPOVMconceptQuantum channel, concept, no evidence levelQuantum channelconceptTensor product of Hilbert spaces, concept, no evidence levelTensor product of Hil…conceptQuantum information, definition, no evidence levelQuantum informationdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Density matrix concept
  2. Hilbert space concept
  3. POVM concept
  4. Quantum channel concept
  5. Tensor product of Hilbert spaces concept
  6. Quantum information definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Quantum_information
About this page
TagW8MJBCX8MP
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Kinddefinition
WritingW0 stub
MSC81P45, 94A15
DefinesQuantum information
UsesHilbert space, Tensor product of Hilbert spaces, Density matrix, Quantum channel, POVM
Changed3 November 2026

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