Quantum information theory studies information processing with quantum states, measurements, channels, entanglement, and quantum entropy.
For normalized maximally entangled , the Choi matrix of is . Choi's theorem says is completely positive exactly when .
A random unitary channel has . Its Choi matrix is a convex combination of maximally entangled pure states.
A quantum channel is unital when . Every random unitary channel is unital, while the converse fails in dimension at least three.
The Werner–Holevo channel is . Its normalized Choi matrix is the maximally mixed state on the antisymmetric subspace.
A quantum channel is strictly contractive in trace distance when it reduces the distance between every pair of distinct density operators by a uniform factor smaller than one.
A finite-dimensional quantum channel is primitive when some power maps every nonzero positive operator to a positive-definite operator. Equivalently, eigenvalue one is simple and no other eigenvalue lies on the unit circle.
A Stinespring dilation represents a quantum channel by an isometry followed by tracing out the environment.
A bipartite state is separable when it is a convex combination of product states; otherwise it is entangled.
Every separable state has positive partial transpose. The condition is also sufficient for separability on and systems.
A channel is entanglement breaking when applying it to one side of every bipartite state always produces a separable state. Equivalently, its Choi matrix is separable, and equivalently it has a measure-and-prepare representation.
Quantum binary hypothesis testing chooses a two-outcome measurement to distinguish two candidate density operators, trading the probabilities of the two kinds of error.
For hypotheses with priors , the optimal success probability isAn optimal measurement projects onto the positive and negative spectral subspaces of the weighted difference.
Entanglement monogamy limits how strongly one subsystem can be entangled with several independent partners.
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Quantum information theory is a field of study that combines principles from quantum mechanics and information theory to understand how information can be stored, processed, and transmitted using quantum systems. It explores the fundamental limits of information processing and seeks to harness quantum phenomena to improve information technology. Key concepts in quantum information theory include: 1. **Qubits**: The fundamental unit of quantum information, analogous to classical bits but capable of existing in superpositions of states.