Steinspring quantum error correction
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This is more for mathematical simplicity than physical insight, but it is always good to declutter our equations a bit if we can. Once we start playing with adding physical systems and increasing the dimension of the underlying Hilbert space, it is convenient to switch from unitaries to isometries. 12 Quantum error correction and fault tolerance.11.4 Some errors can be corrected on some states.11 Decoherence and basic quantum error correction.10.4 Oracles, and our first quantum algorithm.10.2 Hadamard and quantum Fourier transforms.10.1 Quantum Boolean function evaluation.7.13.19 Tricks with a maximally mixed state.7.13.16 Complete positivity of a certain map.7.13.15 Trace preserving and partial trace.7.13.9 Unchanged reduced density operator.7.13.5 The “control” part of controlled-NOT.7.10 The mathematics of “can” and “cannot”.7.8 Completely positive trace-preserving maps.7.5 Stinespring’s dilation and Kraus’s ambiguity.6.8.7 Spectral decompositions and common eigenbases.6.8.6 Distinguishability and the trace distance.6.8.1 Some density operator calculations.
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6.3 Some instructive examples, questions, and remarks.5.9.10 Hadamard transforms in components.5.9.8 Playing with conditional unitaries.5.9.5 Arbitrary controlled-U on two qubits.5.8 Why qubits, subsystems, and entanglement?.5.7.5 Density operators, and other things to come.5.6.1 The Bell states, and the Bell measurement.5.3 Quantum theory, formally (continued).4.12.4 Unitary transformations of measurements.4.9 Distinguishability of non-orthogonal states.4.6 Compatible observables and the uncertainty relation.4.4 Example of an incomplete measurement.4.3 The projection rule, and incomplete measurements.Through this difficult time APS and the Physical Review editorial office are fully equipped and actively working to support researchers by continuing to carry out all editorial and peer-review functions and publish research in the journals as well as minimizing disruption to journal. 3.7.6 Special orthogonal matrix calculations COVID-19 has impacted many institutions and organizations around the world, disrupting the progress of research.3.7.4 Linear algebra of the Pauli vector.3.7.3 Pauli matrix expansion coefficients.3.2 Quantum interference, revisited (still about beam-splitters).3.1 Physics against logic, via beam-splitters.2.12.2 One of the many cross-product identities.2.9.1 Drawing points on the Bloch sphere.2.8 Any unitary operation on a single qubit.2.7.1 From bit-flips to phase-flips, and back again.1.10.8 Polynomial = good exponential = bad.1.7 Computation: deterministic, probabilistic, and quantum.1.2 Quantum interference: the failure of probability theory.0.2 Euclidean vectors and vector spaces.