On the Choi-Jamiolkowski Correspondence in Infinite Dimensions

Abstract

We give a mathematical formulation for the Choi-Jamiolkowski (CJ) correspondence in the infinite-dimensional case in the form close to one used in quantum information theory. We show that there is no need to use a limiting procedure to define "unnormalized maximally entangled state" and the corresponding analog of the Choi matrix since they can be defined rigorously as, in general, nonclosable forms on an appropriate dense subspace. The properties of these forms are discussed in Sec. 2. An important question is: when the CJ form is given by a bounded operator. This is the case for entanglement-breaking channels: we prove this in Sec. 3 along with a version of a result of Wolf et al. characterizing CJ operators which correspond to such channels by giving precise definitions of a separable operator and a relevant integral. In Sec. 4 we obtain explicit expressions for CJ forms and operators defining a general Bosonic Gaussian channel. In Sec. 5 we give a decomposition of the Gaussian CJ form into product of the four principal types and a necessary and sufficient condition for existence of the bounded CJ operator.

Results in Sec. 3,4 are strengthened. Sec. 4 is complemented with treatment of three degenerate cases. Sec. 5 is added where a decomposition of the Gaussian CJ form into product of the four principal types and a necessary and sufficient condition for existence of the bounded CJ operator are given

References in corpus (6)

• The characterization of Gaussian operations and Distillation of Gaussian States
• Multi-mode bosonic Gaussian channels
• Squeezing the limit: Quantum benchmarks for the teleportation and storage of squeezed states
• The entropy gain of infinite-dimensional quantum channels
• The optimal unitary dilation for bosonic Gaussian channels
• Separability and Entanglement-Breaking in Infinite Dimensions

Cited by in corpus (27)

• Non-Gaussian Quantum States and Where to Find Them
• Entanglement witnesses: construction, analysis and classification
• Continuous variable steering and incompatibility via state-channel duality
• Optimal continuous variable quantum teleportation with limited resources
• Non-Markovianity of Gaussian Channels
• Causal Boxes: Quantum Information-Processing Systems Closed under Composition
• Simulation of Gaussian channels via teleportation and error correction of Gaussian states
• Renyi relative entropies of quantum Gaussian states
• Approximate reversal of quantum Gaussian dynamics
• Extendibility of bosonic Gaussian states
• Quantum communication on the bosonic loss-dephasing channel
• Gaussian local unitary equivalence of $n$-mode Gaussian states and Gaussian transformations by local operations with classical communication
• Teleportation-based collective attacks in Gaussian quantum key distribution
• Security of differential phase shift QKD from relativistic principles
• Amendable Gaussian channels:restoring entanglement via a unitary filter
• Exact entanglement cost of quantum states and channels under PPT-preserving operations
• Entanglement Thresholds of Doubly-Parametric Quantum Transducers
• On positive maps in quantum information
• Geometry on the manifold of Gaussian quantum channels
• The structure of general quantum Gaussian observable
• Gaussian intrinsic entanglement: An entanglement quantifier based on secret correlations
• Truncated channel representations for coupled harmonic oscillators
• Witnessing non-Markovianity with Gaussian quantum steering in collision model
• Mapping indefinite causal order processes to composable quantum protocols in a spacetime
• Divergence-free approach for obtaining decompositions of quantum-optical processes
• Typical Gaussian Quantum Information
• From the Choi Formalism in Infinite Dimensions to Unique Decompositions of Generators of Completely Positive Dynamical Semigroups