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arXiv:2002.04657 · Physical Review A

Geometry of generalized Pauli channels

Katarzyna Siudzińska

Published Jun 18, 2020• 7 citations
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Abstract

We analyze the geometry of the generalized Pauli channels constructed from the mutually unbiased bases. The Choi-Jamio\l{}kowski isomorphism allows us to express the Hilbert-Schmidt line and volume elements in terms of the eigenvalues of the generalized Pauli maps. After determining appropriate regions of integration, we analytically compute the volume of generalized Pauli channels and their important subclasses. In particular, we obtain the volumes of the generalized Pauli channels that can be generated by a legitimate generator and are entanglement breaking. We also provide the upper bound for the volume of positive, trace-preserving generalized Pauli maps. This helps us to better understand the structure of high-dimensional sets of quantum channels.

DOI
10.1103/physreva.101.062323

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We analyze the geometry of the generalized Pauli channels constructed from the mutually unbiased bases. The Choi-Jamio\l{}kowski isomorphism allows us to express the Hilbert-Schmidt line and volume elements in terms of the eigenvalues of the generalized Pauli maps. After determining appropriate regions of integration, we analytically compute the volume of generalized Pauli channels and their important subclasses. In particular, we obtain the volumes of the generalized Pauli channels that can be generated by a legitimate generator and are entanglement breaking. We also provide the upper bound for the volume of positive, trace-preserving generalized Pauli maps. This helps us to better understand the structure of high-dimensional sets of quantum channels.
We analyze the geometrical properties of trace-preserving Pauli maps. Using the Choi-Jamio\l{}kowski isomorphism, we express the Hilbert-Schmidt line and volume elements in terms of the eigenvalues of the Pauli map. We analytically compute the volumes of positive, trace-preserving Pauli maps and Pauli channels. In particular, we find the relative volumes of the entanglement-breaking Pauli channels as well as the channels that can be generated by a time-local generator. Finally, we show what part of the Pauli channels are P\mathrm{P}P- and CP\mathrm{CP}CP-divisible, which is related to the notion of Markovianity.
We analyze the geometry of positive and completely positive, trace-preserving Pauli maps that are fully determined by up to two distinct parameters. This includes five classes of symmetric and noninvertible Pauli channels. Using the Hilbert-Schmidt metric in the space of the Choi-Jamio\l{}kowski states, we compute the relative volumes of entanglement breaking, time-local generated, and divisible channels. Finally, we find the shapes of the complete positivity regions in relation to the tetrahedron of all Pauli channels.
In the space of quantum channels, we establish the geometry that allows us to make statistical predictions about relative volumes of entanglement breaking channels among all the Gaussian quantum channels. The underlying metric is constructed using the Choi-Jamio\l{}kowski isomorphism between the continuous-variable Gaussian states and channels. This construction involves the Hilbert-Schmidt distance in quantum state space. The volume element of the one-mode Gaussian channels can be expressed in terms of local symplectic invariants. We analytically compute the relative volumes of the one-mode Gaussian entanglement breaking and incompatibility breaking channels. Finally, we show that, when given the purities of the Choi-Jamio\l{}kowski state of the channel, one can determine whether or not such channel is incompatibility breaking.
Provided a maximally entangled state vector in H⊗H{\mathcal{H}}\otimes{\mathcal{H}}H⊗H, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of CP(H,K){\mathsf CP}({\mathcal{H}},{\mathcal{K}})CP(H,K) (completely positive maps of density operators on H{\mathcal{H}}H tothose on K{\mathcal{K}}K) onto the density operators on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, which extends to a unitary map. Here, H{\mathcal{H}}H and K{\mathcal{K}}K are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both H{\mathcal{H}}H and K{\mathcal{K}}K which produces density operators on either L(H,K){\mathsf L}({\mathcal{H}},{\mathcal{K}})L(H,K) (Hilbert-Schmidt inner product) or K⊗H∗{\mathcal{K}}\otimes{\mathcal{H}}^*K⊗H∗ instead of K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H. In particular, no entangled state input is required. There are no such binatural maps carried on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, but those natural in only K{\mathcal{K}}K are canonically associated with unitary maps H→H∗{\mathcal{H}}\rightarrow{\mathcal{H}}^*H→H∗ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.