The Quantum Formalization Drive

This paper defines the squashed entanglement of a quantum channel as the maximum squashed entanglement that can be registered by a sender and receiver at the input and output of a quantum channel, respectively. A new subadditivity inequality for the original squashed entanglement measure of Christandl and Winter leads to the conclusion that the squashed…

Let $(G,\theta)$ be a Banach--Lie group with involutive automorphism $\theta$, $\g = \fh \oplus \fq$ be the $\theta$-eigenspaces in the Lie algebra $\g$ of $G$, and $H = (G^\theta)_0$ be the identity component of its group of fixed points. An Olshanski semigroup is a semigroup $S \subeq G$ of the form $S = H \exp(W)$, where $W$ is an open $\Ad(H)$-invariant convex cone in $\fq$ and…

The capacity of a classical-quantum channel (or in other words the classical capacity of a quantum channel) is considered in the most general setting, where no structural assumptions such as the stationary memoryless property are made on a channel. A capacity formula as well as a characterization of the strong converse property is given just in parallel…

Positive maps which are not completely positive are used in quantum information theory as witnesses for convex sets of states, in particular as entanglement witnesses and more generally as witnesses for states having Schmidt number not greater than k. It is known that such witnesses are related to k-positive maps. In this article we propose a new proof for…

We derive analytical expressions for the single mode quantum field state at the individual output ports of a beam splitter when a single-photon Fock state and a coherent state are incident on the input ports. The output states turn out to be a statistical mixture between a displaced Fock state and a coherent state. Consequently we are able to find an…

We consider two properties of the set of quantum states as a convex topological space and some their implications concerning the notions of a convex hull and of a convex roof of a function defined on a subset of quantum states. By using these results we analyze two infinite-dimensional versions (discrete and continuous) of the convex roof construction of…

A brief proof of the statement that the zero-set of a nontrivial real-analytic function in $d$-dimensional space has zero measure is provided.

A generic quantum channel can be represented in terms of a unitary interaction between the information-carrying system and a noisy environment. Here, the minimal number of quantum Gaussian environmental modes required to provide a unitary dilation of a multi-mode bosonic Gaussian channel is analyzed both for mixed and pure environment corresponding to the…

We develop a theory of exact transmission of finite-dimensional $C^*$-algebras through quantum channels. These algebras describe hybrid classical-quantum information, allowing the dimension of the quantum system to depend on the classical message. Allowing arbitrary encodings and decodings yields a transmission set of algebra types, ordered by embedding, that…

A computable measure of entanglement establishes the selected mathematical guarantees below. The mission turns those original paper results into precise Lean proof targets.

Practical Quantum Error Mitigation for Near-Future Applications.

General Entanglement Breaking Channels establishes the selected mathematical guarantees below. The mission turns those original paper results into precise Lean proof targets.

Generalized Quantum Signal Processing establishes the selected mathematical guarantees below. The mission turns those original paper results into precise Lean proof targets.

Separability Criterion for Density Matrices establishes the selected mathematical guarantees below. The mission turns those original paper results into precise Lean proof targets.

Reversing quantum dynamics with near-optimal quantum and classical fidelity (Barnum, Knill) establishes the selected mathematical guarantees below. The mission turns those original paper results into precise Lean proof targets.

Potential-Function Proofs for First-Order Methods (Bansal, Gupta) establishes the selected mathematical guarantees below. The mission turns those original paper results into precise Lean proof targets.

Quantum Amplitude Amplification and Estimation (Brassard, Høyer, Mosca, Tapp) establishes the selected mathematical guarantees below. The mission turns those original paper results into precise Lean proof targets.