The Quantum Formalization Drive

A quantum channel is a completely positive, trace-preserving linear map between matrix algebras; it models the most general physical transformation of a quantum system. When a group $G$ acts on the input and output systems, a channel is $G$-covariant if it commutes with the group action. Covariant channels appear whenever a physical process respects a symmetry:…

Operators on $n$ copies of $\mathbb C^d$ that commute with every $U^{\otimes n}$, $U$ unitary, are exactly the linear combinations of permutation operators $V(\sigma)$, $\sigma\in S(n)$ (Schur–Weyl duality). Many protocols in quantum information carry a different symmetry: the first $n-1$ systems transform by $U$ and the last by the complex conjugate $\bar U$. It is the…

Operators made from a complex matrix and its conjugate occur in the non-chiral sector of gauge theories. Their correlation functions involve many possible pairings of matrix entries, and those pairings become difficult to organize as the number of entries grows. Kimura and Ramgoolam use the walled Brauer algebra to label these operators and to describe…

Positive linear maps on complex matrix algebras provide tests for quantum entanglement: applying such a map to one part of a bipartite state can expose a state that cannot be written as a mixture of product states. The distinction between ordinary positivity, positivity with an auxiliary matrix system, and complete positivity is therefore substantive.…

Given two Banach spaces $X$ and $Y$, we introduce and study a concept of norm-attainment in the space of nuclear operators $\mathcal{N}(X,Y)$ and in the projective tensor product space $X \widehat{\otimes}_\pi Y$. We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which…

We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by Koashi and Imoto, we show that such states will have the form of a so-called short quantum…

We present an algorithm for the explicit numerical calculation of SU(N) and SL(N,C) Clebsch-Gordan coefficients, based on the Gelfand-Tsetlin pattern calculus. Our algorithm is well-suited for numerical implementation; we include a computer code in an appendix. Our exposition presumes only familiarity with the representation theory of SU(2).

We describe a new method for approximating an arbitrary $n$ qubit unitary with precision $\varepsilon$ using a Clifford and T circuit with $O(4^{n}n(\log(1/\varepsilon)+n))$ gates. The method is based on rounding off a unitary to a unitary over the ring $\mathbb{Z}[i,1/\sqrt{2}]$ and employing exact synthesis. We also show that any $n$ qubit unitary over the ring …

A state on a tripartite quantum system $A \otimes B \otimes C$ forms a Markov chain if it can be reconstructed from its marginal on $A \otimes B$ by a quantum operation from $B$ to $B \otimes C$. We show that the quantum conditional mutual information $I(A: C | B)$ of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus…

We find a regularized formula for the entanglement-assisted (EA) capacity region for quantum multiple access channels (QMAC). We illustrate the capacity region calculation with the example of the collective phase-flip channel which admits a single-letter characterization. On the way, we provide a first-principles proof of the EA coding theorem based on a…

This paper is essentially a lecture from the author's course on quantum information theory, which is devoted to the result of C. H. Bennett, P. W. Shor, J. A. Smolin and A. V. Thapliyal (quant-ph/0106052) concerning entanglement-assisted classical capacity of a quantum channel. A modified proof of this result is given and relation between…

This paper explores the long journey from projective tensor products of a pair of Banach spaces, passing through the definition of nuclear operators still on the realm of projective tensor products, to the of notion of trace-class operators on a Hilbert space, and shows how and why these concepts (nuclear and trace-class operators, that is) agree in the end.

Collins and Popescu realized a powerful analogy between several resources in classical and quantum information theory. The Collins-Popescu analogy states that public classical communication, private classical communication, and secret key interact with one another somewhat similarly to the way that classical communication, quantum communication, and…

We define classical-quantum multiway channels for transmission of classical information, after recent work by Allahverdyan and Saakian. Bounds on the capacity region are derived in a uniform way, which are analogous to the classically known ones, simply replacing Shannon entropy with von Neumann entropy. For the single receiver case (multiple access…

The entanglement-assisted classical capacity of a noisy quantum channel is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel, assuming that the sender and receiver have access to the resource of shared quantum entanglement, which may be used up by the communication protocol. We show that…