We study a variant of Schumacher compression. The task asks for the minimal memory cost for compressing many copies of a density matrix with known spectrum and unknown eigenbasis, without preserving its purification. For fixed dimension and distinct positive eigenvalues, we obtain the cost through its additive constant. Achievability uses a generalization of Werner's cloning map to $\mathrm{GL}(d,\mathbb C)$ irreducible representations, with finite trace-distance bounds controlled by highest-weight differences and row gaps. The matching converse follows from a quantitative form of Koashi--Imoto incompressibility for irreducible group orbits under a spectral-gap assumption. We also identify the relation between this quantum minimum description length and universal lossless coding overhead, and its connection to free entropy is explained in a companion letter. We provide a Lean certificate for our proofs.
Nonstabilizerness (magic) is the quantum resource that, together with stabilizer operations, makes universal quantum computation possible. However, quantifying nonstabilizerness is difficult. The standard measure sums over exponentially many Pauli operators, and for quantum codes no quantitative theory has been availab...
We determine the optimal universal coefficient in a reduction of multiple to binary quantum hypothesis testing. Specifically, for every finite ensemble in any Hilbert space dimension (finite or infinite), we prove that the error probability of the global pretty good measurement (PGM) is at most four times the sum of th...
We study multi-armed bandits whose arms supply unknown quantum states and whose mean rewards are defined by a known effect $F$. Each fresh copy may be measured by an arbitrary, adaptively chosen POVM. We prove an exact collapse: the measured relative entropy from an arm state to its confusing reward half-space equals t...
We construct quantum codes from codewords of a cyclic code and the minors of a polynomial matrix. The construction uses a two-row, three-column matrix whose minors yield a second matrix satisfying the CSS orthogonality condition. We derive the code dimension from the greatest common divisors of the entries and minors,...
V. Nourozi, R. Smarandache, David G. M. Mitchell· 0 citations
This work constructs finite-block preprocessing schemes whose private information is an entropy difference evaluated at two distributions related by group convolution, and proves nonadditivity at an explicit channel parameter using a two-use encoder, an analytic single-use converse, and certified entropy bounds.
In free probability, Voiculescu's free entropy counts matrix microstates that approximate an operator. It is a non-commutative analog of the Shannon entropy, distinct from any quantum entropy of a density operator, but it has had no physical meaning in quantum theory. We define a finite-dimensional physical free entrop...
Patrick M. Hayden, A. Maloney, Jin-Zhao Wang et al.· 1 citation
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