One-Shot Randomized and Nonrandomized Partial Decoupling

Abstract

We introduce a task that we call partial decoupling, in which a bipartite quantum state is transformed by a unitary operation on one of the two subsystems and then is subject to the action of a quantum channel. We assume that the subsystem is decomposed into a direct-sum-product form, which often appears in the context of quantum information theory. The unitary is chosen at random from the set of unitaries having a simple form under the decomposition. The goal of the task is to make the final state, for typical choices of the unitary, close to the averaged final state over the unitaries. We consider a one-shot scenario, and derive upper and lower bounds on the average distance between the two states. The bounds are represented simply in terms of smooth conditional entropies of quantum states involving the initial state, the channel and the decomposition. Thereby we provide generalizations of the one-shot decoupling theorem. The obtained result would lead to further development of the decoupling approaches in quantum information theory and fundamental physics. I. INTRODUCTIONDecoupling refers to the fact that we may destroy correlation between two quantum systems by applying an operation on one of the two subsystems. It has played significant roles in the development of quantum Shannon theory for a decade, particularly in proving the quantum capacity theorem [1], unifying various quantum coding theorems [2], analyzing a multipartite quantum communication task [3,4] and in quantifying correlations in quantum states [5]. It has also been applied to various fields of physics, such as the black hole information paradox [6], quantum many-body systems [7] and quantum thermodynamics [8,9]. Dupuis et al.[10] provided one of the most general formulations of decoupling, which is often referred to as the decoupling theorem. The decoupling approach simplifies many problems of our interest, particularly when combined with the fact that any purification of a mixed quantum state is convertible to another reversibly [11].All the above studies rely on the notion of random unitary, i.e., unitaries drawn at random from the set of all unitaries acting on the system, which leads to the full randomization over the whole Hilbert space. In various situations, however, the full randomization is a too strong demand. In the context of communication theory, for example, the full randomization leads to reliable transmission of quantum information, while we may be interested in sending classical information at the same time [12], for which the full randomization is more than necessary. In the context of quantum many-body physics, the random process caused by the complexity of dynamics is in general restricted by symmetry, and thus no randomization occurs among different values of conserved quantities. Hence, in order that the random-unitary-based method fits into broader context in quantum information theory and fundamental physics, it would be desirable to generalize the arXiv:1903.05796v2 [quant-ph] 24 Jul 2019 D. Choi-Jamiolkowski represen...

Near-optimal covariant quantum error-correcting codes from random unitaries with symmetries

“…In our analysis, the Haar randomness has only been used in the partial decoupling theorem, and as was noted in Ref. [33], symmetric 2-designs are sufficient for the partial decoupling bounds to hold. Therefore, all our bounds hold for symmetric 2-designs.…”

“…A generalization of decoupling called partial decoupling that is useful for our purpose was studied in Ref. [33], where the unitary U A exhibits further structures. More specifically, we assume that the Hilbert space of A takes the form of a direct-sum-product decomposition…”

“…Let l j and r j be the dimensions of H A l j and H Ar j respectively. The (non-smoothed) partial decoupling theorem [33,Eq. (84)] states that…”

“…We first restate the partial decoupling theorem adapted to our structure of the space, which corresponds to l j = 1 and r j = n j for all 0 ≤ j ≤ n in Ref. [33]. We denote by Π j the projector onto the subspace with Hamming weight j.…”

“…To do so, we use the complementary channel technique [32], which allows one to characterize the error rate of a code by the amount of information leaked into the environment. At a high level, our derivation of the code errors for both U (1) and SU (d) is essentially based on breaking down the error into two components, one characterizing the deviation of the physical state from its average (over the randomness in the encoding) leading to an error which can be bounded using a "partial decoupling" theorem [33] and turns out to be exponentially small, while the other characterizing a polynomially small intrinsic error induced by this average physical state. We show that our random codes almost always saturate certain lower bounds to leading order (up to constant factors, in certain cases exactly), indicating that the symmetric unitaries typically give rise to nearly optimal covariant codes.…”