Characterizing the Structure of Preserved Information in Quantum Processes

Abstract

We introduce a general operational characterization of information-preserving structures -encompassing noiseless subsystems, decoherence-free subspaces, pointer bases, and error-correcting codes-by demonstrating that they are isometric to fixed points of unital quantum processes. Using this, we show that every information-preserving structure is a matrix algebra. We further establish a structure theorem for the fixed states and observables of an arbitrary process, which unifies the Schrödinger and Heisenberg pictures, places restrictions on physically allowed kinds of information, and provides an efficient algorithm for finding all noiseless and unitarily noiseless subsystems of the process. DOI: 10.1103/PhysRevLett.100.030501 PACS numbers: 03.67.Pp, 03.65.Yz, 03.67.Lx, 89.70.ÿa Quantum processes, also known as quantum channels, quantum operations, or completely positive (CP) maps [1,2], are central to the theory and practice of quantum information processing (QIP). They describe how quantum states evolve over a period of time in the presence of noise, or how a device's output depends on its input. They are also complex and unwieldy: to fully specify a quantum process on a d-dimensional system requires d 4 real numbers. Most of these data are irrelevant to what one really wants to know: What information can pass unharmed through the process? Besides being central to QIP, a general answer is broadly relevant to both fundamental physics and quantum technologies, for the information-preserving degrees of freedom are precisely those that may be reliably characterized and exploited. Information-preserving structures (IPS) in quantum processes-what they are and how to find them -are the subject of this Letter.The quest for such structures has a long history in quantum physics. Pointer states (PS), defined in the context of quantum measurement theory, are ''most classical'' states that resist decoherence [3]. QIP science has spurred interest in the preservation of quantum information, leading to the notion of noiseless subsystems (NS) [4] as passive IPS that emerge from the existence of symmetries in the noise, and recover both decoherence-free subspaces (DFS) [5] and PS in special limits. Processes admitting no NS may still preserve information, which can be actively protected using quantum error correction (QEC) [6,7] to create an effective NS. Rapid experimental progress in implementing DFS [8], NS [9], and QEC [10] heightens the need for a complete and constructive characterization of preserved information.In this Letter, we formulate a general operational theory of IPS. The key insight is to identify preserved information with sets of states (or codes) whose mutual distinguishability is left unchanged. We prove that every preserved code can, through error correction, be made noiseless, then show that every maximal noiseless code is isometric [11] to the fixed-point set of the dynamics. This set, in turn, is isometric to a matrix algebra; thus, we conclude that every IPS is an algebra. Finally, we provide an ex...

Simple approach to approximate quantum error correction based on the transpose channel

“…R P defined here is exactly the case for the initial state P /d, where d is the dimension of C. In Ref. [23], R P was shown to be useful for correcting information carried by codes preserved according to an operationally motivated notion. The term transpose channel owes its origin to Ref.…”

Quantum and classical resources for unitary design of open-system evolutions

“…||X − Y || 1 is the quantum total-variation distance, which is a natural measure of distinguishability between quantum states [5,40,41]. Equation 7 is equivalent to requesting that the distance between the maps is small in the induced operator norm:…”

Pointer states via engineered dissipation

“…Mathematically (in a sense that will be made more precise later), this is only possible if the open-system Hamiltonian exhibits a sufficient degree of symmetry, which effectively allows PSs to be eigenstates (fixed points [7]) in the resulting system-plus-bath dilation. This has two implications: On the one hand, for a generic open quantum system, such a symmetry is not typical and at best approximate, thus PSs need not exist -with all the initial preparations of the system being rapidly degraded over comparable time scales.…”

“…As a result, PSs play a fundamental role in investigations of the quantum-to-classical transition and quantum measurement models [2], as well as of general aspects of "most classical" minimum-uncertainty states in quantum-dynamical systems [3][4][5][6]. In the context of quantum information processing, a set of mutually orthogonal PSs (a pointer basis) provides the simplest example of an "information-preserving structure" (IPS) [7]: since arbitrary convex mixtures of PSs are preserved, a pointer basis naturally realizes a robust classical memory. As a result, PSs are also practically attractive in view of their potential for long-lasting storage capabilities.…”

“…While the physical mechanism responsible for the observed initial-statesensitivity is different than in a PS-engineering protocol (stemming, in the experiment, from systematic pulse errors), we shall also address the emergence of such accidental PSs in DD sequences and show how they may be formally related within a unified control-theoretic framework. Second, we also iterate that DD has been invoked as a general strategy for generating IPSs in (nonMarkovian) open quantum systems -decoherence-free subspaces and noiseless subsystem [23,24], which can both be seen as multi-dimensional generalizations of PSs in an appropriate sense [7]. Again, a key difference is that such schemes are not tailored to preserve a target subspace or subsystem specified in advance.…”