Recently, the theory of quantum error control codes has been extended to subsystem codes over symmetric and asymmetric quantum channels -- qubit-flip and phase-shift errors may have equal or different probabilities. Previous work in constructing quantum error control codes has focused on code constructions for symmetric quantum channels. In this paper, we develop a theory and establish the connection between asymmetric quantum codes and subsystem codes. We present families of subsystem and asymmetric quantum codes derived, once again, from classical BCH and RS codes over finite fields. Particularly, we derive an interesting asymmetric and symmetric subsystem codes based on classical BCH codes with parameters $[[n,k,r,d]]_q$, $[[n,k,r,d_z/d_x]]_q$ and $[[n,k',0,d_z/d_x]]_q$ for arbitrary values of code lengths and dimensions. We establish asymmetric Singleton and Hamming bounds on asymmetric quantum and subsystem code parameters; and derive optimal asymmetric MDS subsystem codes. Finally, our constructions are well explained by an illustrative example.
This paper is written on the occasion of the 50th anniversary of the discovery of classical BCH codes and their quantum counterparts were derived nearly 10 years ago.
The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is…
In this paper we extend to asymmetric quantum error-correcting codes (AQECC) the construction methods, namely: puncturing, extending, expanding, direct sum and the (u|u + v) construction. By applying these methods, several families of asymmetric quantum codes can be constructed. Consequently, as an example of application of quantum code expansion developed here, new families of asymmetric quantum codes derived from generalized Reed-Muller (GRM) codes, quadratic residue (QR), Bose-Chaudhuri-Hocquenghem (BCH), character codes and affine-invariant codes are constructed.
It is recently conjectured in quantum information processing that phase-shift errors occur with high probability than qubit-flip errors, hence the former is more disturbing to quantum information than the later one. This leads us to construct asymmetric quantum error controlling codes to protect quantum information over asymmetric channels, $\Pr Z \geq \Pr X$. In this paper we present two generic methods to derive asymmetric quantum cyclic codes using the generator polynomials and defining sets of classical cyclic codes. Consequently, the methods allow us to construct several families of asymm…
Recently, quantum error-correcting codes were proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit flip and phase flip errors. An example for a channel which exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit flips and phase flips can be related to relaxation and dephasing time, respectively. We give systematic constructions of asymmetric quantum stabilizer codes that exploit this asymmetry. Our approach is based on a CSS construction that combines BCH …
It is conjectured that quantum computers are able to solve certain problems more quickly than any deterministic or probabilistic computer. A quantum computer exploits the rules of quantum mechanics to speed up computations. However, it is a formidable task to build a quantum computer, since the quantum mechanical systems storing the information unavoidably interact with their environment. Therefore, one has to mitigate the resulting noise and decoherence effects to avoid computational errors. In this work, I study various aspects of quantum error control codes -- the key component of fault-tol…
We present an analytical approach to construct the Lie algebra of finite-dimensional subsystems of the driven asymmetric top rotor. Each rotational level is degenerate due to the isotropy of space, and the degeneracy increases with rotational excitation. For a given rotational excitation, we determine the nested commutators between drift and drive Hamiltonians using a graph representation. We then generate the Lie algebra for subsystems with arbitrary rotational excitation using an inductive argument.
In many physical systems it is expected that environmental decoherence will exhibit an asymmetry between dephasing and relaxation that may result in qubits experiencing discrete phase errors more frequently than discrete bit errors. In the presence of such an error asymmetry, an appropriately asymmetric quantum code - that is, a code that can correct more phase errors than bit errors - will be more efficient than a traditional, symmetric quantum code. Here we construct fault tolerant circuits to convert between an asymmetric subsystem code and a symmetric subsystem code. We show that, for a mo…
The de Finetti representation theorem for continuous variable quantum system is first developed to approximate an N-partite continuous variable quantum state with a convex combination of independent and identical subsystems, which requires the original state to obey permutation symmetry conditioned on successful experimental verification on k of N subsystems. We generalize the de Finetti theorem to include asymmetric bounds on the variance of canonical observables and biased basis selection during the verification step. Our result thereby enables application of infinite-dimensional de Finetti …
Quantum discord-breaking channels were previously defined as the local channels that act on subsystem $A$ to produce classical-quantum states across system $AB$. However, unlike entanglement, discord is asymmetric. Here, we characterise the discord-breaking channels that act on subsystem $B$, thus completing the overall description of discord-breaking channels. We then introduce the notion of discord-annihilating channels, which act globally on system $AB$ to destroy quantum discord, and find their closed form. Discord-annihilating channels have clear operational description, involving subspac…
Spontaneous synchronisation is a collective phenomenon that can occur in both dynamical classical and quantum systems. Here, we analyse the spontaneous synchronisation dynamics of vibrations assisting energy transfer in a bio-inspired system. We find the emergence of a constant non-zero `synchronisation phase' between synchronised vibrational displacements as the natural frequencies of the oscillators are detuned. This phase difference arises from the asymmetric participation of local modes in the long-lived synchronised state. Furthermore, we investigate the relationships between the synchron…
P-divisibility conditions on subsystems in quantum dynamics
TCG Centres for Research and Education in Science and Technology (TCG CREST)（科学技术研究与教育中心）；Academy of Scientific and Innovative Research (AcSIR)（科学与创新研究学院）；Ramakrishna Mission Vivekananda Educational and Research Institute（罗摩克里希那使命维韦卡南达教育与研究所）
Composite quantum systems can be decomposed into subsystems in many different inequivalent ways. We call a particular decomposition a meronomic reference frame for the system. We apply the ideas of quantum reference frames to characterize meronomic frames, identify tasks that require such frames to accomplish, and show how asymmetric quantum states can be used to embody meronomic frame information.
Continuous variable entanglement can be produced in nonlinear systems or via interference of squeezed states. In many of optical systems, such as parametric down conversion or interference of optical squeezed states, production of two perfectly symmetric subsystems is usually used for demonstrating the existence of entanglement. This symmetry simplifies the description of the concept of entanglement. However, asymmetry in entanglement may arise naturally in a real experiment, or be intentionally introduced in a given quantum information protocol. These asymmetries can emerge from having the ou…
We propose a deterministic scheme for establishing hybrid Einstein-Podolsky-Rosen (EPR) entanglement channel between a macroscopic mechanical oscillator and a magnon mode in a distant yttrium-iron-garnet (YIG) sphere across about ten gigahertz of frequency difference. The system consists of a driven electromechanical cavity which is unidirectionally coupled to a distant electromagnonical cavity inside which a YIG sphere is placed. We find that far beyond the sideband-resolved regime in the electromechanical subsystem, stationary phonon-magnon EPR entanglement can be achieved. This is realized …
The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes t…
We address a scheme to enhance the quantum correlations in cavity opto-magnomechanical system by using the coherent feedback loop in the presence of magnon squeezing. The proposed coherent feedback-control allows a significant enhancement of the entanglement of three bipartite subsystems, i.e., photon-phonon, photon-magnon and phonon-magnon. We also study the Einstein-Podolsky-Rosen steering and one-way steering in the presence of thermal effects without imposing additional conditions of asymmetric losses or noises in the subsystems. Furthermore, we investigate the sensitiveness of the scheme …
The compass model on a square lattice provides a natural template for building subsystem stabilizer codes. The surface code and the Bacon-Shor code represent two extremes of possible codes depending on how many gauge qubits are fixed. We explore threshold behavior in this broad class of local codes by trading locality for asymmetry and gauge degrees of freedom for stabilizer syndrome information. We analyze these codes with asymmetric and spatially inhomogeneous Pauli noise in the code capacity and phenomenological models. In these idealized settings, we observe considerably higher thresholds …
Measurement-based quantum correlation mimics several characteristics of multipartite quantum correlations and at the same time, it reduces the parent system to a smaller subsystem. On the other hand, genuine multipartite entanglement measures can capture certain features of a multisite composite system that are inaccessible via bipartite quantum correlation quantifiers. We merge these two concepts by introducing localizable genuine multimode entanglement for continuous variable systems, both for Gaussian and non-Gaussian multimode parent states. We report a compact form of localizable generali…
Generation of quantum correlations between separate objects is of significance both in fundamental physics and in quantum networks. One important challenge is to create the directional "spooky action-at-a-distanc" effects that Schrödinger called "steering" between two macroscopic and massive objects. Here, we analyze a generic scheme for generating steering correlations in cascaded hybrid systems in which two distant oscillators with effective masses of opposite signs are coupled to a unidirectional light field, a setup which is known to build up quantum correlations by means of quantum back-a…
What is the quantum state of the universe? Although there have been several interesting suggestions, the question remains open. In this paper, I consider a natural choice for the universal quantum state arising from the Past Hypothesis, a boundary condition that accounts for the time-asymmetry of the universe. The natural choice is given not by a wave function (representing a pure state) but by a density matrix (representing a mixed state). I begin by classifying quantum theories into two types: theories with a fundamental wave function and theories with a fundamental density matrix. The Past …
In a quantum universe with a strong arrow of time, we postulate a low-entropy boundary condition (the Past Hypothesis) to account for the temporal asymmetry. In this paper, I show that the Past Hypothesis also contains enough information to simplify the quantum ontology and define a natural initial condition. First, I introduce Density Matrix Realism, the thesis that the quantum state of the universe is objective and impure. This stands in sharp contrast to Wave Function Realism, the thesis that the quantum state of the universe is objective and pure. Second, I suggest that the Past Hypothesis…
Error probability distribution associated with a given Clifford measurement circuit is described exactly in terms of the circuit error-equivalence group, or the circuit subsystem code previously introduced by Bacon, Flammia, Harrow, and Shi. This gives a prescription for maximum-likelihood decoding with a given measurement circuit. Marginal distributions for subsets of circuit errors are also analyzed; these generate a family of related asymmetric LDPC codes of varying degeneracy. More generally, such a family is associated with any quantum code. Implications for decoding highly-degenerate qua…
We present a fault-tolerant universal gate set consisting of Hadamard and controlled-controlled-Z (CCZ) on Bacon-Shor subsystem codes. Transversal non-Clifford gates on these codes are intriguing in that higher levels of the Clifford hierarchy become accessible as the code becomes more asymmetric. For instance, in an appropriate gauge, Bacon-Shor codes on an $m\times m^k$ lattice have transversal $k$-qubit-controlled $Z$. Through a variety of tricks, including intermediate error-correction and non-Pauli recovery, we reduce the overhead required for fault-tolerant CCZ. We calculate pseudothresh…
