Ashikhmin and Litsyn showed that all binary stabilizer codes - pure or impure - of sufficiently large length obey the quantum Hamming bound, ruling out the possibility that impure codes of large length can outperform pure codes with respect to sphere packing. In contrast we show that impure subsystem codes do not obey the quantum Hamming bound for pure subsystem codes, not even asymptotically. We show that there exist arbitrarily long Bacon-Shor codes that violate the quantum Hamming bound.
Subsystem codes are a generalization of noiseless subsystems, decoherence free subspaces, and quantum error-correcting codes. We prove a Singleton bound for GF(q)-linear subsystem codes. It follows that no subsystem code over a prime field can beat the Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the Hamming bound. A number of open problems concern the comparison in performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements…
Free fermions on Hamming graphs $H(d,q)$ are considered and the entanglement entropy for two types of subsystems is computed. For subsets of vertices that form Hamming subgraphs, an analytical expression is obtained. For subsets corresponding to a neighborhood, i.e. to a set of sites at a fixed distance from a reference vertex, a decomposition in irreducible submodules of the Terwilliger algebra of $H(d,q)$ also yields a closed formula for the entanglement entropy. Finally, for subsystems made out of multiple neighborhoods, it is shown how to construct a block-tridiagonal operator which commut…
We provide a generalization for the polygamy constraint of multiparty entanglement in arbitrary dimensional quantum systems. By using the $β$th-power of entanglement of assistance for $0\leq β\leq1$ and the Hamming weight of the binary vector related with the distribution of subsystems, we establish a class of weighted polygamy inequalities of multiparty entanglement in arbitrary dimensional quantum systems. We further show that our class of weighted polygamy inequalities can even be improved to be tighter inequalities with some conditions on the assisted entanglement of bipartite subsystems.
Quantum error correction requires the use of error syndromes derived from measurements that may be unreliable. Recently, quantum data-syndrome (QDS) codes have been proposed as a possible approach to protect against both data and syndrome errors, in which a set of linearly dependent stabilizer measurements are performed to increase redundancy. Motivated by wanting to reduce the total number of measurements performed, we introduce QDS subsystem codes, and show that they can outperform similar QDS stabilizer codes derived from them. We also give a construction of single-error-correcting QDS stab…
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. Particu…
Entanglement of pure states of bipartite quantum systems has been shown to have a unique measure in terms of the von Neumann entropy of the reduced states of either of its subsystems. The measure is established under entanglement manipulation of an asymptotically large number of copies of the states. In this paper, different asymptotic measures of entanglement assigned to arbitrary sequences of bipartite pure states are shown to coincide only when the sequence is information stable, in terms of the quantum spectral information rates of the sequence of subsystem states. Additional bounds on the…
The composite systems can be non-uniquely decomposed into parts (subsystems). Not all decompositions (structures) of a composite system are equally physically relevant. In this paper we answer on theoretical ground why it may be so. We consider a pair of mutually un-coupled modes in the phase space representation that are subjected to the independent quantum amplitude damping channels. By investigating asymptotic dynamics of the degrees of freedom, we find that the environment is responsible for the structures non-equivalence. Only one structure is distinguished by both locality of the environ…
State redistribution is the protocol in which, given an arbitrary tripartite quantum state, with two of the subsystems initially being with Alice and one being with Bob, the goal is for Alice to send one of her subsystems to Bob, possibly with the help of prior shared entanglement. We derive an upper bound on the second order asymptotic expansion for the quantum communication cost of achieving state redistribution with a given finite accuracy. In proving our result, we also obtain an upper bound on the quantum communication cost of this protocol in the one-shot setting, by using the protocol o…
Given any two classical codes with parameters $[n_1,k,d_1]$ and $[n_2,k,d_2]$, we show how to construct a quantum subsystem code in 2-dimensions with parameters $[[N,K,D]]$ satisfying $N\le 2n_1n_2$, $K=k$, and $D=\min(d_1,d_2)$. These quantum codes are in the class of generalized Bacon-Shor codes introduced by Bravyi. We note that constructions of good classical codes can be used to construct quantum codes that saturate Bravyi's bound $KD=O(N)$ on the code parameters of 2-dimensional subsystem codes. One of these good constructions uses classical expander codes. This construction has the addi…
We establish contact between the delocalization properties of pure quantum states, as quantified by their number of principal components, and the average generalized entanglement properties, as quantified by purity measures relative to different observable sets. We find that correlations between products of state vector components with respect to Hamming distance play an important role in the structure of subsystem-based purity measures. In particular, we derive general conditions under which the amount of global multipartite entanglement relates to the inverse participation ratio averaged ove…
Preparation of Hamming-Weight-Preserving Quantum States with Log-Depth Quantum Circuits
State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所处理器重点实验室）；School of Computer Science and Technology, University of Chinese Academy of Sciences（中国科学院大学计算机科学与技术学院）；Shandong University（山东大学）
We introduce the concept of generalized concatenated quantum codes. This generalized concatenation method provides a systematical way for constructing good quantum codes, both stabilizer codes and nonadditive codes. Using this method, we construct families of new single-error-correcting nonadditive quantum codes, in both binary and nonbinary cases, which not only outperform any stabilizer codes for finite block length, but also asymptotically achieve the quantum Hamming bound for large block length.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we solve in the asymptotic long-time regime the master equation for two independent harmonic oscillators interacting with an environment. We give a description of the continuous-variable asymptotic entanglement in terms of the covariance matrix of the considered subsystem for an arbitrary Gaussian input state. Using Peres--Simon necessary and sufficient condition for separability of two-mode Gaussian states, we show that for certain classes of environments the initial state evolves asympto…
Entanglement purification takes a number of noisy EPR pairs and processes them to produce a smaller number of more reliable pairs. If this is done with only a forward classical side channel, the procedure is equivalent to using a quantum error-correcting code (QECC). We instead investigate entanglement purification protocols with two-way classical side channels (2-EPPs) for finite block sizes. In particular, we consider the analog of the minimum distance problem for QECCs, and show that 2-EPPs can exceed the quantum Hamming bound and the quantum Singleton bound. We also show that 2-EPPs can ac…
We derive upper bounds on the quantum Fisher information in interferometry with $N$ subsystems, e.g. two-level atoms or Gaussian modes, in the presence of arbitrarily correlated Gaussian dephasing including independent and collective dephasing. The derived upper bound enables us to analyse the Fisher information asymptotic behaviour when $N\rightarrow\infty$. Dephasing introduces random phases to subsystems dynamics, which lowers the precision of estimating the phase difference $ϕ$ in an interferometer. The method presented uses Bayesian estimation of the random phases and eliminates dephasing…
We study the temporal evolution of the circuit complexity for a subsystem in harmonic lattices after a global quantum quench of the mass parameter, choosing the initial reduced density matrix as the reference state. Upper and lower bounds are derived for the temporal evolution of the complexity for the entire system. The subsystem complexity is evaluated by employing the Fisher information geometry for the covariance matrices. We discuss numerical results for the temporal evolutions of the subsystem complexity for a block of consecutive sites in harmonic chains with either periodic or Dirichle…
We consider a composite open quantum system consisting of a fast subsystem coupled to a slow one. Using the time-scale separation, we develop an adiabatic elimination technique to derive at any order the reduced model describing the slow subsystem. The method, based on an asymptotic expansion and geometric singular perturbation theory, ensures the physical interpretation of the reduced second-order model by giving the reduced dynamics in a Lindblad form and the state reduction in Kraus map form. We give explicit second-order formulas for Hamiltonian or cascade coupling between the two subsyste…
We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order extension of the well-known Coppersmith-Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to ques…
We study the von Neumann entropy asymptotics of pure translation-invariant quasi-free states of d-dimensional fermionic systems. It is shown that the entropic area law is violated by all these states: apart from the trivial cases, the entropy of a cubic subsystem with edge length L cannot grow slower than L^{d-1}ln L. As for the upper bound of the entropy asymptotics, the zero-entropy-density property of these pure states is the only limit: it is proven that arbitrary fast sub-L^d entropy growth is achievable.
We exploit the knowledge of the entanglement spectrum in the ground state of the gapped XXZ spin chain to derive asymptotic exact results for the full counting statistics of the transverse magnetisation in a spin block of length. We found that for a subsystem of even length the full counting statistics is Gaussian, while for odd subsystems it is the sum of two Gaussian distributions. We test our analytic predictions with accurate tensor networks simulations. As a byproduct, we also obtain the symmetry (magnetisation) resolved entanglement entropies.
We consider one-dimensional quantum spin chain, which is called XX model, XX0 model or isotropic XY model in a transverse magnetic field. We study the model on the infinite lattice at zero temperature. We are interested in the entropy of a subsystem [a block of L neighboring spins]. It describes entanglement of the block with the rest of the ground state. G. Vidal, J.I. Latorre, E. Rico, and A. Kitaev showed that for large blocks the entropy scales logarithmically. We prove the logarithmic formula for the leading term and calculate the next term. We discovered that the dependence on the magnet…
The Schrödinger-Langevin (SL) equation is considered as an effective open quantum system formalism suitable for phenomenological applications involving a quantum subsystem interacting with a thermal bath. We focus on two open issues relative to its solutions: the stationarity of the excited states of the non-interacting subsystem when one considers the dissipation only and the thermal relaxation toward asymptotic distributions with the additional stochastic term. We first show that a proper application of the Madelung/polar transformation of the wave function leads to a non zero damping of the…
