Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct several families of QUENTA codes via the Euclidean and the Hermitian construction. Two of the families created have maximal entanglement and have quantum Singleton defect equal to zero or one. Comparing the other families with the codes with the respective quantum Gilbert-Varshamov bound, we show that our codes have a rate that surpasses that bound. At the end, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of maximal entanglement QUENTA codes. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that using our construction it is possible to create an asymptotically family of QUENTA codes that exceeds this bound.
We show how to protect a stream of quantum information from decoherence induced by a noisy quantum communication channel. We exploit preshared entanglement and a convolutional coding structure to develop a theory of entanglement-assisted quantum convolutional coding. Our construction produces a Calderbank-Shor-Steane (CSS) entanglement-assisted quantum convolutional code from two arbitrary classical binary convolutional codes. The rate and error-correcting properties of the classical convolutional codes directly determine the corresponding properties of the resulting entanglement-assisted quan…
Brun, Devetak, and Hsieh [Science 314, 436 (2006)] demonstrated that pre-shared entanglement between sender and receiver enables quantum communication protocols that have better parameters than schemes without the assistance of entanglement. Subsequently, the same authors derived a version of the so-called quantum Singleton bound that relates the parameters of the entanglement-assisted quantum-error correcting codes proposed by them. We present a new entanglement-assisted quantum communication scheme with parameters violating this bound in certain ranges.
We consider classical message transmission under entanglement assistance for compound memoryless and arbitrarily varying quantum channels. In both cases, we prove general coding theorems together with corresponding weak converse bounds. In this way, we obtain single-letter characterizations of the entanglement-assisted classical capacities for both channel models. Moreover, we show that the entanglement-assisted classical capacity does exhibit no strong converse property for some compound quantum channels for the average as well as the maximal error criterion. A strong converse to the entangle…
The entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum MDS (EAQMDS) codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. We show that these EAQMDS codes have much larger minimum distance than the standard quantum…
Entanglement-assisted communication over a random-parameter quantum channel with either causal or non-causal channel side information (CSI) at the encoder is considered. This describes a scenario where the quantum channel depends on the quantum state of the input environment. While Bob, the decoder, has no access to this state, Alice, the transmitter, performs a sequence of projective measurements on her environment and encodes her message accordingly. Dupuis established the entanglement-assisted capacity with non-causal CSI. Here, we establish characterization in the causal setting, and also …
We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglement-assisted quantum error-correcting codes (EAQECCs) over the binary field hold for codes over arbitrary finite fields as well. We also give a Gilbert-Varshamov bound for EAQECCs and constructions of EAQECCs coming from punctured self-orthogonal linear codes which are valid for any finite field.
The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is defined from the orthogonal group of a simplified stabilizer group. From the Poisson summation formula, this duality leads to the MacWilliams identities and linear programming bounds for EAQEC codes. We establish a table of upper and lower bounds on the minimum distance of any maximal-entanglement EAQEC code with length up to 15 channel qubits.
Entanglement-assisted quantum error-correcting codes (EAQECCs) to desired rate, error-correcting capability and maximum shared entanglement are constructed. Thus for a required rate $R$, required error-correcting capability to correct $t$ errors, mds (maximum distance separable) EAQECCs of the form $[[n,r,d;c]]$ with $R=\frac{r}{n}, d\geq (2t+1), c = (n-r), d= (n-r+1)$ are constructed. Series of such codes may be constructed where the rate and the relative distance approach non-zero constants as $n$ approaches infinity. The codes may also be constructed over prime order fields in which modular…
We propose and prove an existential theorem for entanglement-assisted asymmetric quantum error correction. Then we demonstrate its superiority over the conventional one.
The entanglement-assisted stabilizer formalism can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs). In this work, we construct some new entanglement-assisted quantum MDS (EAQMDS) codes with length $n=\frac{q^2+1}5$ from cyclic codes. Compared with all the previously known parameters with the same length, all of them have flexible parameters and larger minimum distance.
The unique and often-weird properties of quantum mechanics allow an information carrier to propagate through multiple trajectories of quantum channels simultaneously. This ultimately leads us to quantum trajectories with an indefinite causal order of quantum channels. It has been shown that indefinite causal order enables the violation of bottleneck capacity, which bounds the amount of the transferable classical and quantum information through a classical trajectory with a well-defined causal order of quantum channels. In this treatise, we investigate this beneficial property in the realm of b…
Bennett et al. showed that allowing shared entanglement between a sender and receiver before communication begins dramatically simplifies the theory of quantum channels, and these results suggest that it would be worthwhile to study other scenarios for entanglement-assisted classical communication. In this vein, the present paper makes several contributions to the theory of entanglement-assisted classical communication. First, we rephrase the Giovannetti-Lloyd-Maccone sequential decoding argument as a more general "packing lemma" and show that it gives an alternate way of achieving the entangl…
Entanglement-assisted quantum (QUENTA) codes are a subclass of quantum error-correcting codes which use entanglement as a resource. These codes can provide error correction capability higher than the codes derived from the traditional stabilizer formalism. In this paper, it is shown a general method to construct QUENTA codes from cyclic codes. Afterwards, the method is applied to Reed-Solomon codes, BCH codes, and general cyclic codes. We use the Euclidean and Hermitian construction of QUENTA codes. Two families of QUENTA codes are maximal distance separable (MDS), and one is almost MDS or alm…
Recently, entanglement-assisted quantum error correcting codes (EAQECCs) have been constructed by cyclic codes and negacyclic codes. In this paper, by analyzing the cyclotomic cosets in the defining set of constacyclic codes, we constructed three classes of new EAQECCs which satisfy the entanglement-assisted quantum Singleton bound. Besides, three classes of EAQECCs with maximal entanglement from constacyclic codes are constructed in the meanwhile.
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…
Construction of quantum codes and entanglement-assisted quantum codes with good parameters via classical codes is an important task for quantum computing and quantum information. In this paper, by a family of one-generator quasi-cyclic codes, we provide quasi-cyclic extended constructions that preserve the self-orthogonality to obtain stabilizer quantum codes. As for the computational results, some binary and ternary stabilizer codes with good parameters are constructed. Moreover, we present methods to construct maximal-entanglement entanglement-assisted quantum codes by means of the class of …
Construction of good quantum codes via classical codes is an important task for quantum information and quantum computing. In this work, by virtue of a decomposition of the defining set of constacyclic codes we have constructed eight new classes of entanglement-assisted quantum maximum distance separable codes.
This paper develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes (EAQECCs) with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error correction performance, high rates, and low decoding complexity. The proposed method produces infinitely many new codes with a wide variety of parameters and entanglement requirements. Our framework encompass…
The zero-error capacity of a classical channel is expressed in terms of the independence number of some graph and its tensor powers. This quantity is hard to compute even for small graphs such as the cycle of length seven, so upper bounds such as the Lovasz theta function play an important role in zero-error communication. In this paper, we show that the Lovasz theta function is an upper bound on the zero-error capacity even in the presence of entanglement between the sender and receiver.
The fully quantum reverse Shannon theorem establishes the optimal rate of noiseless classical communication required for simulating the action of many instances of a noisy quantum channel on an arbitrary input state, while also allowing for an arbitrary amount of shared entanglement of an arbitrary form. Turning this theorem around establishes a strong converse for the entanglement-assisted classical capacity of any quantum channel. This paper proves the strong converse for entanglement-assisted capacity by a completely different approach and identifies a bound on the strong converse exponent …
Motivated by the problem of designing quantum repeaters, we study entanglement distillation between two parties, Alice and Bob, starting from a mixed state and with the help of "repeater" stations. To treat the case of a single repeater, we extend the notion of entanglement of assistance to arbitrary mixed tripartite states and exhibit a protocol, based on a random coding strategy, for extracting pure entanglement. The rates achievable by this protocol formally resemble those achievable if the repeater station could merge its state to one of Alice and Bob even when such merging is impossible. …
The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is the code resulting from exchanging the original code's information qubits with its ebits. To introduce this notion, we show how entanglement-assisted (EA) repetition codes and accumulator codes are dual to each other, much like their classical counterparts, and we give an explicit, general quantum shift-register circuit that encodes both classes of codes.We later show that our constructions are optimal, and this result completes our understanding of these dual classes of codes. We also establish the Gilbert-Varshamov…
Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard quantum error-correcting codes, which can only be constructed from dual-containing codes. However, the parameter of ebits $c$ is usually calculated by computer search. In this paper, we construct four classes of MDS entanglement-assisted quantum error-correcting codes (MDS EAQECCs) based on $k$-Galois LCD MDS codes for some certain code lengths,…
In this paper, we produce two new classes of entanglement-assisted quantum MDS codes (EAQMDS codes) with length $n|q^2-1$ and $n|q^2+1$ via cyclic codes over finite fields of odd characteristic. Among our constructions there are many EAQMDS codes with new parameters which have never been reported. And some of them have great larger minimum distance than known results.
We solve the entanglement-assisted (EA) classical capacity region of quantum multiple-access channels with an arbitrary number of senders. As an example, we consider the bosonic thermal-loss multiple-access channel and solve the one-shot capacity region enabled by an entanglement source composed of sender-receiver pairwise two-mode squeezed vacuum states. The EA capacity region is strictly larger than the capacity region without entanglement-assistance. With two-mode squeezed vacuum states as the source and phase modulation as the encoding, we also design practical receiver protocols to realiz…
In this paper, we produce some new classes of entanglement-assisted quantum MDS codes(EAQMDS codes for short) via generalized Reed-Solomon codes over finite fields of odd characteristic. Among our constructions, there are many EAQMDS codes with new lengths which have never been reported. Moreover, some of them have larger minimum distance than known results.
The entanglement-assisted (EA) formalism allows arbitrary classical linear codes to transform into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this work, we propose a decomposition of the defining set of constacyclic codes. Using this method, we construct four classes of $q$-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical constacyclic MDS codes by exploiting less pre-shared maximally entangled states. We show that a class of $q$-ary EAQMDS have minimum distance upper limit greate…
Entanglement-assisted quantum error correcting codes (EAQECCs) play a significant role in protecting quantum information from decoherence and quantum noise. Recently, constructing entanglement-assisted quantum maximum distance separable (EAQMDS) codes with flexible parameters has received much attention. In this work, four families of EAQMDS codes with a more general length are presented. And the method of selecting defining set is different from others. Compared with all the previously known results, the EAQMDS codes we constructed have larger minimum distance. All of these EAQMDS codes are n…
We give a capacity formula for the classical communication over a noisy quantum channel, when local operations and global permutations allowed in the encoding and bipartite states preshared between the sender and the receiver. The two endpoints of this formula are the Holevo capacity (without entanglement assistance) and the entanglement-assisted capacity (with unlimited entanglement assistance). What's more, we show that the capacity satisfies the strong converse property and thus the formula serves as a sharp dividing line between achievable and unachievable rates of communication. We prove …
This paper proposes new propagation rules on quantum codes in the entanglement-assisted and in quantum subsystem scenarios. The rules lead to new families of such quantum codes whose parameters are demonstrably optimal. To obtain the results, we devise tools to puncture and shorten codes in ways that ensure their Hermitian hulls have certain desirable properties. More specifically, we give a general framework to construct $k$-dimensional generalized Reed-Solomon codes whose Hermitian hulls are $(k-1)$-dimensional maximum distance separable codes.
Superadditivity for Entanglement-Assisted Communication
National Taiwan University（台湾大学）；National Center for Theoretical Sciences（国家理论科学中心）；Hon Hai (Foxconn) Quantum Computing Center（鸿海（富士康）量子计算中心）；RWTH Aachen University（亚琛工业大学）
Entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and fundamental class of codes. They allow for the construction of quantum codes from classical codes by relaxing the duality condition and using pre-shared entanglement between the sender and receiver. However, in general it is not easy to determine the number of shared pairs required to construct an EAQECC. In this paper, we show that this number is related to the hull of the classical code. Using this fact, we give methods to construct EAQECCs requiring desirable amount of entanglement. This leads to design families…
Ask how the quantum compression of ensembles of pure states is affected by the availability of entanglement, and in settings where the encoder has access to side information. We find the optimal asymptotic quantum rate and the optimal tradeoff (rate region) of quantum and entanglement rates. It turns out that the amount by which the quantum rate beats the Schumacher limit, the entropy of the source, is precisely half the entropy of classical information that can be extracted from the source and side information states without disturbing them at all ("reversible extraction of classical informat…
We establish the existence of optimal maximal entanglement entanglement-assisted quantum $[[n,k,d;n-k]]_2$ codes for $(n,k,d)=(14,6,7)$, $(15,7,7)$, $(17,6,9)$, $(17,7,8)$, $(19,7,9)$ and $(20,7,10)$. These codes are obtained from quaternary Hermitian linear complementary dual codes. We also give some observation on the largest minimum weights.
We study entanglement-assisted quantum error-correcting codes (EAQECCs) arising from classical one-point algebraic geometry codes from the Hermitian curve with respect to the Hermitian inner product. Their only unknown parameter is $c$, the number of required maximally entangled quantum states since the Hermitian dual of an AG code is unknown. In this article, we present an efficient algorithmic approach for computing $c$ for this family of EAQECCs. As a result, this algorithm allows us to provide EAQECCs with excellent parameters over any field size.
Entanglement-assisted quantum error correcting codes (EAQECCs) can be derived from arbitrary classical linear codes. However, it is a very difficult task to determine the number of entangled states required. In this work, using the method of the decomposition of the defining set of constacyclic codes, we construct two families of q-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical constacyclic MDS codes by exploiting less pre-shared maximally entangled states. We show that a class of q-ary EAQMDS have minimum distance upper bound greater than q. Some of them have much lar…
Entanglement-assisted quantum error correcting codes (EAQECCs) constructed from Reed-Solomon codes and BCH codes are considered in this work. It is provided a complete and explicit formula for the parameters of EAQECCs coming from any Reed-Solomon code, for the Hermitian metric, and from any BCH code with extension degree $2$ and consecutive cyclotomic cosets, for both the Euclidean and the Hermitian metric. The main task in this work is the computation of a completely general formula for $c$, the minimum number of required maximally entangled quantum states.
Recently, Galindo et al. introduced the concept of asymmetric entanglement-assisted quantum error-correcting codes (AEAQECCs) from Calderbank-Shor-Steane (CSS) construction. In general, it's difficult to determine the required number of maximally entangled states of an AEAQECC, which is associated with the dimension of the intersection of the two corresponding linear codes. Two linear codes are said to be a linear l-intersection pair if their intersection has dimension l. In this paper, all possible linear l-intersection pairs of MDS codes are given. As an application, we give a complete chara…
We study universal quantum codes for entanglement-assisted quantum communication over compound quantum channels. In this setting, sender and receiver do not know the specific channel that will be used for communication, but only know the set that the channel is selected from. We investigate different variations of the problem: uninformed users, informed receiver, informed sender, and feedback assistance. We derive single-letter formulas for all corresponding channel capacities. Our proofs are based on one-shot decoupling bounds and properties of smooth entropies.
With entanglement-assisted (EA) formalism, arbitrary classical linear codes are allowed to transform into EAQECCs by using pre-shared entanglement between the sender and the receiver. In this paper, based on classical cyclic MDS codes by exploiting pre-shared maximally entangled states, we construct two families of $q$-ary entanglement-assisted quantum MDS codes $[[\frac{q^{2}+1}{a},\frac{q^{2}+1}{a}-2(d-1)+c,d;c]]$, where q is a prime power in the form of $am+l$, and $a=(l^2+1)$ or $a=\frac{(l^2+1)}{5}$. We show that all of $q$-ary EAQMDS have minimum distance upper limit much larger than the…
As in classical coding theory, quantum analogues of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane (CSS) construction. However, special requirements in the quantum setting severely limit the structures such quantum codes can have. While the entanglement-assisted stabilizer formalism overcomes this limitation by exploiting maximally entangled states (ebits), excessive reliance on ebits is a substantial obstacle to implementation. This paper gives necessary and sufficient conditions for the …
