Entanglement-Assisted Quantum Communication Beating the Quantum Singleton Bound

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.

] 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. Introduction.-Entanglement is a resource that enables or enhances many tasks in quantum communication. When sender and receiver share a maximally entangled state, quantum teleportation allows the sender to transmit an unknown quantum state by just sending a finite amount of classical information over a noiseless classical channel [1]. When the entangled states are initially distributed over a noisy quantum channel, using local quantum operations and a noiseless classical channel, sender and receiver can extract a smaller number of maximally entangled states with higher fidelity by a distillation process [2]. The correspondence between entanglement distillation protocols and quantum error-correcting codes in a communication scenario has been discussed in [3]. Quantum error-correcting codes (QECCs), however, are somewhat more general in the sense that they allow to recover a quantum state affected by a general quantum channel, provided that a suitable encoding for that channel exists [4]. Hence QECCs can be used both for communication and storage, and they are essential ingredients for fault-tolerant quantum computation [5]. On the other hand, in [6] it has been shown that the performance of QECCs in a communication scenario can be improved when a noisy quantum channel is assisted by entanglement.
In this Letter we present a quantum communication scheme that also uses a noisy quantum channel assisted by entanglement. The main idea it to execute a teleportation protocol in which the classical information is protected using a code and then sent via the noisy quantum channel to the receiver. This allows to use classical error-correcting codes. In some range, our scheme has better parameters than the one proposed in [6], showing that the adaption of the quantum Singleton bound to that class of codes presented in [7] can be violated.
Quantum Teleportation.-We start with a short summary quantum teleportation [1], as illustrated in Fig. 1.
The aim is to transmit an arbitrary quantum state |ϕ in the Hilbert space H ∼ = (C d ) ⊗c of c quantum systems of dimension d (qudits). The protocol is assisted by c copies of a maximally entangled bipartite state which are shared by sender S and receiver R. Applying Heisenberg-Weyl operators X a Z b (or generalized Pauli matrices) on one of the systems, the collection of the resulting states constitutes the generalized Bell basis Here X is a cyclic shift operator in the standard basis and Z its diagonal form. The sender performs a generalized Bell measurement, i. e., a measurement in the generalized Bell basis (2), on the input state |ϕ and c qudits from the maximally entangled states. For each pair of qudits, one obtains a pair (a i , b i ) of classical values. The strings a = (a 1 , a 2 , . . . , a c ) and b = (b 1 , b 2 , . . . , b c ) with c classical symbols each are sent to the receiver, indicated by the double lines in Fig. 1. Depending on the values of a and b, the receiver applies correction operations X a and Z b and obtains the original state |ϕ .
Quantum Error-Correcting Codes.-A standard quantum error-correcting code C of length n is a subspace of the Hilbert space (C d ) ⊗n of n qudits. Usually, d is assumed to be a power of a prime, i.e., d = q = p m for some prime p. A QECC encoding k qudits has dimension q k and is denoted by [[n, k, d]] q . A QECC with minimum distance d = 2t + 1 allows to correct all errors affecting no more than t of the subsystems. When the position of the errors is known, then errors on up to d − 1 subsystems can be corrected [8]. Alternatively, all errors on no more than d − 1 of the subsystems that act non-trivially on the code can be detected. Independent of the dimension q of the subsystems, the parameters of a QECC are constraint by the so-called quantum Singleton bound [4,9] 2d ≤ n − k + 2. ( Codes meeting this bound with equality are called quantum MDS (QMDS) codes. With few exceptions, QMDS codes are only known for length n ≤ q 2 + 1 and minimum distance d ≤ q + 1 [10]. The overall scheme is illustrated in Fig. 2. The unitary encoding operator E maps the input state |ϕ with k qudits and n − k ancilla qudits |0 n−k to the encoding space with n qudits. Those n qudits are sent over a noisy quantum channel N , whose output enters the decoder D. The decoder outputs a quantum state |ϕ ′ with k qudits. When the error can be corrected, the states |ϕ ′ and |ϕ are equal. Entanglement Assisted Quantum Error-Correcting Codes.-An entanglement assisted quantum error-correcting code (EAQECC), denoted by [[n, k, d; c]] q , is a quantum error-correcting code that additionally uses c maximally entangled states. The scheme is illustrated in Fig. 3. One half of each maximally entangled state |Φ ∈ C q ⊗ C q enters the encoding operation E together with the k-qudit state |ϕ to be transmitted and n − k − c ancilla qudits. The other half of each maximally entangled state is assumed to be transmitted error-free. The n qudits output by the encoding operation are sent through the noisy quantum channel N . The receiver applies the decoding operation D to both the n qudits output from the channel and the c noiseless qudits from the maximally entangled states.
In [7], the authors formulate a Singleton bound for the parameters [[n, k, d; c]] q of an EAQECC: This bound can be derived by considering the n qudits sent over the noisy quantum channel N together with the c qubits from the maximally entangled states sent over a noiseless channel as a standard QECC of length n + c in the quantum Singleton bound (3). This, however, ignores the fact that the c additional qudits are assumed to be error-free. The approach in [11] accounts for this additional assumption. In Theorem 6 of [11], the bound (4) has been shown to be valid for the case d ≤ (n + 2)/2. In [12], EAQECCs meeting the bound (4) with equality were constructed. Some of the codes use only c = 1 or c = 2 maximally entangled states, the maximal value is c = q + 1. While the length of the codes is bounded by q 2 + 1 like in the case of QMDS codes, the minimum distance can be as large as d = 2q for some of the codes, compared to d ≤ q + 1 for most QMDS codes without entanglement assistance.
In [6], a construction of EAQECC from any linear code [n, κ, d] q 2 over the finite field F q 2 with q 2 was given. The parameters of the resulting EAQECC are [[n, 2κ − n + c, d; c]] q , where the number c of maximally entangled states depends on the classical code and is at most n − κ. Using a result from [13] it follows that for q ≥ 3, any classical linear code [n, κ, d] q 2 can be converted into an EAQECC that requires the maximal amount of entanglement c = n − κ. The EAQECC has parameters [[n, κ, d; n − κ]] q . From a classical MDS code [n, κ, n − κ + 1] q 2 , we obtain an EAQECC with parameters [[n, k, n − k + 1; n − k]] q , meeting the bound (4) with equality. Assuming the maximal value for c = n − k, the minimum distance of an EAQECC from this construction obeys the bound which is exactly the Singleton bound for classical codes. The bound (5) is also an absolute bound on the minimum distance of any quantum code for the following reason. A quantum code with minimum distance d can correct errors that affect d − 1 errors at known positions. Hence, after tracing out d − 1 of the systems, we will still be able to recover any encoded state of k qudits. The residual state has n − d + 1 qudits, and hence the bound k ≤ n − d + 1 follows.
The New Scheme.-In our scheme, we use the c maximally entangled states in a teleportation protocol to transmit k = c qudits. Each generalized Bell measurement in the teleportation protocol has q 2 possible outcomes, i.e., we have to send a classical string with 2k symbols from an alphabet of size q to the receiver. As we allow for n uses of a quantum channel, we can use a classical code C over an alphabet of size q encoding 2k symbols into n symbols, denoted by [n, 2k, d] q , were again d denotes the minimum distance of the code (for more details about classical error-correcting codes, see for example [14]). The classical string of length n is mapped to one of the q n basis states of the Hilbert space of n qudits and then sent via the noisy quantum channel N to the receiver. The receiver measures the output of the quantum channel in the computational basis and obtains a classical string of length n. Applying error correction for the classical code C, the 2k symbols corresponding to the measurement result from the teleportation protocol are retrieved. The measurement and the classical decoder are depicted together as a quantum-to-classical map D q→c in Fig. 4. The receiver applies the corresponding correction operators X a and Z b to the c qudits from the c maximally entangled states and completes the teleportation protocol.
The decoding operator D and the correction operators can be combined as a quantum map D ′ , see the dashed box in Fig. 4. Furthermore, the sender does not have to perform a measurement in the generalized Bell basis, but may apply a unitary transformation that maps the Bell basis to the standard basis, labeled as "Bell transf." in Fig. 5. Then those 2k qudits can be encoded into n qudits using the quantum map E . Hence, Fig. 5 shows a fully-quantum version of our new scheme, showing that our protocol uses the same type of operations as an EAQECC shown in Fig. 3. We will also use the same notation [[n, k, d; c]] q for the parameters.
On the other hand, note that we are only transmitting basis states over the quantum channel, and therefore the protocol is resilient to arbitrary phase errors. When following the standard teleportation protocol, as shown in Fig. 4, one can replace the quantum channel by a classical channel. The parameters of our scheme are determined by the classical code C. The Singleton bound for classical codes implies the bound on the minimum distance of our scheme. It can be achieved whenever the classical code is an MDS code. In the special case k = c, the bound (4) implies d ≤ n/2 + 1.
Hence, for k < n/4 the bound (4) is more restrictive than the bound for our scheme (see also Fig. 6). Even when more maximally entangled states are used in the original construction of EAQECCs, our scheme has a larger normalized minimum distance δ = d/n for a rate R = k/n below a certain threshold (e.g., R < 1/5 for c = (n − k)/2, see Fig. 6).
Concluding remarks.-Quantum codes based on teleportation have been considered before when studying the entanglement-assisted capacity of quantum channels [19, Section III.E]. It was observed that this results in an entanglement-assisted capacity that is half the classical capacity of the unassisted quantum channel. We are, however, not aware of related results for the finite-length case.
Our scheme beats the quantum Singleton bound (4) for quantum communication schemes with a rate below a certain threshold and uses a smaller amount c of entanglement than the scheme proposed in [6]. On the other hand, when the amount of additional entanglement does not matter, using c = n − k maximally entangled states in the original scheme reaches the absolute bound (5). It is plausible to assume that using c > n − k maximally entangled states would not result in better parameters, as in this case the encoding operation E would map k + c > n qudits to a smaller number of qubits.
We conclude by noting that in order to beat the originally stated quantum Singleton bound for entanglement-assisted quantum-error correcting codes (4), one has to use c ≥ k maximally entangled pairs. This result, together with further bounds relating length n, dimension k, minimum distance d, and the number c of maximally entangled pairs in general entanglement-assisted schemes can be found in [20].