Duality in Entanglement-Assisted Quantum Error Correction

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.Comment: This paper is a compact version of arXiv:1010.550

Semidefinite programming converse bounds for quantum communication

We derive several efficiently computable converse bounds for quantum communication over quantum channels in both the one-shot and asymptotic regime. First, we derive one-shot semidefinite programming (SDP) converse bounds on the amount of quantum information that can be transmitted over a single use of a quantum channel, which improve the previous bound from [Tomamichel/Berta/Renes, Nat. Commun. 7, 2016]. As applications, we study quantum communication over depolarizing channels and amplitude damping channels with finite resources. Second, we find an SDP strong converse bound for the quantum capacity of an arbitrary quantum channel, which means the fidelity of any sequence of codes with a rate exceeding this bound will vanish exponentially fast as the number of channel uses increases. Furthermore, we prove that the SDP strong converse bound improves the partial transposition bound introduced by Holevo and Werner. Third, we prove that this SDP strong converse bound is equal to the so-called max-Rains information, which is an analog to the Rains information introduced in [Tomamichel/Wilde/Winter, IEEE Trans. Inf. Theory 63:715, 2017]. Our SDP strong converse bound is weaker than the Rains information, but it is efficiently computable for general quantum channels.Comment: 17 pages, extended version of arXiv:1601.06888. v3 is closed to the published version, IEEE Transactions on Information Theory, 201

Entanglement required in achieving entanglement-assisted channel capacities

Entanglement shared between the two ends of a quantum communication channel has been shown to be a useful resource in increasing both the quantum and classical capacities for these channels. The entanglement-assisted capacities were derived assuming an unlimited amount of shared entanglement per channel use. In this paper, bounds are derived on the minimum amount of entanglement required per use of a channel, in order to asymptotically achieve the capacity. This is achieved by introducing a class of entanglement-assisted quantum codes. Codes for classes of qubit channels are shown to achieve the quantum entanglement-assisted channel capacity when an amount of shared entanglement per channel given by, E = 1 - Q_E, is provided. It is also shown that for very noisy channels, as the capacities become small, the amount of required entanglement converges for the classical and quantum capacities.Comment: 9 pages, 2 figures, RevTex

On converse bounds for classical communication over quantum channels

We explore several new converse bounds for classical communication over quantum channels in both the one-shot and asymptotic regimes. First, we show that the Matthews-Wehner meta-converse bound for entanglement-assisted classical communication can be achieved by activated, no-signalling assisted codes, suitably generalizing a result for classical channels. Second, we derive a new efficiently computable meta-converse on the amount of classical information unassisted codes can transmit over a single use of a quantum channel. As applications, we provide a finite resource analysis of classical communication over quantum erasure channels, including the second-order and moderate deviation asymptotics. Third, we explore the asymptotic analogue of our new meta-converse, the $\Upsilon$-information of the channel. We show that its regularization is an upper bound on the classical capacity, which is generally tighter than the entanglement-assisted capacity and other known efficiently computable strong converse bounds. For covariant channels we show that the $\Upsilon$-information is a strong converse bound.Comment: v3: published version; v2: 18 pages, presentation and results improve

The quantum capacity with symmetric side channels

We present an upper bound for the quantum channel capacity that is both additive and convex. Our bound can be interpreted as the capacity of a channel for high-fidelity quantum communication when assisted by a family of channels that have no capacity on their own. This family of assistance channels, which we call symmetric side channels, consists of all channels mapping symmetrically to their output and environment. The bound seems to be quite tight, and for degradable quantum channels it coincides with the unassisted channel capacity. Using this symmetric side channel capacity, we find new upper bounds on the capacity of the depolarizing channel. We also briefly indicate an analogous notion for distilling entanglement using the same class of (one-way) channels, yielding one of the few entanglement measures that is monotonic under local operations with one-way classical communication (1-LOCC), but not under the more general class of local operations with classical communication (LOCC).Comment: 10 pages, 4 figure

Entanglement-Assisted Quantum Error-Correcting Codes with Imperfect Ebits

The scheme of entanglement-assisted quantum error-correcting (EAQEC) codes assumes that the ebits of the receiver are error-free. In practical situations, errors on these ebits are unavoidable, which diminishes the error-correcting ability of these codes. We consider two different versions of this problem. We first show that any (nondegenerate) standard stabilizer code can be transformed into an EAQEC code that can correct errors on the qubits of both sender and receiver. These EAQEC codes are equivalent to standard stabilizer codes, and hence the decoding techniques of standard stabilizer codes can be applied. Several EAQEC codes of this type are found to be optimal. In a second scheme, the receiver uses a standard stabilizer code to protect the ebits, which we call a "combination code." The performances of different quantum codes are compared in terms of the channel fidelity over the depolarizing channel. We give a formula for the channel fidelity over the depolarizing channel (or any Pauli error channel), and show that it can be efficiently approximated by a Monte Carlo calculation. Finally, we discuss the tradeoff between performing extra entanglement distillation and applying an EAQEC code with imperfect ebits.Comment: 15 pages, 12 figure

Codes for Simultaneous Transmission of Quantum and Classical Information

We consider the characterization as well as the construction of quantum codes that allow to transmit both quantum and classical information, which we refer to as `hybrid codes'. We construct hybrid codes $[\![n,k{: }m,d]\!]_q$ with length $n$ and distance $d$, that simultaneously transmit $k$ qudits and $m$ symbols from a classical alphabet of size $q$. Many good codes such as $[\![7,1{: }1,3]\!]_2$, $[\![9,2{: }2,3]\!]_2$, $[\![10,3{: }2,3]\!]_2$, $[\![11,4{: }2,3]\!]_2$, $[\![11,1{: }2,4]\!]_2$, $[\![13,1{: }4,4]\!]_2$, $[\![13,1{: }1,5]\!]_2$, $[\![14,1{: }2,5]\!]_2$, $[\![15,1{: }3,5]\!]_2$, $[\![19,9{: }1,4]\!]_2$, $[\![20,9{: }2,4]\!]_2$, $[\![21,9{: }3,4]\!]_2$, $[\![22,9{: }4,4]\!]_2$ have been found. All these codes have better parameters than hybrid codes obtained from the best known stabilizer quantum codes.Comment: 6 page