We revisit a fundamental open problem in quantum information theory, namely, whether it is possible to transmit quantum information at a rate exceeding the channel capacity if we allow for a non-vanishing probability of decoding error. Here, we establish that the Rains information of any quantum channel is a strong converse rate for quantum communication. For any sequence of codes with rate exceeding the Rains information of the channel, we show that the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if we consider codes that perform classical postprocessing on the transmitted quantum data. As an application of this result, for generalized dephasing channels, we show that the Rains information is also achievable, and thereby establish the strong converse property for quantum communication over such channels. Thus, we conclusively settle the strong converse question for a class of quantum channels that have a non-trivial quantum capacity.
We revisit a fundamental open problem in quantum information theory, namely, whether it is possible to transmit quantum information at a rate exceeding the channel capacity if we allow for a non-vanishing probability of decoding error. Here, we establish that the Rains information of any quantum channel is a strong converse rate for quantum communication. For any sequence of codes with rate exceeding the Rains information of the channel, we show that the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if we consider codes that perform classical postprocessing on the transmitted quantum data. As an application of this result, for generalized dephasing channels, we show that the Rains information is also achievable, and thereby establish the strong converse property for quantum communication over such channels. Thus, we conclusively settle the strong converse question for a class of quantum channels that have a non-trivial quantum capacity.
We revisit a fundamental open problem in quantum information theory, namely whether it is possible to transmit quantum information at a rate exceeding the channel capacity if we allow for a non-vanishing probability of decoding error. Here we establish that the Rains information of any quantum channel is a strong converse rate for quantum communication: For any code with a rate exceeding the Rains information of the channel, we show that the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if we consider codes that perform classical post-processing on the transmitted quantum data. Our result has several applications. Most importantly, for generalized dephasing channels we show that the Rains information is also achievable, and thereby establish the strong converse property for quantum communication over such channels. This for the first time conclusively settles the strong converse question for a class of quantum channels that have a non-trivial quantum capacity.
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.
The quantum capacity of a noisy channel quantifies the maximum rate at which quantum information can be transmitted reliably. For general channels, its evaluation requires an optimization over arbitrarily many channel uses. Degradable channels form a central exception: their capacity is given by the single-letter coherent information, while antidegradable channels have zero capacity. Nevertheless, even for these fundamental classes, it has remained open whether communication above capacity becomes possible when a fixed non-maximal error is tolerated. Here we resolve this problem by proving an exponential strong converse for every finite-dimensional degradable and antidegradable channel: at any rate above capacity, the fidelity of every coding scheme decays exponentially with the number of channel uses. As an immediate consequence, we establish the first all-code exponential strong converse for the quantum erasure channel throughout its full parameter range, strengthening previous results that applied only to almost all codes. We also show that exponential strong-converse bounds are preserved under receiver post-processing. This yields efficiently computable semidefinite-programming bounds for arbitrary finite-dimensional channels, improved bounds for Pauli channels, and an exact exponential strong converse for a nondegradable multilevel amplitude-damping family.
A fully general strong converse for channel coding states that when the rate of sending classical information exceeds the capacity of a quantum channel, the probability of correctly decoding goes to zero exponentially in the number of channel uses, even when we allow code states which are entangled across several uses of the channel. Such a statement was previously only known for classical channels and the quantum identity channel. By relating the problem to the additivity of minimum output entropies, we show that a strong converse holds for a large class of channels, including all unital qubit channels, the d-dimensional depolarizing channel and the Werner-Holevo channel. This further justifies the interpretation of the classical capacity as a sharp threshold for information transmission.
A quantum channel can transmit quantum states, classical messages, or classical messages that remain secret from its environment. We prove exponential strong converses for all three tasks over arbitrary finite-dimensional memoryless quantum channels, at their respective regularized capacities and allowing arbitrary entangled inputs across channel uses and joint decoding. Above quantum or classical capacity, entanglement fidelity or decoding success probability, respectively, decays exponentially with the number of channel uses. Above private capacity, fidelity to an ideal shared secret key decays exponentially; in particular, vanishing secrecy error forces decoding success to vanish. The proofs combine one-shot quantum blowing-up lemmas, which convert codes with low success into reliable codes with a controlled loss in code size, with a low-degree polynomial approximation of the channel's Stinespring image projector. The loss is governed by the projective tensor norm of the approximation across the receiver--environment bipartition. For memoryless channels, the polynomial construction gives exponential accuracy while keeping the rate loss arbitrarily small, turning the corresponding weak converses into strong converses.