Abstract
The design of error-correcting codes used in modern communications relies on information theory to quantify the capacity of a noisy channel to send information1. This capacity can be expressed using the mutual information between input and output for a single use of the channel; although correlations between subsequent input bits are used to correct errors, they cannot increase the capacity. For quantum channels, it has been an open question whether entangled input states can increase the capacity to send classical information2. The additivity conjecture3,4 states that entanglement does not help, making practical computations of the capacity possible. Although additivity is widely believed to be true, there is no proof. Here, we show that additivity is false, by constructing a random counter-example. Our results show that the most basic question of classical capacity of a quantum channel remains open, with further work needed to determine in which other situations entanglement can boost capacity.
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Main
In the classical setting, Shannon presented a formal definition of a noisy channel as a probabilistic map from input states to output states. In the quantum setting, the channel becomes a linear, completely positive, trace-preserving map from density matrices to density matrices, modelling noise in the system due to interaction with an environment. Such a channel can be used to send either quantum or classical information. In the first case, a marked violation of operational additivity was recently shown, in that there exist two channels, both having zero capacity to send quantum information no matter how many times it is used, which can be used in tandem to send quantum information5.
Here, we address the classical capacity of a quantum channel. To specify how information is encoded in the channel, we must pick a set of states ρi which we use as input signals with probabilities pi. Then the Holevo formula2 for the capacity is:
where H(ρ)=−Tr(ρln(ρ)) is the von Neumann entropy. The maximum capacity of a channel is the maximum over all input ensembles:
Suppose we have two different channels, and . To compute this capacity, it seems necessary to consider entangled input states between the two channels. Similarly, when using the same channel multiple times, it may be useful to use input states that are entangled across multiple uses of the same channel. The additivity conjecture (see Fig. 1) is the conjecture that this does not help and that instead
The additivity conjecture makes it possible to compute the classical capacity of a quantum channel. Furthermore, Shor4 showed that several different additivity conjectures in quantum information theory are all equivalent. These are the additivity conjecture for the Holevo capacity, the additivity conjecture for entanglement of formation6, strong superadditivity of entanglement of formation7 and the additivity conjecture for minimum output entropy3. Here, we show that all of these conjectures are false, by constructing a counter-example to the last of these conjectures. Given a channel , define the minimum output entropy Hmin by
The minimum output entropy conjecture is that for all channels and , we have
A counter-example to this conjecture would be an entangled input state that has a lower output entropy, and hence is more resistant to noise, than any unentangled state (see Fig. 2).
Our counter-example to the additivity of minimum output entropy is based on a random construction, similar to those Winter and Hayden used to show violation of the maximal p-norm multiplicativity conjecture for all p>1 (refs 8, 9, 10). For p=1, this violation would imply violation of the minimum output entropy conjecture; however, the counter-example found in ref. 9 requires a matrix size that diverges as p→1. We use different system and environment sizes (note that D≪N in our construction below) and make a different analysis of the probability of different output entropies. Other violations are known for p close to 0 (ref. 11).
We define a pair of channels and , which are complex conjugates of each other. Each channel acts by randomly choosing a unitary from a small set of unitaries Ui (i=1…D) and applying that to ρ. This models a situation in which the unitary evolution of the system is determined by an unknown state of the environment. We define
where the Ui are N-by-N unitary matrices, chosen at random from the Haar measure, and the probabilities Pi are chosen randomly as described in Supplementary Information. The Pi are all roughly equal. We pick
In Supplementary Information we prove the following theorem:
For sufficiently large D and for sufficiently large N, there is a non-zero probability that a random choice of Ui from the Haar measure and of Pi (as described in Supplementary Information) will give a channel such that
The size of N depends on D.
For any pure-state input, the output entropy of is at most ln(D) and that of is at most 2ln(D). To prove the above theorem, we first construct an entangled state with a lower output entropy for the channel . The entangled state we use is the maximally entangled state
As shown in Lemma 1 in Supplementary Information, the output entropy for this state is bounded by
We then use the random properties of the channel to show that no product state input can obtain such a low output entropy. Lemmas 2–5 in Supplementary Information show that, with non-zero probability, the entropy is at least ln(D)−δ Smax, for
where c1 is a constant and p1(D)=poly(D). Thus, because for large enough D and for large enough N we have 2δ Smax<ln(D)/D, the theorem follows.
The output entropy can be understood differently: for a given pure-state input, can we determine from the output which of the unitaries Ui† was applied? Recall that
for any unitary U. This means that, for the maximally entangled state, if a unitary Ui† was applied to one subsystem, and was applied to the other subsystem, we cannot determine which unitary i was applied by looking at the output. This is the key idea behind equation (1).
Note that the minimum output entropy of must be less than ln(D) by an amount at least of order 1/D. Suppose U1 and U2 are the two unitaries with the largest li. Choose a state |ψ〉 that is an eigenvector of U1U2†. For this state, we cannot distinguish between the states U1†|ψ〉 and U2|ψ〉, and so
Our randomized analysis bounds how much further the output entropy of the channel can be lowered for a random choice of Ui.
Our work raises the question of how strong a violation of additivity is possible. The relative violation we have found is numerically small, but it may be possible to increase this, and to find new situations in which entangled inputs can be used to increase channel capacity, or novel situations in which entanglement can be used to protect against decoherence in practical devices. The map is similar to that used12 to construct random quantum expanders13,14, raising the possibility that deterministic expander constructions can provide stronger violations of additivity.
Although we have used two different channels, it is also possible to find a single channel such that , by choosing Ui from the orthogonal group. Alternatively, we can add an extra classical input used to ‘switch’ between and (P. Hayden, private communication).
The equivalence of the different additivity conjectures4 means that the violation of any one of the conjectures has profound impacts. The violation of additivity of the Holevo capacity means that the problem of channel capacity remains open, because if a channel is used many times, we must do an intractable optimization over all entangled inputs to find the maximum capacity. However, we conjecture that additivity holds for all channels of the form
Our intuition for this conjecture is that we believe that multi-party entanglement (between the inputs to three or more channels) is not useful, because it is very unlikely for all channels to apply the same unitary; note that the state ΨME has a low minimum output entropy precisely because it is left unchanged as in equation (2) if both channels apply corresponding unitaries. This two-letter additivity conjecture would enable us to restrict our attention to considering input states with a bipartite entanglement structure, possibly opening the way to computing the capacity for arbitrary channels.
References
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Acknowledgements
I thank J. Yard, P. Hayden and A. Harrow. This work was supported by US DOE Contract No. DE-AC52-06NA25396.
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Hastings, M. Superadditivity of communication capacity using entangled inputs. Nature Phys 5, 255–257 (2009). https://doi.org/10.1038/nphys1224
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DOI: https://doi.org/10.1038/nphys1224
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