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

Figure 1: Communicating classical information over a quantum channel.
figure 1

a, A set of states ρi are used with probabilities pi as signal states on the channel . The inputs are unentangled between channels and . The capacity of is equal to that of . b, A set of entangled input states ρi are used on the channel . The question addressed is whether entangling can increase capacity.

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).

Figure 2: Minimum output entropy of a quantum channel.
figure 2

a, A pure state |ψ〉 is input to the channel . Although |ψ〉 is a pure state, the output may be a mixed state. We attempt to minimize the output entropy over all pure input states. b, An entangled input state |ψ〉 is input to the channel . The question addressed is whether this entangled input state can have a lower output entropy for channel than the sum of the minimum output entropies for the two channels.

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 DN 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.