Causal limit on quantum communication
The capacity of a channel is known to be equivalent to the highest rate at which it can generate entanglement. Analogous to entanglement, the notion of quantum causality characterises the temporal aspect of quantum correlations. Despite holding an equally fundamental role in physics, temporal quantum correlations have yet to find their operational significance in quantum communication. Here we uncover a connection between quantum causality and channel capacity. We show the amount of temporal correlations between two ends of the noisy quantum channel, as quantified by a causality measure, implies a general upper bound on its channel capacity. The expression of this new bound is simpler to evaluate than most previously known bounds. We demonstrate the utility of this bound by applying it to a class of shifted depolarizing channels, which results in improvement over previously known bounds for this class of channels.
Determining the rate at which information can be reliably transmitted over a given channel is one of the central tasks of information theory. In a classical setting, Shannon shannon1948mathematical () proved that the capacity of discrete memoryless channels are governed by a simple expression. In a quantum setting, however, such a characterisation of a channels’ ability to transmit information has proved far more elusive. In determining the capacity of a quantum channel, , we have to consider the possibility that in order to achieve the maximal capacity per use of the channel it may be necessary to encode information in states which are entangled across channels. Thus, to determine the actual capacity of a quantum channel, one needs to take the supremum of this quantity over tensor products of an arbitrary number of copies of the channel. In the context of quantum communication, a significant amount of progress has been made on achievable rates for the transmission of quantum information over noisy channels holevo2001evaluating (); lloyd1997capacity (); shor2002quantum (); devetak2005private (). However existing formulae for quantum capacities often involve implicit optimisation problems. In the absence of formulae for the exact capacities, one is forced to rely on bounds for the quantum capacity that are tractable to evaluate takeoka2014squashed (); muller2016positivity (); wang2016semidefinite (); sutter2015approximate (); wang2017semidefinite (); tomamichel2017strong (); berta2017amortization (). The reader is referred to wilde2013quantum () for a review of related results.
In this paper we present novel general upper bounds on the quantum capacities of quantum channels that do not require optimisation and are based on causality considerations derived using a pseudo-density matrix (PDM) formalism introduced in fitzsimons2015quantum (). A PDM is a generalization of the standard density matrix which seeks to capture correlations both in space and time. In quantum mechanics, a density matrix is a probability distribution over pure quantum states but it can alternatively be viewed as a representation of the expectation values for each possible Pauli measurement on the system. For a system composed of multiple spatially separated subsystems, each Pauli operator can be expanded as a tensor product of single-qubit Pauli operators, with one acting on each subsystem. PDMs build on this second view of the standard density matrix, extending the notion of the density matrix into the time domain. The resulting pseudo-density matrix is defined as
where is the expectation value for the product of a set of Pauli measurements. Unlike in the standard density matrix, we do not require the measurements act only on distinct spatially separated subsystems. Rather each measurement can be associated with an instant in time and a particular subsystem, and is taken to project the state of the system onto the eigenspace of the measured observable corresponding to the measurement outcome.
The generalization of states to systems extended across multiple points in time has the result that, unlike density matrices, PDMs can have negative eigenvalues. As the PDM is equivalent to the standard density matrix when the measurements are restricted to a single moment in time, the existence of negative eigenvalues in the PDM acts as a witness to the causal structure of the system. In order to quantify the causal component of such correlations, the notion of a causality monotone was introduced in fitzsimons2015quantum (). We now introduce a function based on the logarithm of the trace norm of the PDM, , which is similar to causality monotones, but sacrifices convexity in favour of additivity when applied to tensor products. This is similar to logarithmic negativity plenio2005logarithmic () in the context of spatial correlations. Analogous to entanglement measures vedral1998entanglement (), satisfies the following important properties:
, with if is positive semi-definite, and for obtained from two consecutive measurements on a single qubit closed system,
is invariant under a local change of basis,
is non-increasing under local operations,
, for any probability distribution , and
Properties 1-3 follow directly from the corresponding properties of the causality monotone proved in fitzsimons2015quantum (), since , and from the monotonicity of the logarithm function. Property 4 also follows from the monotonicity of the logarithm function, since this implies . To prove property 5, we note that , and hence .
Results. Any quantum process can be identified with a corresponding PDM. Consider a qubit-to-qubit channel acting on a single qubit described by an initial state . For such a process , a PDM that involves a single use of the channel and two measurements before and after , has been shown to be given by
where and horsman2017can (); zhao2017geometry (). Here we fix the input to be a maximally mixed state. Then, equation 1 can be easily generalised to describe any quantum channel acting on a collection of qubits
Based on this relation, we use the causality to construct an upper bound on the number of uses of a given channel to approximate the ideal (identity) channel . Since we consider only one-way communication, the most general procedure for combining resource channels together to approximate the ideal channel is to consider parallel uses of the channel preceded by some encoding and followed by some decoding procedure, as shown below in FIG. 1.
We compare the causality across the collection of channels with the causality across the identity channel. As a result of the property 4 of and the fact that for quantum channel capacity consideration it suffices to consider isometric encodings barnum2000quantum (), the causality across the combined channels does not increase under encoding and decoding. We then exploit the additivity of causality to relate to the number of uses of the channel. This leads to our main result that the quantum capacity of channel is upper bounded by ,
The mathematical details for deriving this bound are presented in the next section. Evaluating the causality requires only finding the logarithm of the trace norm of a PDM and can be readily calculated for channels acting on relatively small Hilbert spaces. Importantly, computing this bound does not involve any optimisation. Furthermore, equation 3 implies that any channel with has quantum capacity equal to zero. This reflects the fact that such a channel exhibits correlations which could have been produced by measurements on distinct subsystems of a quantum state, and so the system is necessarily constrained by the no-signalling theorem. On the other hand, when is strictly positive, the correlations between the two ends of the channel cannot be captured by bipartite density matrices, thus signifying information being passed forward in time. We emphasize that the bound has been derived for channels acting on the collection of qubits, nonetheless the result applies to channels with arbitrary input and output dimensions. For the method to apply to such cases, it suffices to restrict the channel to act only on a subspace of dimensional Hilbert space.
As a practical illustration of how the causality method works, we apply it to the class of shifted depolarizing channels. A shifted depolarizing channel generalises the well-studied quantum depolarizing channel king2003capacity (); smith2008additive (). It outputs either the state shifted from the maximally mixed state with probability or the input state. For a single qubit the channel can be defined by . The parameter parametrizes the shift, with vanishing corresponding to a standard depolarizing channel. The PDM associated with the single qubit shifted depolarizing channel can be found using equation 2 from which we obtain an analytic expression for the value of , and hence an upper bound on the quantum capacity of the channel
We can compare this with a simple well-known bound on quantum capacities of Holevo and Werner (HW) which is general, and has a similar form to the causality bound, but requires optimisation holevo2001evaluating (). The causality bound is better or equal to the HW bound (see the Supplementary Information for a proof). As shown in FIG. 2, the shifted depolarising channel constitutes an example for which the causality bound is strictly tighter than the HW bound. Furthermore, the bound also improves upon the best known bound from ouyang2014channel (). In fact, it is tighter for most values of shifts as shown in FIG. 3.
Proof of the bound. In this section, we prove the bound in equation 3. First, we construct the pseudo-density matrix corresponding to a channel obtained through using copies of the resource channel preceded by the encoding channel and followed the decoding channel . Let . Note that
By the reverse triangle inequality,
We can relate the trace distance of two pseudo-density matrices to the diamond norm in the following way:
where denotes the diamond norm wilde2013quantum (). Denoting the distance between and in the diamond norm by and using the upper bound on as well as the positivity of , we get
Taking the logarithm on both sides of the above inequality, we find
We can exploit the relation between the PDM and SWAP matrix, as well as the non-increasing property of the trace norm under the partial trace, to show that the causality does not increase under decoding and encoding. A detailed proof is presented in the Supplementary Information. This gives .
Additivity of with respect to tensor products implies that , and . Hence
Finally, since , where is the number of qubits on which the channel acts, we have
The diamond norm distance can be related to distance in the completely bounded infinity norm (see Supplementary Information for details), which in turn guarantees goes to zero as approaches infinity. Therefore, we obtain the bound
Discussion. We have obtained a bound on quantum capacity using fundamental causality considerations. In doing so, we have introduced a new measure of temporal correlations that is analogous to entanglement logarithmic negativity and possesses desired properties that make it useful for studying channel capacities. Studies of spatial correlations have lead to the formulation of many entanglement monotones with different corresponding applications and operational meanings e.g. distillable entanglement, entanglement cost, squashed entanglement horodecki2009quantum (); christandl2004squashed (). As a temporal counterpart of quantum correlations, our work initiates research on operational significance of causality measures that might prove useful in a wider range of applications. The causality method applies to arbitrary quantum channels and produces non-trivial upper bounds for any channel. However, in contrast to most other of such bounds, it does not require optimisation. Our result could help to understand the communication rate of complex systems for which optimisation methods are computationally too costly, including quantum networks and quantum communication between many parties leung2010quantum (); hayashi2007quantum ().
Acknowledgements– We would like to thank Mark Wilde, Andreas Winter and Artur Ekert for helpful discussions and useful comments on the manuscript. J.F.F. acknowledges support from the Air Force Office of Scientific Research under grant FA2386-15-1-4082. V.V. thanks the Leverhulme Trust, the Oxford Martin School, and Wolfson College, University of Oxford. The authors acknowledge support from Singapore Ministry of Education. This material is based on research funded by the National Research Foundation of Singapore under NRF Award No. NRF-NRFF2013-01. R.P. and V.V. thank EPSRC (UK).
Appendix A Supplementary Information
a.1 Causality under encoding and decoding channels
An important property that we have used in our proof was that the decoding and encoding procedures do not increase causality, so that . To show this, we first establish the following lemma.
Let be a linear map from qubits to qubits. Then
where () means that and are applied to the first and second subsystems of each of the SWAPs respectively.
Let Now the tensor product of qubit SWAPs admits a representation
Similarly evaluating the right hand sign of equation (4) we get
where in the last step we have relabelled the indices. ∎
We are now in a position to prove that .
Let and be encoding and decoding channels and . Then
Consider the trace norm . The decoding procedure is a local operation and therefore from property 4 of , we have
Let encode qubits into qubits. Using Lemma 1
Decompose into its positive and negative part
where both and are positive semi-definite. By the triangle inequality
It has been shown in  that in bounding quantum channel capacity, one can restrict to be an isometry with only one non-zero Kraus operator, which we denote by . Then the expression can be written as
where we used the cyclic property of the trace. Since is a projector,
Applying Hölder’s inequality twice and making use of the fact the infinity norm of a projector equals one, we get
where denotes the infinity norm and is equal to the largest singular value of a matrix . Now, and are orthogonal. Hence
which leads to Finally, since logarithm is a monotonic function, the result follows. ∎
a.2 Causality bound against the partial transpose bound
Let us compare our causality bound to the Holevo and Werner bound. Given a quantum channel , and a transpose map , the Holevo-Werner upper bound on the quantum capacity is
Using the definition of the induced norm this can be written as
Now we can compare this to our bound. In the case of the maximally mixed input the pseudo-density matrix becomes
and therefore the causality bound becomes
Comparing this to the Holevo and Werner’s result it is clear that
and the two are equal when the supremum is achieved at the .
a.3 Limit of infinite uses of a channel
Here we show that in the proof of our bound 3, the error parameter goes to zero in the limit of large . Now,
Consider the spectral decomposition of Hermitian where
where denotes an orthonormal basis, and are the corresponding eigenvalues. Let , then we have
Since is the difference of two linear maps and , by linearity we have
Within the supremum, we have 1-norm of the difference between two quantum states. Recall that there is the inequality that relates the 1-norm of the difference between quantum states to the fidelity between the states. Let
denote the fidelity between two positive semidefinite matrices. If , then Then we have the Fuchs-van de Graaf inequalities 
The above inequality is related to entanglement fidelity of a state with respect to the channel . Let . Then from Schumacher’s formula , we have
where is a purification of . We denote
Hence using the notation for the entanglement fidelity, we have
thus Kretschmann and Werner [29, Proposition 4.3] showed that
where denotes the completely bounded norm induced on the operator infinity norm . Therefore
Since is guaranteed to approach zero as approaches infinity in the channel capacity theorems, here also approaches zero.
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