Abstract
When can noiseless quantum information be sent across noisy quantum devices? And at what maximum rate? These questions lie at the heart of quantum technology, but remain unanswered because of nonadditivity— a fundamental synergy which allows quantum devices (aka quantum channels) to send more information than expected. Previously, nonadditivity was known to occur in very noisy channels with coherent information much smaller than that of a perfect channel; but, our work shows nonadditivity in a simple lownoise channel. Our results extend even further. We prove a general theorem concerning positivity of a channel’s coherent information. A corollary of this theorem gives a simple dimensional test for a channel’s capacity. Applying this corollary solves an open problem by characterizing all qubit channels whose complement has nonzero capacity. Another application shows a wide class of zero quantum capacity qubit channels can assist an incomplete erasure channel in sending quantum information. These results arise from introducing and linking logarithmic singularities in the vonNeumann entropy with quantum transmission: changes in entropy caused by this singularity are a mechanism responsible for both positivity and nonadditivity of the coherent information. Analysis of such singularities may be useful in other physics problems.
Introduction
Entropy is fundamental. As a measure of complexity in a statistical distribution, entropy is widely used in learning theory^{1,2}, economics^{3,4}, and cryptography^{5}. In physics, entropy usually quantifies disorder. It is used to express laws of thermodynamics^{6,7,8}, explore the nature of black holes^{9,10,11}, and study a variety of other physical phenomenon^{12,13,14,15,16,17,18}. Advances in understanding mathematical and computational properties of entropy^{19,20,21,22} have opened the doors for deeper insights in physics and many other areas of study.
One area where entropy provides key insights is information science^{23}. The Shannon entropy not only quantifies the amount of classical information in a source, it also plays a fundamental role in answering a key practical question: when, and at what maximum rate can classical information be sent across noisy communication channels? The maximum Shannon entropy common between a channel’s input and output, called the channel mutual information C^{(1)}, gives an achievable rate at which error correcting codes can recover noiseless information sent across many uses of a noisy channel.
The channel mutual information satisfies a crucial property, additivity: the channel mutual information for two channels used together is the sum of each. This additivity ensures that the channel capacity C, defined as the best possible achievable rate, simply equals the channel mutual information C^{(1)}. More remarkably, additivity implies that the channel capacity completely specifies a classical channel’s ultimate ability to send information. These implications are not only fundamental to our understanding of noisy classical information but also critical to the use of channel capacity as a benchmark for error correcting codes. These codes are essential for storing and sending noiseless classical information across noisy channels^{24,25}.
The physical world is not classical but quantum mechanical. It contains quantum information, which is strikingly different from its classical counterpart^{26,27,28,29}. In practice, noisy quantum devices carry quantum information. These devices, which may send, store, or process information, are modeled mathematically by completely positive tracepreserving maps, also called (noisy) quantum channels. While quantum information can be extremely useful for computing and communication, it is notoriously error prone. Consequently, there are both fundamental and practical reasons to understand when and at what maximum rate can noiseless quantum information be stored, processed, or sent across noisy quantum channels^{30,31,32,33,34,35,36}. Despite dedicated efforts, there is no satisfactory answer to this basic question. The key reason behind this unsatisfactory state of affairs is nonadditivity in the quantum analog^{37,38}, \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}\), of the channel mutual information C^{(1)}: for two noisy quantum channels \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\) and \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\) used in parallel, the channel coherent information \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}\) satisfies an inequality,
which can be strict^{39}. Like C^{(1)}, \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}\) is an entropic quantity, however, it represents an achievable rate for correcting errors in quantum information sent across a noisy quantum channel. A channel’s quantum capacity \({{{{{{{{{\mathcal{Q}}}}}}}}}}\) is defined to be the best possible achievable rate^{30}. Nonadditivity of \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}\) makes \({{{{{{{{{\mathcal{Q}}}}}}}}}}\) difficult to compute^{40,41}, and more markedly it makes \({{{{{{{{{\mathcal{Q}}}}}}}}}}\) an incomplete measure^{42} of a channel’s ability to send quantum information.
The difficulty in computing \({{{{{{{{{\mathcal{Q}}}}}}}}}}\) essentially comes from a strict inequality in (1), found when \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\) and \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\) are tensor products of the same channel \({{{{{{{{{\mathcal{B}}}}}}}}}}\). Low–dimensional channels which display this type of nonadditivity include a variety of very noisy qubit channels including the depolarizing^{39,43}, the dephrasure^{44} and other qubit Pauli^{43,45,46} and generalized erasure channels^{47,48}. As a result, even when a channel \({{{{{{{{{\mathcal{B}}}}}}}}}}\) is relatively simple, its quantum capacity \({{{{{{{{{\mathcal{Q}}}}}}}}}}({{{{{{{{{\mathcal{B}}}}}}}}}})\) must be obtained as the limit n ↦ ∞ of a sequence \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{{\mathcal{B}}}}}}}}}}}^{\otimes n})/n\)^{32,33,34,35,36}. This limit, sometimes called a regularization of\({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}\), can be particularly intractable: there are very noisy high dimensional channels for which each term in this sequence can be larger than the previous one^{49}. In addition, for any integer k there is a channel \(\tilde{{{{{{{{{{\mathcal{B}}}}}}}}}}}\) for which \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({\tilde{{{{{{{{{{\mathcal{B}}}}}}}}}}}}^{\otimes k})=0\) but \({{{{{{{{{\mathcal{Q}}}}}}}}}}(\tilde{{{{{{{{{{\mathcal{B}}}}}}}}}}})\ > \ 0\)^{50}. This type of unbounded nonadditivity makes it hard to even check if a channel’s quantum capacity is strictly positive or zero.
Challenges in computing and checking the positivity of a channel’s quantum capacity can be circumvented in the special case of (anti)degradable channels^{51,52}, PPT channels^{53}, DSPT channels^{54}, and less noisy channels^{55}. However, even if one computes a channel’s quantum capacity, nonadditivity implies that this capacity may be an incomplete measure of the channel’s ability to send quantum information. Instances of nonadditivity, i.e., a strict inequality in (1), have been found when \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\) and \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\) are different channels, each having no quantum capacity. One instance, called superactivation has been found when \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\) is a PPT channel and \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\) is a zero capacity erasure or depolarizing channel^{42,56}. Another instance of nonadditivity has been found where \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\) is a rocket channel and \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\) is an erasure channel^{57}, both channels are again very noisy, \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\) has small quantum capacity while \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\) has none, but together they have coherent information much larger than the sum of quantum capacities of each channel.
In the past, instances of nonadditivity found in very noisy channels have shown that quantum information and channels can display a type of synergy which is absent from their classical counterparts. Nonadditvity has previously not been found in lownoise channels, those with coherent information comparable to the quantum capacity of a perfect (identity) channel with the same input dimension as the channel. By contrast, in certain lownoise channels nonadditivity has been shown to be absent^{58}, and in lownoise Pauli channels nonadditivity has been shown to be of little practical relavance^{59}. While the study of nonadditivity remains of fundamental interest, methods for finding and exploring nonadditivity are scant. In high dimensional and high noise PPT and rocket channels, nonadditivity is found by using the special structure of these channels. Whereas in qubit and other low dimensional but high noise channels, methods based on degenerate quantum codes^{45} and numerical searches^{60} can identify nonadditivity, but these too can falter in simple cases of interest^{47}.
Strategies to check if a general channel has zero or nonzero quantum capacity are limited^{61,62}. To test if a channel has zero quantum capacity, one can check whether the channel is PPT or antidegradable. For special channels these two checks can be done algebraically^{52,62,63}, but in general, they require numerically solving a semidefinite program^{64}. Even if one performs these checks, their results can be inconclusive because there may exist zero capacity channels that are neither PPT nor antidegradable. Testing if a channel has nonzero quantum capacity is tricky. Except in very special circumstances, there are no algebraic tests. Numerics can be used to check if a channel’s coherent information is nonzero. However, these numerics can be unreliable, even for lowdimensional qubit channels^{44,65,66} without unbounded nonadditivity. For high dimensional channels numerics can be expensive^{43,67} in addition to being unreliable^{50}.
Seeking physical and mathematical mechanisms to find and understand positivity and nonadditivity remains an enduring challenge in quantum information science. While this challenge tempers hopes for rapid progress on understanding quantum capacities, it also presents an opportunity to introduce new ideas for addressing this challenge.
In this work, we introduce a simple but key property of the vonNeumann entropy, which we call a logsingularity (see Fig. 1), and show that changes in entropy are caused by logsingularities are a mathematical mechanism responsible for both positivity and nonadditivity of the coherent information. Utilizing this mechanism, (1) we provide an instance of nonadditivity using a zero capacity qubit channel in parallel with a lownoise qutrit channel with \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}/{{{{{{{{{\mathrm{log}}}}}}}}}\,}_{2}3\simeq 0.6\); (2) we prove a general theorem which gives algebraic conditions under which a quantum channel must have strictly positive coherent information. A corollary of this theorem gives a simple, dimensional test for capacity. An application of this corollary provides a characterization of all qubit channels whose complement has nonzero quantum capacity. A separate application of the theorem reveals how a large class of zero capacity qubit channels can assist an incomplete erasure channel in sending quantum information.
Results
Logsingularity
Let ρ(ϵ) denote a density operator that depends on a real positive parameter ϵ, and \(S(\epsilon )={{{{{{{{{\rm{Tr}}}}}}}}}}\left(\right.\rho (\epsilon ){{{{{{{{\mathrm{log}}}}}}}}}\,\rho (\epsilon )\left)\right.\) denote its vonNeumann entropy. If one or several eigenvalues of ρ(ϵ) increase linearly from zero to leading order in ϵ then a small increase in ϵ from zero increase S(ϵ) by \(x \epsilon {{{{{{{{\mathrm{log}}}}}}}}}\,\epsilon \) for some constant x > 0; i.e., \(dS(\epsilon )/d\epsilon \simeq x{{{{{{{{\mathrm{log}}}}}}}}}\,\epsilon\), and we say S(ϵ) has an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity with rate x. For instance, a qubit density operator with spectrum (1 − x_{b}ϵ, x_{b}ϵ), 0 ≤ ϵ ≤ 1 and 0 ≤ x_{b} ≤ 1 has an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity of rate x_{b}. A quart density operator with spectrum (1 − x_{c}ϵ, x_{c}ϵ/3, x_{c}ϵ/3, x_{c}ϵ/3) and 0 ≤ x_{c} ≤ 1, has an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity of rate x_{c} (also see Fig. 1).
The term \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity comes from the behavior where the derivative of S(ϵ) with respect to ϵ is logarithmic in ϵ and this derivative tends to infinity as ϵ tends to zero. While S(ϵ) is continuous, the behavior of continuity bounds on S(ϵ) can be dominated by \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularities in the sense that changes in continuity bounds can be essentially logarithmically in ϵ for small ϵ (see Supplementary Note 2). For very small ϵ, this singularity causes a sharp change in the vonNeumann entropy. Since this sharp change occurs for very small parameter values ϵ, its effects can be prohibitively hard to detect numerically. However, when these effects appear, they dominate the behavior of the vonNeumann entropy and the physics which may directly depend on this entropy.
Understanding of the physics of sending noiseless quantum information across a noisy quantum channel is aided by the channel’s coherent information \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}\). To define a quantum channel and its coherent information, consider an isometry J: a ↦ b ⊗ c that generates a pair of quantum channels \({{{{{{{{{\mathcal{B}}}}}}}}}}:a\mapsto b\) and \({{{{{{{{{\mathcal{C}}}}}}}}}}:a\mapsto c\), where each channel may be called the complement of the other. These channels map an input density operator ρ_{a} to outputs \({\rho }_{b}:= {{{{{{{{{\mathcal{B}}}}}}}}}}({\rho }_{a})={{{{{{{{{{\rm{Tr}}}}}}}}}}}_{c}(J{\rho }_{a}{J}^{{{{{\dagger}}}} })\) and \({\rho }_{c}:= {{{{{{{{{\mathcal{C}}}}}}}}}}({\rho }_{a})={{{{{{{{{{\rm{Tr}}}}}}}}}}}_{b}(J{\rho }_{a}{J}^{{{{{\dagger}}}} })\), respectively. The dimensions d_{b} and d_{c}, of outputs b and c, respectively are the ranks of \({{{{{{{{{\mathcal{B}}}}}}}}}}({I}_{a})\) and \({{{{{{{{{\mathcal{C}}}}}}}}}}({I}_{a})\), respectively. These are the smallest possible output dimensions required to define \({{{{{{{{{\mathcal{B}}}}}}}}}}\) and \({{{{{{{{{\mathcal{C}}}}}}}}}}\) (in the notation of Def. 4.4.4 in^{68}, d_{b} is the Choirank of \({{{{{{{{{\mathcal{C}}}}}}}}}}\) and d_{c} is the Choirank of \({{{{{{{{{\mathcal{B}}}}}}}}}}\)). These definitions make the channel pair setting symmetric with respect to the replacement of one channel in the pair with its complement. The coherent information (or the entropy bias) of \({{{{{{{{{\mathcal{B}}}}}}}}}}\) at ρ_{a}, \({{\Delta }}({{{{{{{{{\mathcal{B}}}}}}}}}},{\rho }_{a}):= S({\rho }_{b})S({\rho }_{c})\), maximized over density operators ρ_{a} gives the channel coherent information\({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{B}}}}}}}}}})\).
When considering an input density operator, ρ_{a}(ϵ), we use a concise notation \({S}_{b}(\epsilon ):= S\left(\right.{\rho }_{b}(\epsilon )\left)\right.\), \({S}_{c}(\epsilon ):= S\left(\right.{\rho }_{c}(\epsilon )\left)\right.\), and \({{\Delta }}(\epsilon ):= {{\Delta }}\left(\right.{{{{{{{{{\mathcal{B}}}}}}}}}},{\rho }_{a}(\epsilon )\left)\right.={S}_{b}(\epsilon ){S}_{c}(\epsilon )\). At ϵ = 0 if ρ_{a}(ϵ) has rank d_{a}, then by definition of d_{b}, rank of ρ_{b}(0) will be d_{b} (see Supplementary Note 3), as a result, S_{b}(ϵ) will not have an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity. A similar argument shows if ρ_{a}(0) is rank d_{a} then S_{c}(ϵ) does not have an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity. When the rank of ρ_{a}(0) is strictly less than d_{a}, then an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity can be present in S_{b}(ϵ) or S_{c}(ϵ) (see Fig. 2), or an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity can be present in both S_{b}(ϵ) and S_{c}(ϵ) in which case the \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity with larger rate is said to be stronger.
Positivity of coherent information
Changes in the vonNeumann entropy caused by \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularities can act as a mechanism which makes \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{B}}}}}}}}}})\ > \ 0\) (see Fig. 3). To illustrate this mechanism consider a convex combination of input density operators \({\hat{\rho }}_{a}\) and σ_{a}:
This convex combination (2) leads to other such combinations,
at the outputs of \({{{{{{{{{\mathcal{B}}}}}}}}}}\) and \({{{{{{{{{\mathcal{C}}}}}}}}}}\), respectively. Let \({\hat{\rho }}_{a}\) be a pure state, then \({{\Delta }}({{{{{{{{{\mathcal{B}}}}}}}}}},{\hat{\rho }}_{a})=0\) i.e., Δ(0) = 0 (see Supplementary Note 3). Assume \({\hat{\rho }}_{a},{\sigma }_{a},\) and the channel pair \(({{{{{{{{{\mathcal{B}}}}}}}}}},{{{{{{{{{\mathcal{C}}}}}}}}}})\) are such that an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity is present in S_{b}(ϵ) but not in S_{c}(ϵ); that is, for a small enough increase in ϵ from zero, S_{b}(ϵ) increases by \( O(\epsilon {{{{{{{{\mathrm{log}}}}}}}}}\,\epsilon )\) but S_{c}(ϵ) has no \(O(\epsilon {{{{{{{{\mathrm{log}}}}}}}}}\,\epsilon )\) increase. Thus, for small enough ϵ, \({{\Delta }}(\epsilon )\simeq  O(\epsilon {{{{{{{{\mathrm{log}}}}}}}}}\,\epsilon ) \ > \ 0\); since \({{\Delta }}(\epsilon )\le {{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{B}}}}}}}}}})\), we conclude \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{B}}}}}}}}}})\ > \ 0\).
In the illustration above, let the channels \({{{{{{{{{\mathcal{B}}}}}}}}}}:a\mapsto b\) and \({{{{{{{{{\mathcal{C}}}}}}}}}}:a\mapsto c\) be defined by an isometry L: a ↦ b ⊗ c of the form,
where \(\{\lefti\right\rangle \}\) represents the standard basis, and \(\leftij\right\rangle\) denotes \(\lefti\right\rangle \otimes j\rangle \in b\otimes c\). Let [ψ] denote the dyad \(\psi \rangle \langle \psi \). A channel input of the form in (2) with \({\hat{\rho }}_{a}=[0]\) and σ = (9[1] + 4[2])/13 leads to channel outputs of the form (3). These outputs ρ_{b}(ϵ) and ρ_{c}(ϵ) have \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularities with rates x_{b} = 9/13 and x_{c} = 6/13, respectively (see Fig. 1). Since x_{b} > x_{c}, \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{B}}}}}}}}}})\ > \ 0\), as shown in Fig. 3.
In general, a \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularitybased mechanism can make \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{B}}}}}}}}}})\ > \ 0\) if there is a channel input ρ_{a}(ϵ) for which Δ(0) = 0, and either there is an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity in S_{b}(ϵ) but not in S_{c}(ϵ) or there is an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity in both S_{b}(ϵ) and S_{c}(ϵ) but the one in S_{b}(ϵ) is stronger, in either case, an argument similar to the one in our illustration above implies that \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{B}}}}}}}}}})\ > \ 0\). An analogous \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularitybased mechanism can make \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{C}}}}}}}}}})\ > \ 0\). In principle, this mechanism can be applied to a quantum channel \({{{{{{{{{\mathcal{B}}}}}}}}}}\), regardless of how small or large \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{B}}}}}}}}}})\) may be. In practice, we find that the above mechanism applies in a general situation presented next.
Theorem 1
If a quantum channel \({{{{{{{{{\mathcal{B}}}}}}}}}}\), with output and environment dimension d_{b} and d_{c} respectively, maps some pure state to an output of rank d_{c} < d_{b}, then \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{B}}}}}}}}}})\ > \ 0\).
This theorem applies quite generally, including cases where d_{b} ≥ d_{c}. In these d_{b} ≥ d_{c} cases, a channel \({{{{{{{{{\mathcal{B}}}}}}}}}}\)’s coherent information is strictly positive if the theorem holds for any subchannel of \({{{{{{{{{\mathcal{B}}}}}}}}}}\). There are simple examples (for instance see Supplementary Note 4) of channels with d_{b} = d_{c} where the above theorem applies.
Theorem 1 can be applied to the incomplete erasure channel \({{{{{{{{{\mathcal{C}}}}}}}}}}(\rho )=\lambda \rho \oplus (1\lambda ){{{{{{{{{{\mathcal{C}}}}}}}}}}}_{1}(\rho )\)^{47} whose output is split into two orthogonal subspaces (see Fig. 4). The channel’s input is sent unchanged to the first subspace with probability λ, else it is sent via a noisy channel \({{{{{{{{{{\mathcal{C}}}}}}}}}}}_{1}\) to the second subspace. This channel \({{{{{{{{{\mathcal{C}}}}}}}}}}\) is relevant for describing noise in experiments where the channel user knows if noise has acted or not.
Suppose the incomplete erasure channel \({{{{{{{{{\mathcal{C}}}}}}}}}}\) has a qubit input and \({{{{{{{{{{\mathcal{C}}}}}}}}}}}_{1}\) is any zero quantum capacity qubit channel with a qubit environment^{63}. Any such qubit channel \({{{{{{{{{{\mathcal{C}}}}}}}}}}}_{1}\) has a noise parameter 0 ≤ p ≤ 1/2; where at p = 0, \({{{{{{{{{{\mathcal{C}}}}}}}}}}}_{1}\) erases its input by taking it to a fixed pure state, making \({{{{{{{{{\mathcal{C}}}}}}}}}}\) a regular erasure channel. This regular erasure channel has zero coherent information, i.e., \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{C}}}}}}}}}})=0\) when λ is below a threshold λ_{0} = 1/2^{69}. As the noise in \({{{{{{{{{{\mathcal{C}}}}}}}}}}}_{1}\) is decreased by continuously increasing p, this threshold is expected to decrease continuously. While ordinary numerics may seem to confirm this expectation, simple use of Theorem 1 shows that for any p > 0, \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{C}}}}}}}}}})\ > \ 0\) for any λ > 0, i.e., an arbitrarily small increase in p from zero shifts the threshold value from λ_{0} = 1/2 to λ_{0} = 0. This discontinuous shift doesn’t appear in standard numerics because for small p, \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{C}}}}}}}}}})\) can be as small as O(e^{−1000})^{47}, a number much beyond ordinary numerical precision. Such discontinuous shifts reveal an unexpected behavior: a channel \({{{{{{{{{{\mathcal{C}}}}}}}}}}}_{1}\) which can’t send quantum information on its own, i.e., it has no quantum capacity, can nonetheless assist an incomplete erasure channel in sending quantum information. Such assistance was found previously for two specific qubit channels \({{{{{{{{{{\mathcal{C}}}}}}}}}}}_{1}\) using arguments tailored for those specific channels^{44,47}. Our argument here generalizes those results and points to \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularities as a generic mathematical cause behind this assistance. This assistance is particularly intriguing because it occurs for a wide variety of zero capacity qubit channels \({{{{{{{{{{\mathcal{C}}}}}}}}}}}_{1}\) but doesn’t occur for arbitrary zero capacity channels. For instance, when \({{{{{{{{{{\mathcal{C}}}}}}}}}}}_{1}\) is a zero quantum capacity erasure channel with arbitrary input dimension and erasure probability μ ≥ 1/2, \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{C}}}}}}}}}})=0\) for 0 ≤ λ ≤ 1 − 1/(2 μ).
To check if a channel \({{{{{{{{{\mathcal{B}}}}}}}}}}:a\mapsto b\) has strictly positive coherent information one may numerically find an input density operator ρ for which the entropy difference \({{\Delta }}({{{{{{{{{\mathcal{B}}}}}}}}}},\rho )\ > \ 0\). This numerical search can be unreliable because \({{\Delta }}({{{{{{{{{\mathcal{B}}}}}}}}}},\rho )\) is generally nonconvex in ρ and \({{\Delta }}({{{{{{{{{\mathcal{B}}}}}}}}}},\rho )\) can be affected by \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularities. The search can also be expensive for channels with large input and output dimensions. A corollary of Theorem 1 gives an algebraic result revealing a wide and simple set of channels with large output dimension and nonzero coherent information:
Corollary 1
Any channel \({{{{{{{{{\mathcal{B}}}}}}}}}}\) has \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{B}}}}}}}}}})\ > \ 0\) if its input dimension d_{a} > 1 and output dimension d_{b} > d_{a}(d_{c} − 1).
This result is easy to apply: given a channel one simply uses its dimensions to check if the channel satisfies the conditions of Corollary 1. For instance, Corollary 1 implies that any channel whose output dimension is larger than its input dimension has a strictly positive \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}\) whenever the channel’s environment is a qubit, i.e., d_{c} = 2.
Qubit channels are extremely useful for characterizing noise in experiments. Despite the vast body of work dedicated to studying the capacity of qubit channels^{63,70,71,72,73,74}, a basic question has remained open: when does the complement of a qubit channel have nonzero quantum capacity? Corollary 1, in conjunction with prior work^{63}, answers this question. If the complement of a qubit channel has output dimension 1 or 2, then conditions under which this complement has nonzero capacity can be found in ref. ^{63}; for all remaining cases, Corollary 1 (see Supplementary Note 4) shows that the complement has strictly positive coherent information and quantum capacity. This positivity result contains as a special case the results of^{65} which showed that any qubit Pauli channel \({{{{{{{{{\mathcal{B}}}}}}}}}}\) with d_{c} = 3 or 4 has a complement with positive channel coherent information.
Nonadditivity of coherent information
A mechanism based on \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularities can give rise to nonadditivity of \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}\). For two (possibly different) quantum channels \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\) and \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\), let
for some density operators \({\rho }_{a1}^{* }\) and \({\rho }_{a2}^{* }\) and let \({{{{{{{{{\mathcal{B}}}}}}}}}}:= {{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\otimes {{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\). Choose ρ_{a}(ϵ) at the input of \({{{{{{{{{\mathcal{B}}}}}}}}}}\) with the property that \({\rho }_{a}(0)={\rho }_{a1}^{* }\otimes {\rho }_{a2}^{* }\) and S_{b}(ϵ) has a stronger \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity than S_{c}(ϵ), then a small enough increase in ϵ from zero will increase Δ(ϵ) from \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1})+{{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2})\) by \( O(\epsilon {{{{{{{{\mathrm{log}}}}}}}}}\,\epsilon )\) indicating a strict inequality in (1). This \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularitybased mathematical mechanism responsible for nonadditivity requires S_{b}(ϵ) to have an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity. As stated earlier, this requirement can be satisfied if ρ_{a}(0) has less than full rank. A condition satisfied by several channels with zero and nonzero coherent information. This mechanism will now be used in an explicit instance of nonadditivity using two channels, one with zero and another with large positive coherent information.
To present this instance of nonadditivity, we introduce a lownoise qutrit channel \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\) whose coherent information is comparable to that of a qutrit identity channel. This channel’s superoperator \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}(\rho )={{{{{{{{{{\rm{Tr}}}}}}}}}}}_{c1}({J}_{1}\rho {J}_{1}^{{{{{\dagger}}}} })\) comes from an isometry J_{1}: a1 ↦ b1 ⊗ c1 of the form,
where 0 ≤ s ≤ 1. Since an exchange of s with 1 − s can be achieved by local unitaries in a1, b1, and c1, we restrict ourselves to 0 ≤ s ≤ 1/2. The channel coherent information \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1})\) is given by its entropy difference \({{\Delta }}({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1},{\rho }_{a1}^{* })\) where \({\rho }_{a1}^{* }=(1w)[0]+w[1]\), 0 < w < 1 (see Supplementary Note 5). At s = 0, \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1})=1\) and decreases monotonically with the noise parameter s to become ≃0.695 at s = 1/2. These values of \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1})\) bound the quantum capacity of \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\) from below and they are comparable to the quantum capacity, \({{{{{{{{{\mathrm{log}}}}}}}}}\,}_{2}3\), of the qutrit identity channel.
A \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularitybased argument stated earlier shows that using \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\) in parallel with \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\), a zero quantum capacity qubit amplitude damping channel with damping probability p ≥ 1/2, results in nonadditivity, i.e., a strict inequality in (1) for all 0 < s ≤ 1/2 and \(1/2\ \le \ p\ < \ \bar{p}(s)\) (see Fig. 5 and Methods Section). This nonadditivity has several interesting features. First, it shows the existence of previously unknown nonadditivity when using a lownoise channel. Second, the nonadditivity reported here fosters a more nuanced understanding of quantum capacity. Even in a setting using very simple lowdimensional channels, the quantum capacity provides an incomplete description of a channel’s ability to send quantum information. As shown here, despite having no quantum capacity, the qubit amplitude damping channel \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\) does posses a separate ability to assist transmission of quantum information when used in parallel with a simple qutrit channel \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\). Third, the nonadditivity here is robust against amplitude damping noise: additional amplitude damping noise beyond p = 1/2 does not immediately destroy this nonadditive effect which survives till \(p\ < \ \bar{p}(s)\). Fourth, this instance of nonadditivity has a very wide range: it is present over the entire parameter space of the qutrit channel \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\), except at a single point s = 0. Fifth, numerical techniques, which are commonly used to find nonadditivity, can easily miss this wide range of nonadditivity which appears because of changes in entropy caused by \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularities.
Discussion
We have discussed logarithmic (\({{{{{{{{\mathrm{log}}}}}}}}}\,\)) singularities that occur quite generally in the vonNeumann entropy of any density operator being moved linearly from the boundary to the interior of the set of density operators. In the region where this singularity occurs, it dominates the behavior of the vonNeumann entropy. This kind of dominance can be used to extract insights about physics which depends on this entropy. We have investigated the physics of sending quantum information. Our investigation leads to an insight that \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularities act as a mathematical source behind both positivity and nonadditivity in a channel’s coherent information. An analysis of \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularities could potentially be useful in other areas where the vonNeumann entropy plays a central role. One area of this type is the study of continuous variable channels. Capacities of these channels remain an active area of research^{75,76,77,78,79,80,81}, and these capacities also display a variety of exotic behaviour^{82}, including superactivation^{83,84,85}. Extending our \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity ideas to investigate such exotic behavior would be an interesting direction of future work.
Checking if any general channel has nonzero or zero quantum capacity is a fundamental but hard problem. While some general methods have been proposed to solve this problem^{61}, they are not always easy to apply and don’t necessarily lead to new channels with zero or positive quantum capacity. By contrast, the algebraic \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularitybased method proposed here is easy to apply and it unearths a variety of channels with positive quantum capacity. Using it, we give Theorem 1 which reveals certain general conditions for strict positivity of a channel’s coherent information. Corollary 1 of Theorem 1 unearths a wide variety of channels with strictly positive coherent information. Corollary 1 only makes use of a channel’s dimensions. Extending this corollary, for example by showing \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{B}}}}}}}}}})\ > \ 0\) for some d_{b} > d_{c}, would be an interesting direction of future research. Yet another direction would be to supplement our mathematical \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity reasoning with more physical arguments. Such reasoning may help explain the positivity of \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}\) found in the incomplete erasure channel and clarify why the simplest zero capacity channels behave differently from others when used as part of the incomplete erasure channel. This clarification may provide further insights into the transmission of quantum information. Another source of insight may be a quantitative analysis of bounds on the quantum capacity^{59,64,86,87,88,89,90,91} of the incomplete erasure channel or some other channel where a \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularitybased mechanism is responsible for strict positivity of the channel’s coherent information. In certain cases, \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularities can dominate continuitybased bounds on the coherent information and it would be interesting to see if such effects also appear in continuity bounds on a channel’s quantum capacity^{86}.
Using \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularities, we have shown that the coherent information and quantum capacity of several channels is nonzero. It follows that the twoway quantum capacity^{30} and the private capacity^{36} of these same channels are also nonzero. These observations comes from the simple fact that the quantum capacity of a channel is a lower bound on the channel’s private and twoway quantum capacities. These other capacities are even less understood than the quantum capacity and our \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularitybased analysis could prove useful in their investigation. For instance, the reverse coherent information^{92}, which is a lower bound on the twoway quantum capacity, may yield to a \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularitybased analysis, similar to the one performed here. Admittedly, our \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularitybased method for showing positivity of capacity does not solve the general problem of finding all channels with strictly positive capacity. Results concerning unbounded nonadditivity^{49,50} temper hopes about the existence of an easy to apply but a completely general method for checking positivity of the quantum capacity. Our work nonetheless points out that such tempering need not hinder progress in finding new, interesting, and potentially insightful instances of channels with strictly positive quantum capacity.
Another aspect of our findings is how changes in the vonNeumann entropy caused by \({{{{{{{{\mathrm{log}}}}}}}}}\,\) singularities is a mathematical mechanism responsible for nonadditivity. Prior search for such mechanisms have focussed on the structure of special channels^{42,56,57} used for obtaining nonadditivity or on the use of certain tailored quantum codes^{45,60,93}. We open another direction by showing how a fundamental property of the vonNeumann entropy can lead to nonadditivity. This \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity property can be analyzed to find nonadditivity in channels with large, small, or no quantum capacity. The algebraic nature of this analysis allows us to identify nonadditivity over wide ranges of a channel’s parameter, without the need for traditional numerics^{43,46,67}. While our work opens one path, it is not the only path forward. We leave open the exciting but challenging possibility of finding other mathematical and physical principles that may explain nonadditive effects in quantum information science.
Unlike prior work, nonadditivity of the coherent information reported here occurs in a lownoise channel. From a fundamental physics perspective, nonadditivity using the product of one lownoise channel with another lowdimensional but zero capacity qubit amplitude damping channel is surprising because it implies that even in such a simple setting the quantum capacity of the amplitude damping channel does not fully characterize its resourcefulness for sending quantum information.
This simple example adds to the collection of exotic channels from which further physics can be extracted. In principle, this lowdimensional and lownoise channel can be experimentally realized. Given the practical relevance of lownoise channels, our finding of nonadditivity in such channels suggests that nonadditivity is not just a fundamental curiosity but a potential resource for quantum technologies.
Methods
Proof of Theorem 1
A \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularitybased mechanism responsible for making a channel’s coherent information strictly positive is the key ingredient in the proof of Theorem 1. Assume d_{c} < d_{b} and \({{{{{{{{{\mathcal{B}}}}}}}}}}\) maps some pure state [ψ]_{a} to an output \({{{{{{{{{\mathcal{B}}}}}}}}}}({[\psi ]}_{a})\) of rank d_{c}. Consider an input density operator ρ_{a}(ϵ) of the form in eq. 2 where \({\hat{\rho }}_{a}\) is the pure state [ψ]_{a} and σ_{a} = I_{a}/d_{a}. The outputs ρ_{b}(ϵ) and ρ_{c}(ϵ) have the form in (3) where \({\hat{\rho }}_{b}\) and \({\hat{\rho }}_{c}\) have the same rank d_{c}, σ_{c}, and σ_{b} have ranks d_{c} and d_{b}, respectively (see para 3 in Methods Section). As a result S_{b}(ϵ) has an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity while S_{c}(ϵ) doesn’t, consequently \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{\mathcal{B}}}}}}}}}})\ > \ 0\). The absence of an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity in S_{c}(ϵ) follows from the fact that at ϵ = 0, ρ_{c}(ϵ) is a rank d_{c} operator \({\hat{\rho }}_{c}\). To notice the presence of an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity in S_{b}(ϵ), it is helpful to rewrite ρ_{b}(ϵ) in eq. 3, as
where
At ϵ = 0, ρ_{b}(ϵ) is \({\hat{\rho }}_{b}\), which has d_{b} − d_{c} zero eigenvalues. Corresponding to these zero eigenvalues is an eigenspace of dimension d_{b} − d_{c} > 0. Let P_{0} be a projector onto this eigenspace and
Since σ_{b} has rank d_{b}, the operator \({\tilde{\omega }}_{b}\) is positive definite on the support of P_{0}, thus \({\tilde{\omega }}_{b}\) has (d_{b} − d_{c}) strictly positive eigenvalues {e_{i}}. Elementary results from perturbation theory (for instance see Section 5.2 in ref. ^{94}) show that all d_{b} − d_{c} zero eigenvalues of ρ_{b}(ϵ) at ϵ = 0 become nonzero for positive ϵ, and to leading order in ϵ these eigenvalues increase linearly such that the i^{th} such eigenvalue is simply ϵe_{i}. As a result, S_{b}(ϵ) has an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity.
Nonadditivity
A \({{{{{{{{\mathrm{log}}}}}}}}}\,\)singularitybased mechanism is responsible for nonadditivity (1) when \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\) and its complement \({{{{{{{{{{\mathcal{C}}}}}}}}}}}_{1}\) are channels defined by J_{1} in (6) and \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\), along with its complement \({{{{{{{{{{\mathcal{C}}}}}}}}}}}_{2}\), are defined an isometry \({J}_{2}:{{{{{{{{{{\mathcal{H}}}}}}}}}}}_{a2}\mapsto {{{{{{{{{{\mathcal{H}}}}}}}}}}}_{b2}\otimes {{{{{{{{{{\mathcal{H}}}}}}}}}}}_{c2}\) of the form,
Here \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\) represents a qubit amplitude damping channel with damping probability p. We shall be interested in the parameter region 1/2 ≤ p ≤ 1 where \({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2}\) is antidegradable and \({{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2})={{{{{{{{{\mathcal{Q}}}}}}}}}}({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2})=0\)^{63}. Consider the channel pair \({{{{{{{{{\mathcal{B}}}}}}}}}}={{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1}\otimes {{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2},{{{{{{{{{\mathcal{C}}}}}}}}}}={{{{{{{{{{\mathcal{C}}}}}}}}}}}_{1}\otimes {{{{{{{{{{\mathcal{C}}}}}}}}}}}_{2}\), with channel input
where
0 ≤ ϵ ≤ 1, and w is chosen such that at ϵ = 0,
i.e., \({{\Delta }}(0)={{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{1})+{{{{{{{{{{\mathcal{Q}}}}}}}}}}}^{(1)}({{{{{{{{{{\mathcal{B}}}}}}}}}}}_{2})\). For any ϵ > 0 and s > 0, an eigenvalue \(\left(\right.(1p)w\left)\right.\epsilon\) of ρ_{b}(ϵ) and an eigenvalue (pkw)ϵ,
of ρ_{c}(ϵ) increases linearly from zero to leading order in ϵ. Thus S_{b}(ϵ) has an \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity of rate (1 − p)w and S_{c}(ϵ) has a \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity of rate pkw. The \(\epsilon\ {{{{{{{{\mathrm{log}}}}}}}}}\,\)singularity in S_{b}(ϵ) is stronger when \(p\ < \ \bar{p(s)}=1/(1+k)\) (plotted in Fig. 5). This stronger singularity implies nonadditivity, i.e., a strict inequality in eq. 1.
Data availability
No data sets were generated during this study.
Code availability
Source code for the plots in this study are available on a Github repository, https://github.com/vsiddhu/logSing.
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Acknowledgements
The author thanks Robert B. Griffiths, Graeme Smith, Costin Bădescu, Yang Gao, Michael Widom, and Mark M. Wilde for their useful comments. This work was partially supported by NSF CAREER Award CCF 1652560 and NSF Grant PHY 1915407.
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Siddhu, V. Entropic singularities give rise to quantum transmission. Nat Commun 12, 5750 (2021). https://doi.org/10.1038/s41467021259540
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DOI: https://doi.org/10.1038/s41467021259540
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