11/27/2023 Sarah Maria Hagen

Written by Sarah Maria Hagen

A recent publication by Felix Leditzky, IQUIST faculty member, prompts a somewhat unusual question: "What does a certain family of quantum channels have in common with a platypus?" The semi-aquatic mammal is likely more well known than the concept to which it is compared: A quantum channel is used to model noisy communication between parties in a quantum information protocol.

Leditzky worked with four other researchers, including Graeme Smith, guest speaker at the IQUIST seminar last April, to investigate a class of quantum channels with unique properties, lending itself to a comparison with a platypus. Their work investigating the information-theoretic properties of this channel, specifically its various capacities (that is, the maximum amount of information that can be sent through the channel), was published across two papers this year. The first of these can be found in IEEE Transactions on Information Theory ("The Platypus of the Quantum Channel Zoo") and the other in Physical Review Letters (PRL) ("Generic Nonadditivity of Quantum Capacity in Simple Channels").

Sarah Hagen, a 3^{rd} year graduate student in Physics, sat down with Assistant professor Leditzky, who joined the Department of Mathematics at Illinois in January 2021, to learn more about his recent publications. The interview below has been lightly edited for clarity.

**SH: What is behind the comparison of a quantum channel to a platypus?**

FL: The platypus is endemic to Australia; when Europeans first arrived there, they literally thought someone had *stitched *a beak onto a rodent. By "stitching" two different channels together in our paper, we form the quantum channel we study, which I will discuss in more detail later.

In addition, the channel has surprising properties that we don't usually see together in a single family of channels. This unique behavior is the subject of our publications. Similarly, a platypus is a mammal, but it lays eggs and has poisonous fangs and a beak. At the end of the day, however, this comparison, which was Graeme [Smith]'s clever idea, should be taken with a grain of salt.

**SH: If the platypus comparison came later, how did this project start?**

FL: We've been thinking about this channel for a while; for me, this started while I was a postdoc in Canada. Vikesh [Siddhu] discovered this channel – a specific example – in previous work. These publications are the result of all five of us authors thinking about this interesting example for many years and working out the different results about its capacities for the standalone channel (in the IEEE paper) and how the behavior of these capacities does or does not change when considering the channel alongside other channels (in the PRL paper).

**SH: What does it mean to talk about the classical, private, and quantum capacities of this family of platypus channels?**

FL: In general, channel capacity measures how much information can be sent using the channel in question. We consider three different types of capacities depending on which type of information we are interested in transmitting. These types of information exist in a hierarchy: quantum capacity never exceeds private capacity, which in turn never exceeds classical capacity. This becomes intuitive when considering the meaning of each capacity.

Classical capacity can be thought of as a state discrimination problem. Recall that, in the simplest case, sending classical information conveys the value of a bit (0 or 1) faithfully. To transmit classical information using quantum states, such that the encoded information may be retrieved by a quantum measurement, we need to send orthogonal states that can be perfectly distinguished. A channel that preserves the orthogonality of quantum states is, therefore, one with maximal classical capacity. An example of a channel with maximum classical capacity is a dephasing channel, which can be associated with a non-trivial (orthogonal) basis in which classical information can be sent without any loss.

Private capacity considers how well we can establish a secret key or bit string between two parties that may be up against an eavesdropper seeking to gain information about their secret bits (just like in the setting of quantum key distribution). In order to incorporate the eavesdropper, we model its actions as an environment. As a result, our quantum channel becomes part of an open system – in thermodynamics, this is called the heat bath. In an open system, assuming the unitary evolution axiom of quantum mechanics to be true, the entire system (eavesdropper included) is evolving. However, the two parties trying to communicate private information only have access to a subsystem, which we look at to measure the maximal private (secret) information that can be sent.

We take a similar approach to calculate the quantum capacity of a channel. Here the capacity is measured by quantifying the degradation of entanglement once one part of the system has been sent through the channel in question. In general, channels try to decohere quantum systems, so you can think of the channel as a noise model. Measuring the decoherence is part of determining its quantum capacity.

One final important element is that we always calculate these capacities while considering many uses of the channel by the parties in question. Specifically, we make the i.i.d. (independent and identically distributed [channel uses]) assumption, which means that we view each transmission of a state through the channel as applying the same noise model repeatedly, which could be either in series or parallel. Then the capacity is the maximum amount of average information conveyed through many uses of the channel.

**SH: Considering the eavesdropper/environment picture defines new classes of quantum channels that you use to construct the special platypus channel. What are these and how can you "stitch" these together?**

FL: First, let's focus on how the eavesdropper influences the two parties that want to share information: the sender and the receiver. The sender might leak information to the eavesdropper and this leakage can be represented by another channel. Now our picture includes channels from the sender to both the receiver and eavesdropper, like two edges of a triangle.

There is a special class of channels in which the receiver can determine what the eavesdropper receives. Because the receiver can essentially apply another channel to his outcome (the third edge of the triangle) in order to model the eavesdropper's picture, this type of channel is called degradable. A degradable channel means that Bob is essentially the stronger party – as opposed to the eavesdropper. On the other hand, it is possible for the eavesdropper to be in the stronger position. When the eavesdropper can model what Bob receives instead, this type of channel is called anti-degradable.

In order to form our platypus channel, we take a specific example of a degradable channel acting on the 0 and 1 states and "stitch" it with an anti-degradable channel that acts on the 1 and 2 states. This stitching is possible because the two channels map the 1 state to the same output; the rest of the channel simply adopts the respective channels actions on the 0 and 2 states. We must make sure that the resulting channel still preserves the orthogonality of the input states, but in total we now have a channel that acts on a qutrit, a three-level system made up of 0, 1, and 2.

**SH: Your main results are about the (non)additivity of this channel in different settings. Can you detail these concepts further?**

LH: In order to think of additivity, we need to continue the previous discussion of using a channel multiple times, which was introduced earlier when discussing capacities. In general, we can think of a more complex system with many parts that could be correlated. These potentially entangled subsystems can be used to increase the amount of entanglement conveyed across the channel, when using it many times. If by doing this, on average, we *increase *the correlations established across the channel, this channel exhibits *super*additivity (or non-additivity). Conversely, additivity occurs when, while using the channel multiple times, even in a strategic way, only as much correlation on average occurs as would through a single use of the channel.

An example of a super additive channel for quantum information is the depolarizing channel. Superadditivity can be both good and bad. In some sense, a superadditive channel is good news because your rates of distributing entanglement across it are higher whenusing the channel multiple times. But for us, this creates an unbounded optimization problem because you now need to consider more and more channel uses and systems sent through that channel to figure out what is best.

We can also consider a channel that is used together with another channel; instead of just multiple versions of the same single channel, you are now sending different systems simultaneously through different channels. In this case, you can once again look at additivity versus superadditivity. The latter occurs when using the channels together allows for more information transmission than when using the sum of the capacities of the individual channels.

**SH: How does this work fit in with previous research?**

FL: Studying superadditivity of quantum channels is a part of quantum information science that is very old, that is to say, from the 90s. At the same time that people were looking into quantum error correction, people started looking into ways of determining the capacity of a quantum channel. Although this is not an easy task, a lot of work has been done in this area, and I would say that most of the open questions are the truly big ones.

In these works, the family of channels (the platypus channel) we study exhibits interesting behavior regarding its different capacities when viewing the channel used alone (1^{st} paper) or in conjunction with other well-understood channels (2^{nd} paper). We believe that by understanding this behavior – and how it arises – we can increase our understanding of how to calculate the capacity of channels in general.

**SH: Now that the theoretical foundation has been laid, could you summarize what you and your collaborators discovered about this family of platypus channels?**

FL: The platypus channel we present is parametrized by a parameter, which in turn yields a whole family of channels. We also consider higher-dimensional variants of the platypus channel, but the qutrit channel we consider explicitly remains the best way to think about it.

As I described earlier, to form the platypus channel we stitched degradable and anti-degradable channels together – both channels are *additive*. Although it belongs to neither of these classes, nor any other known additive channel classes, our resulting channel is also *additive* for each of its capacities (when considering exclusive use of the platypus channel). In the quantum case, the additivity result relies on a physically motivated conjecture about an entropy optimization problem that we strongly believe is true.

However, when we consider the platypus channel used alongside well-studied qubit channels, we see that these capacities become *superadditive*;* *that is, the additivity of the channel is not preserved when combined with channels including the erasure channel, depolarizing channels, qubit Pauli channels, the amplitude damping channel, and even randomly chosen qubit channels!

Taken together, this means that the platypus family of channels is *weakly additive* because it remains additive when it is the only channel we use but becomes superadditive when taken together with other channels. This is unique and somewhat surprising behavior, especially considering that the channels that made up the platypus channels were additive.

The higher dimensional variants of the qutrit platypus channel we consider in our second paper yield the same results – the platypus channel is superadditive, while this superadditivity is heightened for the higher dimensions.

**SH: Finally, what's next in this line of research?**

FL: This family of platypus channels represents a special case of a quantum channel in that its capacities are still computable, although the channel itself does not belong to any of the classes for which capacities are easily computable. As of now we cannot reproduce this property, but we would like to identify a sufficient mathematical criterion that gives rise to it. For example, we do not know if the "stitching" is what leads the channel to exhibit this unique behavior, or if it is another mathematical element in the definition of this family of channels. In this manner, one main open problem is to work to better understand the stitching itself. In general, we would also like to find more examples like this one: simple channels that exhibit rich behavior.