# A family of degradable fermionic Gaussian channels

Quantum computing is the future. Or the present, if you believe that some *real* quantum computers are commercially available already. Because it is hard to admit that any computer system whose design and theoretical basis depend on quantum effects for their operation may be on the market today, even though IBM says otherwise. The trick in this case being that, because quantum computers are extremely fragile, you buy them via the company’s ‘cloud’, meaning that you do not have them *physically* but can access them through the Internet.

From a fundamental point of view, the term quantum computing implies the use of quantum effects that have no classical analogue to process information. In a classical-physics computer, information is held in bits, which can have two alternative values (0 and 1). In a quantum computer the 0 and 1 values are held simultaneously in an entangled state. This unit of information is the quantum bit or qubit. Much more information can be held in this way and, *in principle*, it is possible to do a parallel processing of the information. Thus, quantum computers would be much faster than conventional machines and capable of performing calculations that could not realistically be done otherwise.

Let’s put some numbers on that last statement. For example, a computer register made of three classical physical bits can store any of eight numbers, namely 000, 001, 010, 100, 011, 101, 110 and 111, but it can obviously store only one of these numbers at a time. A register made of three qubits could actually store all eight numbers simultaneously. The key point about a quantum register is that if it can be made to change to perform a computation, the processing occurs on all possible numbers in the register simultaneously. Also, increasing the number of qubits in the register increases de numbers exponentially:* n* qubits can store 2^{n} numbers (IBM’s 20-qubit machine would store 1,048,576 numbers). Consequently, a working quantum computer would have the potential for massive amounts of parallel processing. It is as if the computer were operating simultaneously in many parallel universes.

But, and this is a big *but*, as we know, decoherence exists. In general, decoherence is any process in which a quantum mechanical state of a system is altered by the interaction between the system and its environment. It follows that decoherence must be a ubiquitous phenomenon in nature, responsible for the emergence of classical behaviour from the quantum substrate. Hence, when building a physical quantum computer, how would it be possible to access the information without collapsing the quantum state? The answer is that we would need quantum channels, *paths* along which quantum information can be transmitted reliably.

Again, the quantum world is weirder than you think. Quantum channels are not strictly-speaking physical channels, like a wire or a pipe, but mathematical formalisms substantiated on some physical information-carrier system. To be precise, a quantum channel is a completely positive, trace-preserving map on a suitable algebra of operators.

Therefore, the properties of quantum channels and their capacity to transmit classical or quantum information is central to quantum information processing. Channel capacities are difficult to compute since, in general, they require an optimization over entangled inputs to many channels in parallel and are only known for a few channels. Fortunately, these complications do not arise for the quantum capacity of degradable channels (*degradable* being a tag that depends on a linear algebra property of the channel operator expressed as a matrix; its exact meaning is not important for what follows), as degradable quantum channels are among the only channels whose quantum and private classical capacities are known ^{1}. Simply put, degradable channels are interesting because they have a simple formula that characterizes their quantum capacity.

On the other hand, the most natural information carrier in solid-state systems are electrons or, in general, fermions. Impressive experimental advances (e.g., edge channels, moving quantum dots, quantum dot arrays) demonstrate that electrons can be cleanly and individually transported in well-controlled semiconductor systems, providing fermionic quantum channels over sample-scale distances. These may serve for on-chip information transfer, like between different registers of a quantum processor. This progress makes the study of fermionic quantum channels all the more interesting, as they are also useful to describe the storage (transmission in time) of quantum information using fermionic systems.

Fermionic quantum channels have mostly been studied for non-interacting fermions, leading to the notion of fermionic Gaussian channels (roughly speaking, a Gaussian channel is that in which the output equals the input plus the noise, the noise assumed to be independent of the signal). So, the question is, how degradable are fermionic Gaussian channels?

This is exactly the question that Eliška Greplová and Ikerbasque Research Professor Géza Giedke (DIPC) investigate^{2}, and arrive to a most interesting conclusion: there is only a family of degradable fermionic Gaussian channels.

The researchers exploit fermionic phase-space methods to analyze the degradability of the fermionic Gaussian channels. They derive a simple standard form that simplifies further analysis. With the phase-space characterization of quantum channels in this form, they give a full characterization of all degradable fermionic Gaussian channels and show that there is only one family of such channels, the single-mode attenuation channel.

The simple structure of fermionic Gaussian channels allows a straightforward answer to the question of which n → n channels are degradable. This is in contrast to the available characterization for bosons, where the full degradability description is restricted to the one-mode case or to the notion of weak degradability.

This result may be of use in bounding the private and quantum capacity of some non-degradable channels or non-Gaussian channels.

So here we are. Still exploring the basics. If you read all of the above, would you easily buy there are commercial quantum computers already in the market? IBM’s uses electronics, you know, fermions.

*Author: César Tomé López is a science writer and the editor of Mapping Ignorance*

## References

- T.S. Cubitt et al (2008) The structure of degradable quantum channels
*Journal of Mathematical Physics*doi: 10.1063/1.2953685 ↩ - Eliška Greplová & Géza Giedke (2018) Degradability of Fermionic Gaussian Channels
*Physical Review Letters*doi: 10.1103/PhysRevLett.121.200501 ↩