Transforming quantum operations: Quantum supermaps

Abstract

We introduce the concept of quantum supermap, describing the most general transformation that maps an input quantum operation into an output quantum operation. Since quantum operations include as special cases quantum states, effects, and measurements, quantum supermaps describe all possible transformations between elementary quantum objects (quantum systems as well as quantum devices). After giving the axiomatic definition of supermap, we prove a realization theorem, which shows that any supermap can be physically implemented as a simple quantum circuit. Applications to quantum programming, cloning, discrimination, estimation, information-disturbance trade-off, and tomography of channels are outlined.PACS numbers: 03.65.Ta, 03.67.-aThe input-output description of any quantum device is provided by the quantum operation of Kraus [1], which yields the most general probabilistic evolution of a quantum state. Precisely, the output state ρ out is given by the quantum operation E applied to the input state ρ in as followswhere p(E|ρ in ) is the probability of E occurring on state ρ in , when E is one of a set of alternative transformations, such as in a quantum measurement. Owing to its physical meaning, a quantum operation E must be a linear, trace non-increasing, completely positive (CP) map (see, e.g.[2]). The most general form of such a map is known as Kraus formwhere the operators E j satisfy the bound1. Tracepreserving maps, i.e. those achieving the bound, are a particular kind of quantum operations: they occur deterministically and are referred to as quantum channels.In general it is convenient to consider two different input and output Hilbert spaces H in and H out , respectively. In this way, the concept of quantum operation can be used to treat also quantum states, effects, and measurements, which describe the properties of elementary quantum objects such as quantum systems and measuring devices. Indeed, states can be described as quantum operations with one-dimensional H in , i.e. with Kraus operators E j given by ket-vectors √ p j |ψ j ∈ H out , thus yielding the output state ρ out = E(1) = j p j |ψ j ψ j |.A quantum effect 0 ≤ P ≤ I [3] corresponds instead to a quantum operation with one-dimensional H out , i.e. with Kraus operators given by bra-vectorsMore generally, any quantum measurement can be viewed as a particular quantum operation, namely as a quantum-to-classical channel [4]. Channels, states, effects, and measurements are all special cases of quantum operations. What about then considering maps between quantum operations themselves? They would describe the most general kind of transformations between elementary quantum objects. For example a programmable channel [5] would be a map of this type, with a quantum state at the input and a channel at the output. Or else, a device that optimally clones a set of unknown unitary gates would be a map from channels to channels. We will call such a general class of quantum maps quantum supermaps, as they transform CP maps (sometimes referred to as superoper...

Probabilistic theories with purification

“…A concrete example of non-causal theory is the theory studied in Refs. [32,33], where the states are quantum operations, and the transformations are "supermaps" transforming quantum operations into quantum operations. In this case, transforming a "state" means inserting the corresponding quantum operation in a larger circuit, and the sequence of two such transformations is not a causal sequence.…”

Causally nonseparable processes admitting a causal model

“…The CJ-isomorphism can also be used to represent 'superoperators' or 'processes' which map quantum maps to quantum maps, quantum states or probabilities [3,14,[17][18][19]. In this paper, we will focus on processes mapping two quantum operations x x a and h y b -corresponding to the CJ representation of Alice's and Bob's CP maps-to a probability (see figure 1).…”

Questioning the rules of the game

“…[22] for the general framework and Refs. [31,32] for two explicit examples of noncausal theories). The second reason is that we want to stress that causality is an essential ingredient in our derivation.…”