Even though coherent control of quantum operations appears to be achievable in practice, it is still not yet well understood. Among theoretical challenges, standard completely positive trace preserving (CPTP) maps are known not to be appropriate to represent coherently controlled quantum channels. We introduce here a graphical language for coherent control of general quantum channels inspired by practical quantum optical setups involving polarising beam splitters (PBS). We consider different situations of coherent control and disambiguate CPTP maps by considering purified channels, an extension of Stinespring’s dilation. First, we show that in classical control settings, the observational equivalence classes of purified channels correspond to the standard definition of quantum channels (CPTP maps). Then, we propose a refinement of this equivalence class generalising the "half quantum switch" situation, where one is allowed to coherently control which quantum channel is applied; in this case, quantum channel implementations can be distinguished using a so-called transformation matrix. A further refinement characterising observational equivalence with general extended PBS-diagrams as contexts is also obtained. Finally, we propose a refinement that could be used for more general coherent control settings.
Even though coherent control of quantum operations appears to be achievable in practice, it is still not yet well understood. Among theoretical challenges, standard completely positive trace preserving (CPTP) maps are known not to be appropriate to represent coherently controlled quantum channels. We introduce here a graphical language for coherent control of general quantum channels inspired by practical quantum optical setups involving polarising beam splitters (PBS). We consider different situations of coherent control and disambiguate CPTP maps by considering purified channels, an extension of Stinespring’s dilation. First, we show that in classical control settings, the observational equivalence classes of purified channels correspond to the standard definition of quantum channels (CPTP maps). Then, we propose a refinement of this equivalence class generalising the "half quantum switch" situation, where one is allowed to coherently control which quantum channel is applied; in this case, quantum channel implementations can be distinguished using a so-called transformation matrix. A further refinement characterising observational equivalence with general extended PBS-diagrams as contexts is also obtained. Finally, we propose a refinement that could be used for more general coherent control settings.
Even though coherent control of quantum operations appears to be achievable in practice, it is still not yet well understood. Among theoretical challenges, standard completely positive trace preserving (CPTP) maps are known not to be appropriate to represent coherently controlled quantum channels. We introduce here a graphical language for coherent control of general quantum channels inspired by practical quantum optical setups involving polarising beam splitters (PBS). We consider different situations of coherent control and disambiguate CPTP maps by considering purified channels, an extension of Stinespring's dilation.First, we show that in classical control settings, the observational equivalence classes of purified channels correspond to the standard definition of quantum channels (CPTP maps). Then, we propose a refinement of this equivalence class generalising the "half quantum switch" situation, where one is allowed to coherently control which quantum channel is applied; in this case, quantum channel implementations can be distinguished using a so-called transformation matrix. A further refinement characterising observational equivalence with general extended PBS-diagrams as contexts is also obtained. Finally, we propose a refinement that could be used for more general coherent control settings.
We introduce the PBS-calculus to represent and reason on quantum computations involving coherent control of quantum operations. Coherent control, and in particular indefinite causal order, is known to enable multiple computational and communication advantages over classically ordered models like quantum circuits. The PBS-calculus is inspired by quantum optics, in particular the polarising beam splitter (PBS for short). We formalise the syntax and the semantics of the PBS-diagrams, and we equip the language with an equational theory, which is proved to be sound and complete: two diagrams are representing the same quantum evolution if and only if one can be transformed into the other using the rules of the PBS-calculus. Moreover, we show that the equational theory is minimal. Finally, we consider applications like the implementation of controlled permutations and the unrolling of loops.
We introduce the PBS-calculus to represent and reason on quantum computations involving polarising beam splitters (PBS for short). PBS-diagrams can be used to represent various schemes including quantum-controlled computations, which are known to have multiple computational and communication advantages over classically ordered models like quantum circuits. The PBS-calculus is equipped with an equational theory, which is proved to be sound and complete: two diagrams are representing the same quantum evolution if and only if one can be transformed into the other using the rules of the PBS-calculus. Moreover, we show that the equational theory is minimal. Finally, we show that any PBS-diagram involving only unitary matrices can be transformed into a diagram without feedback loop.
Coherent control of quantum computations can be used to improve some quantum protocols and algorithms. For instance, the complexity of implementing the permutation of some given unitary transformations can be strictly decreased by allowing coherent control, rather than using the standard quantum circuit model. In this paper, we address the problem of optimising the resources of coherently controlled quantum computations. We refine the PBS-calculus, a graphical language for coherent control which is inspired by quantum optics. In order to obtain a more resource-sensitive language, it manipulates abstract gates - that can be interpreted as queries to an oracle - and more importantly, it avoids the representation of useless wires by allowing unsaturated polarising beam splitters. Technically the language forms a coloured PROP. The language is equipped with an equational theory that we show to be sound, complete, and minimal. Regarding resource optimisation, we introduce an efficient procedure to minimise the number of oracle queries of a given diagram. We also consider the problem of minimising both the number of oracle queries and the number of polarising beam splitters. We show that this optimisation problem is NP-hard in general, but introduce an efficient heuristic that produces optimal diagrams when at most one query to each oracle is required.
Abstract In this chapter we visit the ‘church of the larger Hilbert space’: we take a closer look at the relationship of pure and mixed quantum states via purification, and discuss a similar connection between completely positive and trace-preserving (CPTP) maps and unitary dynamics on a larger Hilbert space. We show that such CPTP maps, which represent quantum channels, are equivalent to operations admitting a Kraus decomposition, as well as to unitary dynamics on a larger Hilbert space with factoring initial condition, the famous Stinespring dilation of quantum operations. In this context we introduce the Choi-Jamiołkowski isomorphism, which expresses a duality between maps between two Hilbert spaces and states on tensor products of these Hilbert spaces, and we review several important examples of quantum channels such as the dephasing channel, amplitude-damping channel, and depolarizing channel. Finally, we examine operations that are not possible in quantum theory, focusing on the no-cloning theorem.
Supermaps between quantum channels (completely positive trace-preserving (CPTP) maps of matrix algebras) were introduced in [Chiribella et al., EPL 83(3) (2008)]. In this work we generalise to supermaps between channels of any type; by channels we mean CPTP maps of finite-dimensional C*-algebras. Channels include POVMs, quantum instruments, classically controlled families of quantum channels, classical channels, quantum multimeters, and more. We show that deterministic supermaps between channels of any type can be realised using simple circuits, recovering the previous realisation theorems of [Chiribella et al., EPL 83(3) (2008)] (for deterministic supermaps between quantum channels) and [Bluhm et al. (2024)] (for deterministic supermaps between quantum multimeters) as special cases. To prove this realisation theorem we use the graphical calculus of the 2-category of finite-dimensional 2-Hilbert spaces; the paper includes an accessible and elementary introduction to this graphical calculus, and no prior knowledge of category theory is expected of the reader.