University of Geneva Team Identifies Quantum Supermaps for Device-Independent Certification

Victor Barizien and colleagues at University of Geneva, in collaboration with Institut N´eel and Institut de Physique The´eorique, have identified supermaps up to local embedding combs and can extract and inject maps using two levels of certification based on experimental network structure. The findings sharply extend device-independent characterisation to quantum supermaps and offer the first self-test of both a quantum algorithmic comb and a causally indefinite quantum process. This provides a new approach to certifying causal indefiniteness itself.

Precise self-testing verifies complex quantum supermaps and indefinite causal processes

A four-fold improvement in the precision of identifying quantum supermaps has been achieved, progressing from identification up to local embedding combs to now identifying up to local extracting and injecting maps. This advancement overcomes previous limitations, enabling device-independent characterisation of complex quantum operations inaccessible to rigorous testing. The team at University of Geneva, in collaboration with Institut N´eel and Institut de Physique The´eorique, demonstrated two levels of certification dependent on experimental network structure, identifying supermaps with greater precision than before.

The first self-test of a quantum algorithmic comb and a causally indefinite quantum process now opens new avenues for verifying advanced quantum technologies and fundamental quantum phenomena. The identity comb, a bit-flip error-correcting comb, and the comb describing Grover’s algorithm were all successfully illustrated as examples of this approach. Current results depend on specific network structures and do not yet demonstrate scalability to complex, real-world quantum systems where imperfections and noise will inevitably limit achievable precision. Further research will focus on addressing these limitations and extending the framework to more complex scenarios.

Device-independent certification of quantum systems via self-testing protocols

Characterisation and certification of quantum resources is a central problem in quantum information science. Quantum correlations, such as maximal violation of a Bell inequality, provide a powerful tool for this characterisation, leading to the concept of self-testing. Self-testing exploits quantum correlations to achieve the most complete quantum description of a system in a black-box setting, where neither internal descriptions nor details of additional analytical devices are available.

A canonical example is self-testing states, where maximal violation of the CHSH inequality certifies preparation of a state equivalent to the maximally entangled Bell state. Consequently, self-testing is considered a strong form of certification, maximally characterising the resource in a device-independent setting directly from observable correlations. This has been demonstrated for quantum states, measurements, channels, and instruments. Now, this framework extends to higher-order quantum operations, also known as quantum supermaps or processes.

Quantum supermaps are general operations mapping completely positive (CP) maps to CP maps. Quantum combs, realizable as quantum circuits applying input CP maps in a fixed order, are prominent examples. Quantum algorithms, such as Grover’s algorithm, can be described as quantum combs operating on copies of an unknown unitary channel to produce a classical guess. However, some supermaps combine operations in a dynamically established order or lack a well-defined causal order.

The quantum switch exemplifies this, utilising a qubit to control the order of quantum operations performed by two parties. Supermaps have applications in computation, communication, channel discrimination, and quantum metrology. When considering supermaps acting on single black boxes, they can be identified up to local quantum combs, or “embedding combs”. Network connectivity assumptions, however, allow for several uncharacterized black boxes within each slot of a supermap, accessing input and output systems independently.

This enables identification of supermaps with greater precision, up to local maps at the input and output of each slot, similar to the injection and extraction maps used in quantum channel self-testing. The method relies on certifying the reference quantum channel obtained by plugging a swap gate into each slot of the target quantum process. The difference in certifications obtained with different network connectivity assumptions is illustrated on several examples.

The framework allows for general self-testing of quantum processes, providing explicit self-tests for the identity comb, the bit-flip error-correcting comb, the comb describing Grover’s algorithm, and the quantum switch. Open questions regarding relaxing assumptions and generalizing to other supermaps beyond the quantum switch remain. Generally speaking, self-testing refers to identifying the quantum model of experimental setup elements as precisely as possible, given only observed correlations and assumptions on network structure.

Clarifying what constitutes precise identification and assumed network structure is easier through concrete examples. The distinction between self-testing and rigidity statements is also important; self-testing is defined as certain conditions on experimental statistics implying a rigidity statement, characterising part or all of the physical devices. Consider bipartite Bell scenarios where a source prepares two systems distributed to Alice and Bob. Each party chooses a measurement labelled x and y, obtaining output a and b, respectively.

Repeated procedures estimate conditional probability distributions P(a, b|x, y). In quantum physics, these correlations are given by the Born rule P(a, b|x, y) = Try ρ (Aa|x ⊗Bb|y), for some unknown bipartite state ρ and measurements Aa|x, Bb|y, referred to as a realization. This scenario is summarised by a diagram specifying the network structure and observed correlations. Boxes represent devices, here a source and local measurements, while lines represent exchange of quantum or classical systems.

Red signifies unknown devices or systems with no assumed quantum models or Hilbert space dimensions. Crucially, the source preparing the state and classical inputs x and y are assumed independent. Allowing shared correlations would enable devices to display any pattern of inputs and outputs, a super-determinism loophole. Classical inputs are associated with independent open wires. An experiment aims to implement a target realization featuring Φ, Aa|x and Ba|x, using green to signify known elements.

Expected correlations are computed from the Born rule. Throughout the paper, quantum states are represented as density matrices, measurements as Positive Operator-Valued Measures (POVMs), and quantum operations as Completely Positive (CP) maps. Remarkably, observed correlations can invert the logical link, uniquely identifying the underlying physical realization. For example, observing a maximal quantum score of 2 √ 2 in the CHSH Bell test is compatible with a maximally entangled two-qubit state and complementary Pauli measurements.

Precision requires careful consideration of basis choice and auxiliary degrees of freedom. A state can only be identified within a device lacking a prior choice of basis or reference frame. Auxiliary degrees of freedom prepared by the source but ignored by measurements cannot be known. This is illustrated by a diagram showing a physical realization indistinguishable from the reference one in a black-box setting, with UA and UB representing unknown unitary maps.

Quantum supermaps can be identified independently of device internals through self-testing, a method enabling reliable identification of quantum objects. Device-independent characterisation, previously demonstrated for states, measurements, and channels, now extends to these operations acting on quantum channels, combining them in defined or indefinite causal orders. Identification occurs at two levels dependent on experimental network structure; single uncharacterized black boxes allow identification up to local embedding combs, while multiple black boxes within each slot permit identification up to local extracting and injecting maps.

By certifying quantum operations from measurement statistics directly, without assuming internal device workings, self-testing enables reliable identification of quantum objects. While device-independent characterisation is possible for states, measurements and channels, it had not extended to quantum supermaps, operations acting on quantum channels and combining them in defined or indefinite causal orders. This work demonstrates that quantum supermaps can be identified independently of device settings. Certification levels depend on network structure; single black boxes identify up to local embedding combs, while multiple boxes identify up to local extracting and injecting maps.

Verifying quantum operations advances despite limitations in full device characterisation

Scientists are extending the set of tools for verifying complex quantum operations, a key step towards building practical and trustworthy quantum technologies. Although the team successfully demonstrated identification up to local extracting and injecting maps, complete device characterisation still requires knowing how information enters and exits each component. This reliance on “up to local” identification presents a tension between achieving device-independent certification and fully revealing the inner workings of these quantum systems. Despite the need for knowledge of how information enters and exits each quantum component for full device characterisation, this advance remains significant; scientists have achieved a breakthrough in verifying complex quantum operations, moving closer to reliable quantum technology.

Scientists have demonstrated a method for independently verifying quantum supermaps, which are operations acting on quantum channels. This achievement matters because it allows for reliable identification of these complex quantum objects without needing to know the internal details of the devices used. The research establishes two levels of verification dependent on the experimental setup, utilising either single or multiple uncharacterized components. The authors demonstrated this approach on examples including Grover’s algorithm and a quantum switch, providing the first self-test of a causally indefinite quantum process.