Transforming quantum operations: quantum supermaps (Chiribella, D'Ariano and Perinotti, 2008) introduced the supermap formalism — the most general physically realizable transformation that takes an unknown quantum operation as input and produces a quantum operation as output — and proposed, as a new open problem, the information–disturbance trade-off for quantum operations: how much information can be extracted about an unknown operation (for example by estimating an unknown unitary channel) while the operation handed on remains as close as possible to the undisturbed one, with channel-based cryptography as the envisioned operational payoff. The flagship instance — an unknown unitary drawn at random according to the Haar measure — was completely solved by the same group in 2010 (Bisio, Chiribella, D'Ariano and Perinotti), including the optimal trade-off curve and an explicit optimal network. What remains open, and is left explicitly as future work by that solution, is the general problem: characterize the optimal estimation-versus-disturbance trade-off, and construct the optimal supermaps achieving it, for ensembles of quantum operations beyond Haar-covariant unitary families — restricted and non-covariant sets of unitaries, and noisy or otherwise non-unitary channel ensembles. The sources additionally raise, as an intriguing possibility and motivation rather than as a listed future-work item, the prospect of channel-based cryptographic protocols built on this trade-off.

Background

A quantum operation (Kraus operation) is the most general probabilistic evolution of a quantum system: a linear, trace-non-increasing, completely positive map on density operators, of which a quantum channel is the deterministic (trace-preserving) special case. Whereas the classical information–disturbance principle concerns measurements on unknown quantum states — any information gained about a non-orthogonal ensemble of states necessarily disturbs them, which underlies quantum cryptography — the 2008 supermaps paper asked what happens when the unknown object is itself a dynamics, i.e. an operation. The natural language for this is the supermap: a higher-order transformation that accepts an input operation (a black box with a fixed input and output system) and produces an output operation, in the same way an ordinary channel transforms states. Chiribella, D'Ariano and Perinotti proved a realization theorem: every admissible supermap is physically implementable by plugging the unknown black box into a fixed quantum circuit, and the single-use case is equivalently described by the quantum comb / quantum network formalism (a decomposition of the circuit into its ordered slots for the black box), with process POVMs (a 'tester': a POVM — the general form of a quantum measurement, given by positive operators summing to the identity — whose outcomes label which operations ran in the circuit) capturing the purely information-extracting case. Related impossibility context: the no-programming theorem of Nielsen and Chuang (1997) shows that no finite quantum program can implement all unitary channels exactly on a single device, so exact deterministic storage of an unknown operation is ruled out and only approximate or probabilistic strategies are available; by contrast, exact cloning of unitary channels is possible in the comb formalism with sufficiently many uses, one of several ways in which channel-level manipulation differs structurally from state-level manipulation. Within this formalism the paper posed the information–disturbance trade-off for quantum operations: interpose a supermap on an unknown operation, gain information about it (e.g. an estimate of an unknown unitary UU), and simultaneously output an operation that is the most similar possible to UU; and asked whether the resulting trade-off supports new cryptographic protocols based on channels rather than states. The paradigmatic instance — estimating a completely unknown dd-dimensional unitary drawn from the Haar measure (the unique unitary-invariant distribution on the unitary group) while outputting a channel maximally close to it — was completely solved in 2010 by Bisio, Chiribella, D'Ariano and Perinotti: the optimal trade-off curve between estimation fidelity and channel fidelity was derived in closed form (coinciding, non-trivially, with the curve for estimating a maximally entangled state, although the Choi isomorphism — the canonical one-to-one correspondence between quantum channels and bipartite states — does not make the two problems operationally equivalent), and an explicit optimal quantum network — a superposition-inducing interaction followed by a Bell measurement (a joint measurement in the maximally entangled basis) and unitary feed-forward — was constructed. That solution, however, relies essentially on Haar (group) covariance: symmetry lets the optimization over all networks be parallelized and solved by group representation theory. The same paper explicitly leaves open the trade-off for non-universal, restricted sets of unitaries (it singles out two mutually unbiased bases of qubit unitaries — bases whose vectors give equal-magnitude overlaps with each other — as used in two-way quantum key distribution) and the connection to cloning of unitary channels; the cryptographic application appears as motivation — the optimal network is noted to already constitute a non-trivial eavesdropping attack on such a two-way protocol — rather than as an explicitly listed open item. Subsequent work has filled in surrounding territory without solving the general problem: optimal learning of unitary transformations without a disturbance constraint (Bisio, Chiribella, D'Ariano, Facchini and Perinotti, 2010); optimal processing of NN uses of unitary channels into a single use (Bisio, D'Ariano, Perinotti and Sedlák, 2014); a universal optimal trade-off for approximate measurements on states (Hashagen and Wolf, 2019), whose universality phenomenon has no known channel-level counterpart; uncertainty relations for process-measurements (Xiao, Sengupta, Yang and Gour, 2021), a channel-level incompatibility principle distinct from the estimation-versus-disturbance curve; a bidirectional-QKD analysis of a restricted two-unitary instance built from the 2010 optimal network (Abdul Salam, Shamsul Shaari and Mancini, 2024); and a general framework for optimal cloning of channels with super-replication criteria and a link to Bayesian channel estimation (Sekatski, Guryanova, Kothakonda and Skotiniotis, 2025).

Problem Statement

For ensembles (prior distributions) of unknown quantum operations beyond the solved Haar-covariant unitary case — restricted or non-covariant sets of unitaries, or noisy/non-unitary channel ensembles — determine the optimal information–disturbance trade-off at the level of supermaps: in the single-use setting of the source formulation, in which one application of the unknown operation is interposed into the supermap, characterize the full achievable region of pairs (information gained about the operation, e.g. estimation or discrimination fidelity; fidelity of the simultaneously produced output channel to the undisturbed operation) under explicitly stated figures of merit with operational meaning for both coordinates, construct the supermap (quantum network) attaining the boundary of this region, and prove optimality. A complete answer provides a general characterization covering such ensembles, not a solution for one self-selected instance: the two classes named above are conjunctively required, so covering only one class, or only a single example within a class, does not resolve the problem, while the source's own non-orthogonal restricted sets (two mutually unbiased bases of qubit unitaries) are one required case within this scope. Note in this regard that ensembles whose members can be perfectly distinguished without disturbance (e.g. sets of mutually orthogonal unitaries) yield a trivial region (perfect information at zero disturbance), so such degenerate cases do not by themselves resolve the problem. As motivation, the sources raise the question whether the trade-off can underpin the security of channel-based cryptographic primitives (e.g. two-way protocols whose key is encoded in non-orthogonal unitary transformations); this prospect is context for the problem rather than a required deliverable. The common objective is the quantitative extension of the information–disturbance principle from quantum states to quantum operations in the general, non-covariant setting; the Haar-uniform unitary ensemble, already solved by Bisio et al. (2010), serves as the consistency check any general answer must reproduce.

References

  1. G. Chiribella, G. M. D'Ariano, P. Perinotti, "Transforming quantum operations: Quantum supermaps", EPL 83, 30004 (2008). DOI: 10.1209/0295-5075/83/30004; arXiv:0804.0180; https://arxiv.org/abs/0804.0180
  2. A. Bisio, G. Chiribella, G. M. D'Ariano, P. Perinotti, "Information-disturbance tradeoff in estimating a unitary transformation", Physical Review A 82, 062305 (2010). DOI: 10.1103/PhysRevA.82.062305; arXiv:1006.5665; https://arxiv.org/abs/1006.5665
  3. A. Bisio, G. Chiribella, G. M. D'Ariano, S. Facchini, P. Perinotti, "Optimal quantum learning of a unitary transformation", Physical Review A 81, 032324 (2010). DOI: 10.1103/PhysRevA.81.032324; arXiv:0903.0543; https://arxiv.org/abs/0903.0543
  4. A. Bisio, G. M. D'Ariano, P. Perinotti, M. Sedlák, "Optimal processing of reversible quantum channels", Physics Letters A 378, 1797–1808 (2014). DOI: 10.1016/j.physleta.2014.04.042; arXiv:1308.3254; https://arxiv.org/abs/1308.3254
  5. A.-L. K. Hashagen, M. M. Wolf, "Universality and Optimality in the Information–Disturbance Tradeoff", Annales Henri Poincaré 20, 219–258 (2019). DOI: 10.1007/s00023-018-0724-0; arXiv:1802.09893; https://arxiv.org/abs/1802.09893
  6. Y. Xiao, K. Sengupta, S. Yang, G. Gour, "Uncertainty principle of quantum processes", Physical Review Research 3, 023077 (2021). DOI: 10.1103/PhysRevResearch.3.023077; arXiv:2004.05315; https://arxiv.org/abs/2004.05315
  7. N. R. S. Abdul Salam, J. Shamsul Shaari, S. Mancini, "Information disturbance tradeoff in bidirectional QKD", Physica Scripta 99, 105135 (2024). DOI: 10.1088/1402-4896/ad7912; arXiv:2403.19156; https://arxiv.org/abs/2403.19156
  8. P. Sekatski, Y. Guryanova, N. B. T. Kothakonda, M. Skotiniotis, "Cloning Quantum Channels" (2025). arXiv:2509.08059; https://arxiv.org/abs/2509.08059
  9. M. A. Nielsen, I. L. Chuang, "Programmable Quantum Gate Arrays", Physical Review Letters 79, 321–324 (1997). DOI: 10.1103/PhysRevLett.79.321; arXiv:quant-ph/9703032; https://arxiv.org/abs/quant-ph/9703032