Alexander Müller-Hermes

• Piana, Francesca La & Müller-Hermes, Alexander (2026). Annihilating and breaking Lorentz cone entanglement. Linear Algebra and its Applications. ISSN 0024-3795. 739, p. 68–103. doi: 10.1016/j.laa.2026.03.012. Full text in Research Archive Show summary

  Linear maps between finite-dimensional ordered vector spaces with orders induced by proper cones CA and CB are called entanglement breaking if their partial application sends the maximal tensor product K⊗maxCA into the minimal tensor product K⊗minCB for any proper cone K. We study the larger class of Lorentz-entanglement breaking maps where K is restricted to be a Lorentz cone of any dimension, i.e., any cone over a Euclidean ball. This class of maps appeared recently in the study of asymptotic entanglement annihilation and it is dual to the linear maps factoring through Lorentz cones. Our main results establish connections between these classes of maps and operator ideals studied in the theory of Banach spaces. For operators u:X→Y between finite-dimensional normed spaces X and Y we consider so-called central maps which are positive with respect to the cones CA=CX and CB=CY. We show how to characterize when such a map factors through a Lorentz cone and when it is Lorentz-entanglement breaking by using the Hilbert-space factorization norm γ2 and its dual γ2⁎. We also study the class of Lorentz-entanglement annihilating maps whose local application sends the Lorentzian tensor product CA⊗LCA into the minimal tensor product CB⊗minCB. When CA is a cone over a finite-dimensional normed space and CB is a Lorentz cone itself, the central maps of this kind can be characterized by the 2-summing norm π2. Finally, we prove interesting connections between these classes of maps for general cones, and we identify examples with particular properties, e.g., cones with an analogue of the 2-summing property.

• Aubrun, Guillaume & Müller-Hermes, Alexander (2025). Asymptotic Tensor Powers of Banach Spaces. Annales de l'Institut Fourier. ISSN 0373-0956. p. 1–27. doi: 10.5802/aif.3736. Full text in Research Archive Show summary

  We study the asymptotic behaviour of large tensor powers of normed spaces and of operators between them. We define the tensor radius of a finite-dimensional normed space X as the limit of the sequence A k 1 / k , where A k is the equivalence constant between the projective and injective norms on X ⊗ k . We show that Euclidean spaces are characterized by the property that their tensor radius equals their dimension. Moreover, we compute the tensor radius for spaces with enough symmetries, such as the spaces ℓ p n . We also define the tensor radius of an operator T as the limit of the sequence B k 1 / k , where B k is the injective-to-projective norm of T ⊗ k . We show that the tensor radius of an operator whose domain or range is Euclidean is equal to its nuclear norm, and give some evidence that this property might characterize Euclidean spaces.

• Aubrun, Guillaume & Müller-Hermes, Alexander (2025). Limit formulas for norms of tensor power operators. Journal of Functional Analysis. ISSN 0022-1236. 289(10). doi: 10.1016/j.jfa.2025.111113. Full text in Research Archive Show summary

  Given an operator φ : X → Y between Banach spaces, we consider its tensor powers φ⊗k as operators from the k-fold injective tensor product of X to the k-fold projective tensor product of Y . We show that after taking the kth root, the operator norm of φ⊗k converges to the 2-dominated norm γ∗2 (φ), one of the standard operator ideal norms.

• Aubrun, Guillaume; Hermes, Alexander Mueller- & Plavala, Martin (2024). Monogamy of entanglement between cones. Mathematische Annalen. ISSN 0025-5831. 391, p. 1591–1609. doi: 10.1007/s00208-024-02935-4. Full text in Research Archive
• Belzig, Paula; Christandl, Matthias & Hermes, Alexander Mueller- (2024). Fault-Tolerant Coding for Entanglement-Assisted Communication. IEEE Transactions on Information Theory. ISSN 0018-9448. 70(4), p. 2655–2673. doi: 10.1109/TIT.2024.3354319. Full text in Research Archive Show summary

  Channel capacities quantify the optimal rates of sending information reliably over noisy channels. Usually, the study of capacities assumes that the circuits which the sender and receiver use for encoding and decoding consist of perfectly noiseless gates. In the case of communication over quantum channels, however, this assumption is widely believed to be unrealistic, even in the long-term, due to the fragility of quantum information, which is affected by the process of decoherence. Christandl and Müller-Hermes have therefore initiated the study of fault-tolerant channel coding for quantum channels, i.e. coding schemes where encoder and decoder circuits are affected by noise, and have used techniques from fault-tolerant quantum computing to establish coding theorems for sending classical and quantum information in this scenario. Here, we extend these methods to the case of entanglement-assisted communication, in particular proving that the fault-tolerant capacity approaches the usual capacity when the gate error approaches zero. A main tool, which might be of independent interest, is the introduction of fault-tolerant entanglement distillation. We furthermore focus on the modularity of the techniques used, so that they can be easily adopted in other fault-tolerant communication scenarios.

• Aubrun, Guillaume; Davidson, Kenneth R.; Müller-Hermes, Alexander; Paulsen, Vern I. & Rahaman, Mizanur (2024). Completely bounded norms of k-positive maps. Journal of the London Mathematical Society. ISSN 0024-6107. 109(6). doi: 10.1112/jlms.12936. Full text in Research Archive
• Hermes, Alexander Mueller-; Nechita, Ion & Reeb, David (2023). A refinement of Reznick’s Positivstellensatz with applications to quantum information theory. Quantum. doi: 10.22331/q-2023-05-12-1001. Full text in Research Archive
• Carlen, Eric & Müller-Hermes, Alexander (2023). Characterizing Schwarz maps by tracial inequalities. Letters in Mathematical Physics. ISSN 0377-9017. 113(1). doi: 10.1007/s11005-023-01636-4. Full text in Research Archive
• Aubrun, Guillaume & Müller-Hermes, Alexander (2022). Annihilating Entanglement Between Cones. Communications in Mathematical Physics. ISSN 0010-3616. doi: 10.1007/s00220-022-04621-5. Full text in Research Archive Show summary

  Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform the k-fold maximal tensor product of a cone C1 into the k-fold minimal tensor product of a cone C2, and the pair (C1,C2) is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that (C1,C2) is resilient if either C1 or C2 is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.

• Christandl, Matthias & Müller-Hermes, Alexander (2022). Fault-Tolerant Coding for Quantum Communication. IEEE Transactions on Information Theory. ISSN 0018-9448. 70(1), p. 282–317. doi: 10.1109/TIT.2022.3169438. Full text in Research Archive Show summary

  Designing encoding and decoding circuits to reliably send messages over many uses of a noisy channel is a central problem in communication theory. When studying the optimal transmission rates achievable with asymptotically vanishing error it is usually assumed that these circuits can be implemented using noise-free gates. While this assumption is satisfied for classical machines in many scenarios, it is not expected to be satisfied in the near term future for quantum machines where decoherence leads to faults in the quantum gates. As a result, fundamental questions regarding the practical relevance of quantum channel coding remain open. By combining techniques from fault-tolerant quantum computation with techniques from quantum communication, we initiate the study of these questions. We introduce fault-tolerant versions of quantum capacities quantifying the optimal communication rates achievable with asymptotically vanishing total error when the encoding and decoding circuits are affected by gate errors with small probability. Our main results are threshold theorems for the classical and quantum capacity: For every quantum channel T and every ϵ > 0 there exists a threshold p(ϵ, T) for the gate error probability below which rates larger than C-ϵ are fault-tolerantly achievable with vanishing overall communication error, where C denotes the usual capacity. Our results are not only relevant in communication over large distances, but also on-chip, where distant parts of a quantum computer might need to communicate under higher levels of noise than affecting the local gates.

• Fawzi, Omar; Müller-Hermes, Alexander & Shayeghi, Ala (2022). A Lower Bound on the Space Overhead of Fault-Tolerant Quantum Computation. In Braverman, Mark (Eds.), 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Schloss Dagstuhl, Leibniz-Zentrum für Informatik. ISSN 9783959772174. p. 1–20. doi: 10.4230/LIPIcs.ITCS.2022.68. Full text in Research Archive Show summary

  The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation stating that arbitrarily long quantum computations can be performed with a polylogarithmic overhead provided the noise level is below a constant level. A recent work by Fawzi, Grospellier and Leverrier (FOCS 2018) building on a result by Gottesman (QIC 2013) has shown that the space overhead can be asymptotically reduced to a constant independent of the circuit provided we only consider circuits with a length bounded by a polynomial in the width. In this work, using a minimal model for quantum fault tolerance, we establish a general lower bound on the space overhead required to achieve fault tolerance. For any non-unitary qubit channel N and any quantum fault tolerance schemes against i.i.d. noise modeled by N, we prove a lower bound of max {Q(N)−1n, αN log T} on the number of physical qubits, for circuits of length T and width n. Here, Q(N) denotes the quantum capacity of N and αN > 0 is a constant only depending on the channel N. In our model, we allow for qubits to be replaced by fresh ones during the execution of the circuit and in the case of unital noise, we allow classical computation to be free and perfect. This improves upon results that assumed classical computations to be also affected by noise, and that sometimes did not allow for fresh qubits to be added. Along the way, we prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude damping noise resolving a conjecture by Ben-Or, Gottesman and Hassidim (2013).

• Hermes, Alexander Mueller- (2021). Decomposable Pauli diagonal maps and tensor squares of qubit maps. Journal of Mathematical Physics. ISSN 0022-2488. doi: 10.1063/5.0049533. Full text in Research Archive
• Müller-Hermes, Alexander (2021). Cutting Cakes and Kissing Circles. The Mathematical intelligencer. ISSN 0343-6993. 43, p. 40–44. doi: 10.1007/s00283-021-10071-z. Full text in Research Archive
• Christandl, Matthias; Müller-Hermes, Alexander & Wolf, Michael (2019). When Do Composed Maps Become Entanglement Breaking? Annales Henri Poincaré - A Journal of Theoretical and Mathematical Physics. ISSN 1424-0637. 20, p. 2295–2322. doi: 10.1007/s00023-019-00774-7. Full text in Research Archive
• Müller-Hermes, Alexander & Perry, Christopher (2019). All unital qubit channels are 4-noisy operations. Letters in Mathematical Physics. ISSN 0377-9017. 109, p. 1–9. doi: 10.1007/s11005-018-1104-x. Full text in Research Archive
• Müller-Hermes, Alexander & Nechita, Ion (2018). Operator Schmidt ranks of bipartite unitary matrices. Linear Algebra and its Applications. ISSN 0024-3795. 557, p. 174–187. doi: 10.1016/j.laa.2018.07.018. Full text in Research Archive
• Müller-Hermes, Alexander (2018). Decomposability of linear maps under tensor powers. Journal of Mathematical Physics. ISSN 0022-2488. 59. doi: 10.1063/1.5045559. Full text in Research Archive
• Müller-Hermes, Alexander & Franca, Daniel Stilck (2018). Sandwiched Rényi Convergence for Quantum Evolutions. Quantum. 2. doi: 10.22331/q-2018-02-27-55. Full text in Research Archive
• Huber, Marcus; Lami, Ludovico; Lancien, Cecilia & Müller-Hermes, Alexander (2018). High-Dimensional Entanglement in States with Positive Partial Transposition. Physical Review Letters. ISSN 0031-9007. 121. doi: 10.1103/PhysRevLett.121.200503. Full text in Research Archive
• Christandl, Matthias & Müller-Hermes, Alexander (2017). Relative Entropy Bounds on Quantum, Private and Repeater Capacities. Communications in Mathematical Physics. ISSN 0010-3616. 353, p. 821–852. doi: 10.1007/s00220-017-2885-y. Full text in Research Archive
• Müller-Hermes, Alexander & Reeb, David (2017). Monotonicity of the Quantum Relative Entropy Under Positive Maps. Annales Henri Poincaré - A Journal of Theoretical and Mathematical Physics. ISSN 1424-0637. 18, p. 1777–1788. doi: 10.1007/s00023-017-0550-9. Full text in Research Archive
• Müller-Hermes, Alexander; Franca, Daniel Stilck & Wolf, Michael (2016). Entropy production of doubly stochastic quantum channels. Journal of Mathematical Physics. ISSN 0022-2488. 57(2). doi: 10.1063/1.4941136. Full text in Research Archive
• Müller-Hermes, Alexander; Reeb, David & Wolf, Michael (2016). Positivity of linear maps under tensor powers. Journal of Mathematical Physics. ISSN 0022-2488. 57(1). doi: 10.1063/1.4927070. Full text in Research Archive
• Müller-Hermes, Alexander; Franca, Daniel Stilck & Wolf, Michael (2016). Relative entropy convergence for depolarizing channels. Journal of Mathematical Physics. ISSN 0022-2488. 57(2). doi: 10.1063/1.4939560. Full text in Research Archive
• Müller-Hermes, Alexander & Szehr, Oleg (2015). Spectral variation bounds in hyperbolic geometry. Linear Algebra and its Applications. ISSN 0024-3795. 482, p. 131–148. doi: 10.1016/j.laa.2015.05.020. Full text in Research Archive
• Müller-Hermes, Alexander; Reeb, David & Wolf, Michael (2014). Quantum Subdivision Capacities and Continuous-Time Quantum Coding. IEEE Transactions on Information Theory. ISSN 0018-9448. 61(1), p. 565–581. doi: 10.1109/TIT.2014.2366456. Full text in Research Archive

View all works in NVA

• Müller-Hermes, Alexander (2025). Limit formulas for norms of tensor power operators and entanglement annihilating maps. Full text in Research Archive
• Müller-Hermes, Alexander (2025). Fault-tolerant coding for quantum communication. Full text in Research Archive
• Müller-Hermes, Alexander (2024). Limit formulas for norms of tensor power operators and entanglement annihilating maps. Full text in Research Archive
• Müller-Hermes, Alexander (2022). Capacities of quantum channels. Full text in Research Archive
• Müller-Hermes, Alexander (2022). Annihilating Entanglement Between Cones. Full text in Research Archive
• Müller-Hermes, Alexander (2022). Monogamy of entanglement between cones. Full text in Research Archive
• Müller-Hermes, Alexander (2022). Fault-tolerant Coding for Quantum Communication. Full text in Research Archive

View all works in NVA