The axiomatic and the operational approaches to resource theories of magic do not coincide

Heimendahl, Arne; Heinrich, Markus R.; Gross, David J.

doi:10.48550/arxiv.2011.11651

Cited by 2 publications

(1 citation statement)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We believe that our result could be phrased and proved in terms other approaches to contextuality [35][36][37][38][39], as long as one formalizes such adaptive measurement protocols and transformations between correlations within them. However, if one works with axiomatically rather than operationally defined transformations as in [40] the proof no longer carries over automatically, as it could happen that operational transformations form a proper subset of axiomatically defined ones, as happens for instance in resource theories of entanglement [41] and magic [42].…”

mentioning

confidence: 99%

Neither Contextuality nor Nonlocality Admits Catalysts

Karvonen

2021

Preprint

View full text Add to dashboard Cite

We show that the resource theory of contextuality does not admit catalysts. As a corollary, we observe that the same holds for non-locality. This adds a further example to the list of "anomalies of entanglement", showing that non-locality and entanglement behave differently as resources. We also show that catalysis remains impossible even if instead of classical randomness we allow some more powerful behaviors to be used freely in the free transformations of the resource theory. Introduction. Contextuality [1] and non-locality [2, 3] play a prominent role in a wide variety of applications of quantum mechanics, with non-locality being used for example in quantum key-distribution [4], certified randomness [5] and randomness expansion [6].Similarly, contextuality powers quantum computation in some computational models [7][8][9][10][11][12] and even increases expressive power in quantum machine learning [13]. Consequently, it is vital to understand how non-locality and contextuality behave as resources.In this Letter, we show that neither contextuality nor non-locality admit catalysts: that is, there are no correlations that can be used to enable an otherwise impossible conversion between correlations and still recovered afterwards. To state this slightly more precisely, let us write d, e, f . . . for various correlations (whether classical or not), d ⊗ e for having independent instances of d and e, and d e (read as "d simulates e") for the existence of a conversion d → e. Then our results state that, in suitably formalized resource theories of contextuality and non-locality, whenever d ⊗ e d ⊗ f , then e f already. This gives a strong indication that contextuality (and non-locality) are resources that get spent when you use them: there is no way of using a correlation d to achieve a task you could not do otherwise while keeping d intact. As entanglement theory famously allows for catalysts [14],this can be seen as yet another "anomaly of non-locality" [15] and thus further testament to the fact that non-locality and entanglement are different resources. [16] We do this by working in precisely defined resource theories of contextuality and nonlocality. These are not strictly speaking quantum resource theories [17], but resource theories in a more general sense [18, 19], as we allow resources such as PR-boxes [20] that are not quantum realizable. The kinds of conversions between correlations we have in mind capture the intuitive idea of using one system to simulate another one, and have been studied in earlier literature [21][22][23][24][25][26][27][28][29]. These roughly correspond to the local operations and shared randomness-paradigm ((LOSR)) or to wirings and prior-to-input classical communication

show abstract

mentioning

confidence: 99%

Neither Contextuality nor Nonlocality Admits Catalysts

Karvonen

2021

Preprint

View full text Add to dashboard Cite

show abstract

Counting stabiliser codes for arbitrary dimension

Singal

Che

Hsu

et al. 2023

Quantum

View full text Add to dashboard Cite

In this work, we compute the number of [[n,k]]d stabilizer codes made up of d-dimensional qudits, for arbitrary positive integers d. In a seminal work by Gross \cite{Gross2006} the number of [[n,k]]d stabilizer codes was computed for the case when d is a prime (or the power of a prime, i.e., d=pm, but when the qudits are Galois-qudits). The proof in \cite{Gross2006} is inapplicable to the non-prime case. For our proof, we introduce a group structure to [[n,k]]d codes, and use this in conjunction with the Chinese remainder theorem to count the number of [[n,k]]d codes. Our work overlaps with \cite{Gross2006} when d is a prime and in this case our results match exactly, but the results differ for the more generic case. Despite that, the overall order of magnitude of the number of stabilizer codes scales agnostic of whether the dimension is prime or non-prime. This is surprising since the method employed to count the number of stabilizer states (or more generally stabilizer codes) depends on whether d is prime or not. The cardinality of stabilizer states, which was so far known only for the prime-dimensional case (and the Galois qudit prime-power dimensional case) plays an important role as a quantifier in many topics in quantum computing. Salient among these are the resource theory of magic, design theory, de Finetti theorem for stabilizer states, the study and optimisation of the classical simulability of Clifford circuits, the study of quantum contextuality of small-dimensional systems and the study of Wigner-functions. Our work makes available this quantifier for the generic case, and thus is an important step needed to place results for quantum computing with non-prime dimensional quantum systems on the same pedestal as prime-dimensional systems.

show abstract

The axiomatic and the operational approaches to resource theories of magic do not coincide

Cited by 2 publications

References 34 publications

Neither Contextuality nor Nonlocality Admits Catalysts

Neither Contextuality nor Nonlocality Admits Catalysts

Counting stabiliser codes for arbitrary dimension

Contact Info

Product

Resources

About