Abstract
In the stabilizer formalism of quantum computation, the Gottesman–Knill theorem shows that universal fault-tolerant quantum computation requires the resource called magic (nonstabilizerness). Thus stabilizer states serve as “classical states,” and states beyond them are necessary for genuine quantum computation. Characterization, detection, and quantification of magic states are basic issues in this context. In the paradigm of quantum measurement, symmetric informationally complete positive operator valued measures (SIC-POVMs, further abbreviated as SICs) play a prominent role due to their structural symmetry and remarkable features. However, their existence in all dimensions, although strongly supported by extensive theoretical and numerical evidence, remains an elusive open problem (Zauner’s conjecture). A standard method for constructing SICs is via the orbit of the Heisenberg–Weyl group on a fiducial state, and most known SICs arise in this way. A natural question arises regarding the relation between stabilizer states and fiducial states. In this paper, we connect them by showing that they are on two extremes with respect to the \(p\)-norms of characteristic functions of quantum states. This not only reveals a simple path from stabilizer states to SIC fiducial states, showing quantitatively that they are as far away as possible from each other, but also provides a simple reformulation of Zauner’s conjecture in terms of extremals for the \(p\)-norms of characteristic functions. A convenient criterion for magic states and some interesting open problems are also presented.
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We are very grateful to the referee for helpful comments and suggestions.
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This work was supported by the National Key R&D Program of China (grant No. 2020YFA0712700) and the National Natural Science Foundation of China (grant Nos. 11875317 and 61833010).
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 213, pp. 505–522 https://doi.org/10.4213/tmf10334.
Appendix
Here, we present the detailed proof of the basic properties of \(M_p(\rho)\).
Property 1 is evident from the definition by noting that any Clifford operator permutes the Heisenberg–Weyl operators via conjugation, i.e., \(VWV^\dagger\in\mathcal P_d\), for all \(V\in\mathcal C_d\), \(W\in\mathcal P_d\).
Property 2 follows from the classical Minkowski inequality.
Property 3 follows from the abstract Parseval theorem by noting that \(\bigl\{\frac{1}{\sqrt d}D_{k,l}\colon k,l\in\mathbb{Z}_d\bigr\}\) constitutes an orthonormal basis for the operator space \(L(\mathbb{C}^d)\), which implies that
For property 4, we first note that \(|c_{k,l}(\rho)| \le 1\), which follows from the Cauchy–Schwarz inequality as
Next we show that if the lower bound in inequality (10) is attained by the pure state \(\rho=|\psi\rangle\langle\psi|\), then \(\rho\) must be a stabilizer state. In fact, suppose the lower bound in inequality (10) is attained by a pure state \(\rho\); then the inequality in (A.1) is saturated if and only if \(\lambda_{k,l}(\rho)=0\) or 1 (noting that \(0\le\lambda_{k,l}(\rho)\le 1)\). By property 3,
Now, we proceed to establish the right-hand inequality in (10). By writing
Conversely, if the upper bound in inequality (10) is attained by the state \(\rho=|\psi\rangle\langle \psi|\), then by the equality condition of the Hölder inequality, we conclude that all \(\lambda_{k,l}(\rho)=|c_{k,l}(\rho)|=| \operatorname{tr} (D_{k,l}\rho)|\) are equal for \((k,l)\neq(0,0)\). By property 3,
For property 5, noting that \(\lambda_{k,l}(\rho)\in[0,1]\) and \(p>2\), we have
When \(p>2\), by the (reverse) Hölder inequality, we have
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Feng, L., Luo, S. From stabilizer states to SIC-POVM fiducial states. Theor Math Phys 213, 1747–1761 (2022). https://doi.org/10.1134/S004057792212008X
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DOI: https://doi.org/10.1134/S004057792212008X