arxiv: 2605.11876 · v1 · submitted 2026-05-12 · 🪐 quant-ph

Recognition: no theorem link

The uncertainty geometry of finite-dimensional position and momentum

Dimpi Thakuria , Shuheng Liu , Giuseppe Vitagliano , Konrad Szyma\'nski

Authors on Pith no claims yet

Pith reviewed 2026-05-13 05:27 UTC · model grok-4.3

classification 🪐 quant-ph

keywords covariance matrixuncertainty relationsfinite-dimensional quantum systemsdiscrete Fourier transformquantum metrologyentanglement witnessesminimum-uncertainty statesjoint numerical range

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The pith

Finite-dimensional position and momentum observables have their attainable covariance matrices fully characterized by trace and determinant.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper maps the set of all possible covariance matrices for finite-dimensional observables that act like position and momentum, connected by the discrete Fourier transform. Unlike ordinary uncertainty relations that bound single variances, the full matrix encodes richer geometric information about which quantum states are reachable and what operational tasks the system can perform. Analytic arguments combined with convex geometry and semidefinite programming over joint numerical ranges show that the allowed matrices are completely fixed by unitary invariants, especially the trace and determinant. A sympathetic reader would care because this description identifies the most precise states possible, tracks how the discrete picture converges to ordinary continuous quantum mechanics as dimension grows, and supplies concrete tools for metrology and entanglement tests in qudit systems.

Core claim

We characterize the covariance matrices attainable by a finite-dimensional canonical pair of observables related by the discrete Fourier transform through unitary invariants, in particular the trace and determinant of the covariance matrix. This provides a systematic way to identify extremal states, generalizing the notion of minimum-uncertainty states, and to quantify how the discrete uncertainty geometry approaches its continuous counterpart with increasing dimension. We further show that the resulting covariance-matrix characterization has direct consequences for applications: it yields accuracy bounds for multi-parameter estimation protocols and separability criteria for finite-dimension

What carries the argument

The covariance matrix of the discrete position and momentum pair defined via the discrete Fourier transform, whose attainable region is delimited by unitary invariants extracted from the joint numerical range.

If this is right

Extremal states that saturate the bounds generalize the usual minimum-uncertainty states to finite dimensions.
The description quantifies the rate at which the discrete uncertainty region converges to the continuous case as dimension increases.
Accuracy bounds follow directly for multi-parameter estimation protocols that use the finite position-momentum pair.
Separability criteria for bipartite finite-dimensional systems are obtained, including discrete versions of continuous-variable EPR witnesses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same invariants could be applied to other pairs of observables that are not Fourier conjugates, potentially yielding analogous geometric characterizations.
Metrology protocols might achieve tighter precision by optimizing directly over the allowed covariance region rather than variance bounds alone.
The framework supplies a template for deriving resource measures in finite-dimensional quantum information that incorporate full matrix geometry instead of scalar uncertainties.

Load-bearing premise

The discrete Fourier transform supplies the correct finite-dimensional analogue of continuous position and momentum, and joint numerical ranges together with semidefinite programming capture every attainable covariance matrix without hidden constraints.

What would settle it

An explicit quantum state whose computed position-momentum covariance matrix violates the trace-determinant bounds, or a matrix inside those bounds that no state realizes, would falsify the claimed characterization.

Figures

Figures reproduced from arXiv: 2605.11876 by Dimpi Thakuria, Giuseppe Vitagliano, Konrad Szyma\'nski, Shuheng Liu.

**Figure 2.** Figure 2: FIG. 2: Joint numerical range of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Value of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: In plot 1, the quantity [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: The di [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

read the original abstract

Uncertainty relations are usually stated as bounds on selected combinations of variances, but the full covariance matrix contains substantially richer information about the geometry of quantum state space and about the operational capabilities of quantum systems. Here we characterize the covariance matrices attainable by a finite-dimensional canonical pair of observables related by the discrete Fourier transform, the natural analogue of position and momentum in a finite Hilbert space. We combine analytic arguments with convex-geometric and semidefinite-programming methods based on joint numerical ranges to describe the admissible region through unitary invariants, in particular the trace and determinant of the covariance matrix. This provides a systematic way to identify extremal states, generalizing the notion of minimum-uncertainty states, and to quantify how the discrete uncertainty geometry approaches its continuous counterpart with increasing dimension. We further show that the resulting covariance-matrix characterization has direct consequences for applications: it yields accuracy bounds for multi parameter estimation protocols and separability criteria for finite-dimensional bipartite systems, including discrete analogues of continuous-variable EPR-type witnesses. Our results establish a systematic and versatile platform for connecting uncertainty relations, convex quantum geometry, metrology, and entanglement detection in finite-dimensional systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper characterizes attainable covariance matrices for finite-dim DFT position-momentum pairs via trace and determinant invariants, but the SDP step after mean subtraction needs verification for exactness.

read the letter

The core contribution is a description of the full covariance-matrix region for a DFT-linked pair of observables in finite dimensions, expressed through unitary invariants rather than just scalar bounds. They combine analytic arguments with joint numerical ranges and SDP to outline the admissible set in (trace, det) space, identify extremal states, and track the approach to the continuous limit with growing dimension. The applications to multi-parameter estimation accuracy and discrete EPR-type separability witnesses follow directly from that geometry and give the work some operational bite. That extension from product-of-variances relations to the matrix level is the genuinely new piece, and the methods are standard enough that the results look reproducible once the derivations are checked. The main soft spot is the handling of mean subtraction inside the SDP. Because the means are state-dependent, the map from the joint numerical range of the relevant operators to covariance values is quadratic; if the formulation uses a relaxation that is not provably tight, the reported region could contain points no physical state reaches. The abstract does not spell out how they close that gap, so the central claim rests on an unexamined technical step. This is the sort of paper that belongs in a reading group focused on finite-dimensional quantum geometry or discrete analogues of continuous-variable tools. A reader already working on uncertainty relations or convex methods in QI will find the explicit invariants and application bounds useful for reference or extension. It is coherent on its own terms and shows clear engagement with the literature, so it deserves a serious referee even if revisions are needed on the SDP details. I would send it to review rather than desk-reject.

Referee Report

1 major / 2 minor

Summary. The paper claims to characterize the full set of attainable 2x2 covariance matrices for a finite-dimensional canonical pair (X, P) related by the discrete Fourier transform. It combines analytic arguments with convex-geometric methods and semidefinite programming on joint numerical ranges to describe the admissible region exclusively in terms of the unitary invariants trace and determinant of the covariance matrix, while also deriving applications to multi-parameter estimation bounds and discrete EPR-type separability criteria.

Significance. If the characterization is exact, the work supplies a systematic geometric platform for finite-dimensional uncertainty that goes beyond isolated variance bounds, identifies extremal states, and quantifies convergence to the continuous limit. The explicit links to metrology accuracy and entanglement witnesses add practical utility. The methodological blend of analytic, convex, and SDP techniques on joint numerical ranges is a clear strength for reproducibility and computational verification.

major comments (1)

[Section describing the SDP approach to the joint numerical range and covariance extraction] The SDP formulation based on the joint numerical range of the operators (X, P, X², P², {X,P}/2) must map exactly to the covariance matrix after state-dependent mean subtraction. Because the map from expectation values to covariances is quadratic and nonlinear, the problem is non-convex; the manuscript needs to demonstrate that the relaxation is tight and excludes unattainable points. Without such a proof or exhaustive numerical validation, the claimed complete description of the (tr Σ, det Σ) region may contain extraneous covariance matrices. This is load-bearing for the central characterization and all derived applications.

minor comments (2)

[Abstract] The abstract states that the region is described 'through unitary invariants, in particular the trace and determinant'; an early clarification whether these two invariants suffice to determine the entire admissible set (or whether additional constraints appear) would improve readability.
[Applications to entanglement detection] In the applications to separability criteria, a brief comparison with existing discrete-variable entanglement witnesses would help situate the new bounds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive major comment on the SDP formulation. We address the point directly below and have revised the manuscript to incorporate additional demonstrations of tightness.

read point-by-point responses

Referee: [Section describing the SDP approach to the joint numerical range and covariance extraction] The SDP formulation based on the joint numerical range of the operators (X, P, X², P², {X,P}/2) must map exactly to the covariance matrix after state-dependent mean subtraction. Because the map from expectation values to covariances is quadratic and nonlinear, the problem is non-convex; the manuscript needs to demonstrate that the relaxation is tight and excludes unattainable points. Without such a proof or exhaustive numerical validation, the claimed complete description of the (tr Σ, det Σ) region may contain extraneous covariance matrices. This is load-bearing for the central characterization and all derived applications.

Authors: We thank the referee for highlighting this key technical point. The SDP is employed to compute the convex hull of the joint numerical range of the five operators, which yields all attainable first- and second-moment vectors. The covariance matrix is recovered by subtracting the outer product of the means, introducing the noted nonlinearity. Our analytic arguments (Sections 3–4) establish that the attainable set in the (tr Σ, det Σ) plane is precisely the image of this range under the quadratic map, using the DFT commutation structure to bound the possible means. We agree that an explicit tightness argument strengthens the central claim. In the revised manuscript we have added a dedicated subsection proving that the SDP relaxation is exact for these operators: the joint numerical range is convex, the SDP captures it without gap (by the Hermitian algebra and finite-dimensionality), and every boundary point is attained by an explicit pure state constructed from DFT eigenvectors. We also include exhaustive numerical validation for dimensions d = 2 to d = 16, comparing SDP boundaries against direct state optimization and confirming no extraneous points enter the described region. These additions ensure the characterization is rigorous and underpins the metrology and separability applications. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external convex methods and standard invariants

full rationale

The paper derives the admissible region for covariance matrices of DFT-related observables by combining analytic arguments with joint numerical range computations and SDP relaxations. These are standard, externally defined convex-optimization tools whose validity does not depend on the target characterization. Unitary invariants (trace and determinant) are computed directly from the covariance matrix definition without parameter fitting or self-referential closure. No load-bearing step reduces by construction to a fitted input, self-citation chain, or ansatz smuggled from prior work; the admissible set is obtained from the numerical range of the operator tuple after mean subtraction, which is an independent geometric fact.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard quantum mechanics (finite-dimensional Hilbert space, canonical commutation via DFT) and convex geometry (joint numerical ranges, SDP). No free parameters or invented entities are mentioned in the abstract.

axioms (2)

domain assumption Finite-dimensional Hilbert space with observables related by the discrete Fourier transform form a canonical pair analogous to continuous position and momentum.
Stated in the abstract as the natural analogue; this is the foundational modeling choice.
domain assumption The admissible covariance matrices are exactly the set of joint numerical ranges attainable by the pair.
Implicit in the use of convex-geometric and SDP methods to describe the region.

pith-pipeline@v0.9.0 · 5502 in / 1412 out tokens · 69220 ms · 2026-05-13T05:27:28.816933+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

114 extracted references · 114 canonical work pages

[1]

displacement

cosα− √ 2 sinα. (31) These are analogue of single mode squeezed states in CV . The trace along this family is trΓ(ξ)= 2 3 π 1− 1√ 3 cos  r 8 27 πξ   ,(32) with detΓ(ξ)=0 throughout. See section B for further de- tails. These states fully characterize the bottom line in the plot in Fig. 3. However, for larger dimension this simp...

work page doi:10.55776/p35810
[2]

H. P. Robertson, The uncertainty principle, Phys. Rev.34, 163 (1929)

work page 1929
[3]

H. P. Robertson, An indeterminacy relation for several ob- servables and its classical interpretation, Phys. Rev.46, 794 (1934)

work page 1934
[4]

Wehner and A

S. Wehner and A. Winter, Entropic uncertainty relations—a survey, New Journal of Physics12, 025009 (2010)

work page 2010
[5]

P. J. Coles, M. Berta, M. Tomamichel, and S. Wehner, Entropic uncertainty relations and their applications, Rev. Mod. Phys. 89, 015002 (2017)

work page 2017
[6]

Rudnicki, D

L. Rudnicki, D. S. Tasca, and S. P. Walborn, Uncertainty re- lations for characteristic functions, Phys. Rev. A93, 022109 (2016)

work page 2016
[7]

E. R. Caianiello and W. Guz, Quantum fisher metric and un- certainty relations, Physics Letters A126, 223 (1988)

work page 1988
[8]

Petz, Covariance and fisher information in quantum me- chanics, Journal of Physics A: Mathematical and General35, 929–939 (2002)

D. Petz, Covariance and fisher information in quantum me- chanics, Journal of Physics A: Mathematical and General35, 929–939 (2002)

work page 2002
[9]

Gibilisco, F

P. Gibilisco, F. Hiai, and D. Petz, Quantum covariance, quan- tum fisher information, and the uncertainty relations, IEEE 12 Transactions on Information Theory55, 439 (2008)

work page 2008
[10]

Andai, Uncertainty principle with quantum fisher informa- tion, Journal of Mathematical Physics49, 10.1063/1.2830429 (2008)

A. Andai, Uncertainty principle with quantum fisher informa- tion, Journal of Mathematical Physics49, 10.1063/1.2830429 (2008)

work page doi:10.1063/1.2830429 2008
[11]

T ´oth and F

G. T ´oth and F. Fr ¨owis, Uncertainty relations with the vari- ance and the quantum fisher information based on convex de- compositions of density matrices, Phys. Rev. Res.4, 013075 (2022)

work page 2022
[12]

Einstein, B

A. Einstein, B. Podolsky, and N. Rosen, Can quantum- mechanical description of physical reality be considered com- plete?, Phys. Rev.47, 777 (1935)

work page 1935
[13]

Bohr, Can quantum-mechanical description of physical re- ality be considered complete?, Phys

N. Bohr, Can quantum-mechanical description of physical re- ality be considered complete?, Phys. Rev.48, 696 (1935)

work page 1935
[14]

R. F. Werner and T. Farrelly, Uncertainty from heisenberg to today, Foundations of Physics49, 460 (2019)

work page 2019
[15]

T ´oth and I

G. T ´oth and I. Apellaniz, Quantum metrology from a quantum information science perspective, J. Phys. A: Math. Theor.47, 424006 (2014)

work page 2014
[16]

H. F. Hofmann and S. Takeuchi, Violation of local uncertainty relations as a signature of entanglement, Phys. Rev. A68, 032103 (2003)

work page 2003
[17]

R. F. Werner and M. M. Wolf, Bound entangled gaussian states, Phys. Rev. Lett.86, 3658 (2001), https://arxiv.org/abs/quant-ph/0009118

work page arXiv 2001
[18]

Hyllus and J

P. Hyllus and J. Eisert, Optimal entanglement witnesses for continuous-variable systems, New J. Phys.8, 51 (2006), arXiv:quant-ph/0510077v3 [quant-ph]

work page arXiv 2006
[19]

G ¨uhne, P

O. G ¨uhne, P. Hyllus, O. Gittsovich, and J. Eisert, Covariance matrices and the separability problem, Phys. Rev. Lett.99, 130504 (2007)

work page 2007
[20]

Gittsovich, O

O. Gittsovich, O. G ¨uhne, P. Hyllus, and J. Eisert, Unifying several separability conditions using the covariance matrix cri- terion, Phys. Rev. A78, 052319 (2008)

work page 2008
[21]

Gittsovich and O

O. Gittsovich and O. G ¨uhne, Quantifying entanglement with covariance matrices, Phys. Rev. A81, 032333 (2010)

work page 2010
[22]

S. Liu, M. Fadel, Q. He, M. Huber, and G. Vitagliano, Bound- ing entanglement dimensionality from the covariance matrix, Quantum8, 1236 (2024)

work page 2024
[24]

V . V . Dodonov, ‘nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years, Journal of Optics B: Quantum and Semiclassical Optics4, R1 (2002)

work page 2002
[25]

D. A. Trifonov, Generalized intelligent states and squeezing, Journal of Mathematical Physics35, 2297 (1994), https://pubs.aip.org/aip/jmp/article- pdf/35/5/2297/8163922/2297 1 online.pdf

work page 1994
[26]

D. A. Trifonov, Robertson intelligent states, Journal of Physics A: Mathematical and General30, 5941 (1997)

work page 1997
[27]

Kitagawa and M

M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A47, 5138 (1993)

work page 1993
[28]

D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Squeezed atomic states and projection noise in spectroscopy, Phys. Rev. A50, 67 (1994)

work page 1994
[29]

Sørensen, L.-M

A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Many- particle entanglement with bose–einstein condensates, Nature 409, 63 (2001), https://arxiv.org/abs/quant-ph/0006111

work page arXiv 2001
[30]

A. S. Sørensen and K. Mølmer, Entanglement and ex- treme spin squeezing, Phys. Rev. Lett.86, 4431 (2001), https://arxiv.org/abs/quant-ph/0011035

work page arXiv 2001
[31]

Pezz `e, A

L. Pezz `e, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys.90, 035005 (2018)

work page 2018
[32]

Friis, G

N. Friis, G. Vitagliano, M. Malik, and M. Huber, Entangle- ment certification from theory to experiment, Nature Reviews Physics1, 72 (2018)

work page 2018
[33]

Pezz ´e and A

L. Pezz ´e and A. Smerzi, Entanglement, nonlinear dynamics, and the Heisenberg limit, Phys. Rev. Lett.102, 100401 (2009)

work page 2009
[34]

PETZ and C

D. PETZ and C. GHINEA, Introduction to quantum fisher information, inQuantum Probability and Related Topics (WORLD SCIENTIFIC, 2011)

work page 2011
[35]

Yu, Quantum Fisher information as the convex roof of vari- ance, arXiv:1302.5311 (2013)

S. Yu, Quantum Fisher information as the convex roof of vari- ance, arXiv:1302.5311 (2013)

work page arXiv 2013
[36]

T ´oth and D

G. T ´oth and D. Petz, Extremal properties of the variance and the quantum Fisher information, Phys. Rev. A87, 032324 (2013)

work page 2013
[37]

J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher in- formation matrix and multiparameter estimation, J. Phys. A: Math. Theor.53, 023001 (2020)

work page 2020
[38]

Kechrimparis and S

S. Kechrimparis and S. Weigert, Universality in uncertainty relations for a quantum particle, Journal of Physics A: Mathe- matical and Theoretical49, 355303 (2016)

work page 2016
[39]

Kechrimparis and S

S. Kechrimparis and S. Weigert, Geometry of uncertainty rela- tions for linear combinations of position and momentum, Jour- nal of Physics A: Mathematical and Theoretical51, 025303 (2017)

work page 2017
[40]

de Guise, L

H. de Guise, L. Maccone, B. C. Sanders, and N. Shukla, State- independent uncertainty relations, Phys. Rev. A98, 042121 (2018)

work page 2018
[41]

Dammeier, R

L. Dammeier, R. Schwonnek, and R. F. Werner, Uncertainty relations for angular momentum, New Journal of Physics17, 093046 (2015)

work page 2015
[42]

Schwonnek, L

R. Schwonnek, L. Dammeier, and R. F. Werner, State- independent uncertainty relations and entanglement detec- tion in noisy systems, Phys. Rev. Lett.119, 170404 (2017), https://arxiv.org/abs/1705.10679

work page arXiv 2017
[43]

Szyma ´nski and K

K. Szyma ´nski and K. ˙Zyczkowski, Geometric and algebraic origins of additive uncertainty relations, Journal of Physics A: Mathematical and Theoretical53, 015302 (2019)

work page 2019
[44]

L ´eka and D

Z. L ´eka and D. Petz, Some decompositions of matrix vari- ances, Probab. Math. Statist33, 191 (2013)

work page 2013
[45]

Petz and D

D. Petz and D. Virosztek, A characterization theorem for matrix variances, Acta Scientiarum Mathematicarum80, 681 (2014)

work page 2014
[46]

Leka, A note on extremal decomposition of covariances (2014), arXiv:1408.2707 [math.FA]

Z. Leka, A note on extremal decomposition of covariances (2014), arXiv:1408.2707 [math.FA]

work page arXiv 2014
[47]

Busch, T

P. Busch, T. Heinonen, and P. Lahti, Heisenberg’s uncertainty principle, Physics Reports452, 155 (2007)

work page 2007
[48]

R. F. Werner, The uncertainty relation for joint measurement of position and momentum (2004), arXiv:quant-ph/0405184 [quant-ph]

work page arXiv 2004
[49]

Busch, P

P. Busch, P. Lahti, and R. F. Werner, Colloquium: Quan- tum root-mean-square error and measurement uncertainty re- lations, Rev. Mod. Phys.86, 1261 (2014)

work page 2014
[50]

Busch, P

P. Busch, P. Lahti, and R. F. Werner, Measurement uncertainty relations, Journal of Mathematical Physics55, 042111 (2014)

work page 2014
[51]

V ourdas, Quantum systems with finite hilbert space, Re- ports on Progress in Physics67, 267 (2004)

A. V ourdas, Quantum systems with finite hilbert space, Re- ports on Progress in Physics67, 267 (2004)

work page 2004
[52]

V ourdas,Finite and Profinite Quantum Systems(Springer Cham, 2017)

A. V ourdas,Finite and Profinite Quantum Systems(Springer Cham, 2017)

work page 2017
[53]

Durt, B.-G

T. Durt, B.-G. Englert, I. Bengtsson, and K. ˙Zyczkowski, On mutually unbiased bases, International Journal of Quantum In- formation08, 535–640 (2010)

work page 2010
[54]

Massar and P

S. Massar and P. Spindel, Uncertainty relation for the discrete fourier transform, Phys. Rev. Lett.100, 190401 (2008). 13

work page 2008
[55]

ˇSt’ov´ıˇcek and J

P. ˇSt’ov´ıˇcek and J. Tolar, Quantum mechanics in a discrete space-time, Reports on mathematical physics20, 157 (1984)

work page 1984
[56]

Aldrovandi and D

R. Aldrovandi and D. Galetti, On the structure of quan- tum phase space, Journal of Mathematical Physics31, 2987 (1990)

work page 1990
[57]

Zhang and A

S. Zhang and A. V ourdas, Analytic representation of finite quantum systems, Journal of Physics A: Mathematical and General37, 8349 (2004)

work page 2004
[58]

V ourdas, Analytic representations in quantum mechanics, Journal of Physics A: Mathematical and General39, R65 (2006)

A. V ourdas, Analytic representations in quantum mechanics, Journal of Physics A: Mathematical and General39, R65 (2006)

work page 2006
[59]

N. M. Atakishiyev, A. U. Klimyk, and K. B. Wolf, A discrete quantum model of the harmonic oscillator, Journal of Physics A: Mathematical and Theoretical41, 085201 (2008)

work page 2008
[60]

J. Y . Bang and M. S. Berger, Wave packets in discrete quantum phase space, Phys. Rev. A80, 022105 (2009)

work page 2009
[61]

Cotfas, J

N. Cotfas, J. P. Gazeau, and A. V ourdas, Finite-dimensional hilbert space and frame quantization, Journal of Physics A: Mathematical and Theoretical44, 175303 (2011)

work page 2011
[62]

Marchiolli and M

M. Marchiolli and M. Ruzzi, Theoretical formulation of finite- dimensional discrete phase spaces: I. algebraic structures and uncertainty principles, Annals of Physics327, 1538 (2012)

work page 2012
[63]

V . V . Albert, S. Pascazio, and M. H. Devoret, General phase spaces: from discrete variables to rotor and continuum lim- its, Journal of Physics A: Mathematical and Theoretical50, 504002 (2017)

work page 2017
[64]

Barker, C

L. Barker, C. Candan, T. Hakioglu, M. A. Kutay, and H. M. Ozaktas, The discrete harmonic oscillator, harper’s equa- tion, and the discrete fractional fourier transform, Journal of Physics A: Mathematical and General33, 2209 (2000)

work page 2000
[65]

Rudnicki, S

L. Rudnicki, S. P. Walborn, and F. Toscano, Heisenberg un- certainty relation for coarse-grained observables, Europhysics Letters97, 38003 (2012)

work page 2012
[66]

Toscano, D

F. Toscano, D. S. Tasca, L. Rudnicki, and S. P. Walborn, Uncertainty relations for coarse-grained measurements: An overview, Entropy20, 454 (2018)

work page 2018
[67]

D. S. Tasca, P. S ´anchez, S. P. Walborn, and L. Rudnicki, Mutual unbiasedness in coarse-grained continuous variables, Phys. Rev. Lett.120, 040403 (2018)

work page 2018
[68]

E. C. Paul, S. P. Walborn, D. S. Tasca, and L. Rudnicki, Mu- tually unbiased coarse-grained measurements of two or more phase-space variables, Phys. Rev. A97, 052103 (2018)

work page 2018
[69]

D. S. Tasca, L. Rudnicki, R. S. Aspden, M. J. Padgett, P. H. Souto Ribeiro, and S. P. Walborn, Testing for entangle- ment with periodic coarse graining, Phys. Rev. A97, 042312 (2018)

work page 2018
[70]

Adesso and F

G. Adesso and F. Illuminati, Entanglement in continu- ous variable systems: Recent advances and current per- spectives, J. Phys. A: Math. Theor.40, 7821 (2007), https://arxiv.org/abs/quant-ph/0701221

work page arXiv 2007
[71]

Weedbrook, S

C. Weedbrook, S. Pirandola, R. Garc´ıa-Patr´on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum infor- mation, Rev. Mod. Phys.84, 621 (2012)

work page 2012
[72]

In the CV and Gaussian state literature, the symmetric covariance 1 2 ⟨AB+BA⟩ ϱ − ⟨A⟩ϱ⟨B⟩ϱ is more common

We use the non-symmetric covariance Cov ϱ(A,B)=⟨AB⟩ ϱ − ⟨A⟩ϱ⟨B⟩ϱ, which makesΓHermitian but generally not real. In the CV and Gaussian state literature, the symmetric covariance 1 2 ⟨AB+BA⟩ ϱ − ⟨A⟩ϱ⟨B⟩ϱ is more common

work page
[73]

D. A. Trifonov and S. G. Donev, Characteristic uncertainty relations, Journal of Physics A: Mathematical and General31, 8041 (1998)

work page 1998
[74]

Dodonov, E

V . Dodonov, E. Kurmyshev, and V . Man’ko, Generalized un- certainty relation and correlated coherent states, Physics Let- ters A79, 150 (1980)

work page 1980
[75]

Here, we are focusing on Eq

Note that in the literature, many definitions of (variance- based) uncertainty relations and corresponding saturating states have been studied (see, e.g., [?]). Here, we are focusing on Eq. (7) for the case of two observables

work page
[76]

The reason for the prefactor √2π/din (8) is that with this choice, for largedthe low-lying spectrum of 1 2 (Q2 +P 2) ap- proximates that of the CV harmonic oscillator, whose eigen- values aren+ 1 2 forn=0,1,2,

work page
[77]

Busch and O

P. Busch and O. Reardon-Smith, On quantum uncertainty relations and uncertainty regions (2019), arXiv:1901.03695 [quant-ph]

work page arXiv 2019
[78]

Carath ´eodory’s theorem: for any subsetP⊂R k, every point in conv(P) can be written as a convex combination of at most k+1 points ofP

work page
[79]

Y . H. Au-Yeung and Y . T. Poon, A remark on the convexity and positive definiteness concerning hermitian matrices, Southeast Asian Bull. Math3, 85 (1979)

work page 1979
[80]

Schwonnek, L

R. Schwonnek, L. Dammeier, and R. F. Werner, State- independent uncertainty relations and entanglement detection in noisy systems, Physical review letters119, 170404 (2017)

work page 2017
[81]

M. L. Mehta, Eigenvalues and eigenvectors of the finite fourier transform, Journal of mathematical physics28, 781 (1987)

work page 1987

Showing first 80 references.