Library of spin anticoherent states


❯❯  Library of spin anticoherent states with degeneracy

❯❯  Library of 3-d spherical designs with degeneracy


Anticoherent spin-j states to order t are states

$$|\psi\rangle\in\mathbb{C}^{2j+1}\quad\textrm{such that}\quad\langle\psi|(\mathbf{J}\boldsymbol{\cdot}\mathbf{n})^{k}|\psi\rangle\quad\textrm{is independent of }\mathbf{n}\textrm{ for }k=1,\ldots,t$$

These states are equivalent to symmetric N-qubit states (N=2j) with maximally mixed t-qubit reductions in the symmetric sector; see Phys. Rev. A 90, 032314 (2014) and Phys. Rev. Lett. 114, 080401 (2015) for more details. Multiqubit symmetric states can always be written in the form

$$|\psi\rangle=\mathcal{N}\sum_{\sigma\in S_{N}}|\epsilon_{\sigma(1)}\rangle\otimes|\epsilon_{\sigma(2)}\rangle\otimes\cdots\otimes|\epsilon_{\sigma(N)}\rangle$$

where to each single-qubit state (for i=1,…,N)

$$|\epsilon_{i}\rangle=\cos(\tfrac{\theta_i}{2})|0\rangle+\sin(\tfrac{\theta_i}{2})e^{i\varphi_i}|1\rangle$$

one can associate a point on the Bloch sphere. Therefore, multiqubit symmetric states can be described by a set of N points on the Bloch sphere. This is the celebrated Majorana representation.

One can wonder whether the notions of anticoherent spin state to order t and 3-dimensional spherical t-design are related. Some relationships were indeed found in J. Phys. A: Math. Theor. 43 (2010) 255307, where it was conjectured that any spin state is anticoherent to order t if and only if its Majorana points form a 3-d spherical t-design. This conjecture was soon after proven false, see J. Phys. A: Math. Theor. 44 (2011) 343002. In fact, there are anticoherent states to order t that do not define a 3-d spherical t-design and, conversely, there are 3-d spherical t-designs that do not correspond to anticoherent states to order t. For example, the spin-5/2 state (expressed in the standard angular momentum basis)

$$|\psi\rangle=\sqrt{\tfrac{3}{8}}|\tfrac{5}{2},-\tfrac{5}{2}\rangle+\sqrt{\tfrac{5}{8}}|\tfrac{5}{2},\tfrac{3}{2}\rangle$$

is anticoherent to order 1 but its Majorana points do not form a spherical 1-design. Conversely, the spin-5/2 state

$$|\psi\rangle=\tfrac{9}{\sqrt{206}}|\tfrac{5}{2},-\tfrac{5}{2}\rangle+5\sqrt{\tfrac{5}{206}}|\tfrac{5}{2},\tfrac{3}{2}\rangle$$

has Majorana points forming a spherical 1-design but is not anticoherent to order 1.

Nonetheless, numerical results seem to suggest that the pairs (t,j) for which a spin-j state anticoherent to order t exists coincide with those for which a 2j-points spherical t-design exists in three dimensions. It has been conjectured (see the Conclusion of Phys. Scr. 90 (2015) 108008) that the highest possible degree of anticoherence matches the maximal degree t for an N-point spherical t-design. We were able to find anticoherent states for almost all N≤100 (all N≤100 except 29,87 and 94), whose order matches the maximal degree t for an N-point spherical t-design. Despite many efforts, we were unable to find an anticoherent state to order 6 of 29 qubits, eventhough there exists a spherical 6-design made of 29 points. The state that comes closest to an anticoherent state to order 6 that we could find has a purity-based measure of anticoherence (see Phys. Rev. A 96, 032304 (2017) for more details)

$$\mathcal{A}_t(|\psi\rangle)=\frac{t+1}{t}\left(1-\mathrm{Tr}[\rho_t^2]\right)\quad\mathrm{with}\quad \rho_t=\mathrm{Tr}_{\neg t}[|\psi\rangle\langle\psi|]$$

that is approximately equal to 1−3x10-8≠1. This raises the question of the validity of the conjecture. It might be that the maximal degree t for an N-point spherical design is only an upper bound for the maximal order of anticoherence of an N/2-spin state. Further work is needed to confirm or refute this conjecture.

It should be noted that as soon as we impose a minimal degeneracy d to one of the points defining a spherical design or a symmetric multiqubit state, the simultaneous existence of anticoherent states and spherical designs for fixed N, t and d is not observed. For example, we find that there is no spherical 2-design of 7 points with one point 2-fold degenerated while there is a spin-7/2 state anticoherent to order 2 with one Majorana point 2-fold degenerated, i.e. the state

$$|\psi\rangle=\sqrt{\tfrac{7}{10}}|\tfrac{7}{2},-\tfrac{3}{2}\rangle+\sqrt{\tfrac{3}{10}}|\tfrac{7}{2},\tfrac{7}{2}\rangle$$

References

Here below we give putative optimal anticoherent states to order t for all N≤100 except 29,87 and 94. These states were found by maximizing the purity-based measure of anticoherence At and using as numerical criterion that a state is anticoherent if its measure of anticoherence approaches 1 by less than 10-25.
The data are presented both as a list of cartesian coordinates for each Majorana point (acs.* files), as is done by Hardin and Sloane for 3d spherical designs, and as a list of (complex) components in the Dicke states basis (acsdk.* files, first/second column corresponds to real/imaginary part). States marked by (*) are those whose order is less than the maximal degree t for an N-point spherical t-design in 3d.

~~~

Library of spin anticoherent states (2020)
Note : a spin j-state is equivalent to a symmetric N-qubit state with N=2j. Here, we give the values of N instead of j. 

Found by J. Martin.
acs.2.1.txt and acsdk.2.1.txt contain a 2-qubit anticoherent state to order 1

Found by J. Martin.
acs.3.1.txt and acsdk.3.1.txt contain a 3-qubit anticoherent state to order 1

Found by J. Martin.
acs.4.2.txt and acsdk.4.2.txt contain a 4-qubit anticoherent state to order 2

Found by J. Martin.
acs.5.1.txt and acsdk.5.1.txt contain a 5-qubit anticoherent state to order 1

Found by J. Martin.
acs.6.3.txt and acsdk.6.3.txt contain a 6-qubit anticoherent state to order 3

Found by J. Martin.
acs.7.2.txt and acsdk.7.2.txt contain a 7-qubit anticoherent state to order 2

Found by J. Martin.
acs.8.3.txt and acsdk.8.3.txt contain a 8-qubit anticoherent state to order 3

Found by J. Martin.
acs.9.2.txt and acsdk.9.2.txt contain a 9-qubit anticoherent state to order 2

Found by J. Martin.
acs.10.3.txt and acsdk.10.3.txt contain a 10-qubit anticoherent state to order 3

Found by J. Martin.
acs.11.3.txt and acsdk.11.3.txt contain a 11-qubit anticoherent state to order 3

Found by J. Martin.
acs.12.5.txt and acsdk.12.5.txt contain a 12-qubit anticoherent state to order 5

Found by J. Martin.
acs.13.3.txt and acsdk.13.3.txt contain a 13-qubit anticoherent state to order 3

Found by J. Martin.
acs.14.4.txt and acsdk.14.4.txt contain a 14-qubit anticoherent state to order 4

Found by J. Martin.
acs.15.3.txt and acsdk.15.3.txt contain a 15-qubit anticoherent state to order 3

Found by J. Martin.
acs.16.5.txt and acsdk.16.5.txt contain a 16-qubit anticoherent state to order 5

Found by J. Martin.
acs.17.4.txt and acsdk.17.4.txt contain a 17-qubit anticoherent state to order 4

Found by J. Martin.
acs.18.5.txt and acsdk.18.5.txt contain a 18-qubit anticoherent state to order 5

Found by J. Martin.
acs.19.4.txt and acsdk.19.4.txt contain a 19-qubit anticoherent state to order 4

Found by J. Martin.
acs.20.5.txt and acsdk.20.5.txt contain a 20-qubit anticoherent state to order 5

Found by J. Martin.
acs.21.4.txt and acsdk.21.4.txt contain a 21-qubit anticoherent state to order 4

Found by J. Martin.
acs.22.5.txt and acsdk.22.5.txt contain a 22-qubit anticoherent state to order 5

Found by J. Martin.
acs.23.5.txt and acsdk.23.5.txt contain a 23-qubit anticoherent state to order 5

Found by J. Martin.
acs.24.7.txt and acsdk.24.7.txt contain a 24-qubit anticoherent state to order 7

Found by J. Martin.
acs.25.5.txt and acsdk.25.5.txt contain a 25-qubit anticoherent state to order 5

Found by J. Martin.
acs.26.6.txt and acsdk.26.6.txt contain a 26-qubit anticoherent state to order 6

Found by J. Martin.
acs.27.5.txt and acsdk.27.5.txt contain a 27-qubit anticoherent state to order 5

Found by J. Martin.
acs.28.6.txt and acsdk.28.6.txt contain a 28-qubit anticoherent state to order 6

Found by J. Martin.
acs.29.5.txt and acsdk.29.5.txt contain a 29-qubit anticoherent state to order 5 (*)

Found by J. Martin.
acs.30.7.txt and acsdk.30.7.txt contain a 30-qubit anticoherent state to order 7

Found by J. Martin.
acs.31.6.txt and acsdk.31.6.txt contain a 31-qubit anticoherent state to order 6

Found by J. Martin.
acs.32.7.txt and acsdk.32.7.txt contain a 32-qubit anticoherent state to order 7

Found by J. Martin.
acs.33.6.txt and acsdk.33.6.txt contain a 33-qubit anticoherent state to order 6

Found by J. Martin.
acs.34.7.txt and acsdk.34.7.txt contain a 34-qubit anticoherent state to order 7

Found by J. Martin.
acs.35.6.txt and acsdk.35.6.txt contain a 35-qubit anticoherent state to order 6

Found by J. Martin.
acs.36.8.txt and acsdk.36.8.txt contain a 36-qubit anticoherent state to order 8

Found by J. Martin.
acs.37.7.txt and acsdk.37.7.txt contain a 37-qubit anticoherent state to order 7

Found by J. Martin.
acs.38.7.txt and acsdk.38.7.txt contain a 38-qubit anticoherent state to order 7

Found by J. Martin.
acs.39.7.txt and acsdk.39.7.txt contain a 39-qubit anticoherent state to order 7

Found by J. Martin.
acs.40.8.txt and acsdk.40.8.txt contain a 40-qubit anticoherent state to order 8

Found by J. Martin.
acs.41.7.txt and acsdk.41.7.txt contain a 41-qubit anticoherent state to order 7

Found by J. Martin.
acs.42.8.txt and acsdk.42.8.txt contain a 42-qubit anticoherent state to order 8

Found by J. Martin.
acs.43.7.txt and acsdk.43.7.txt contain a 43-qubit anticoherent state to order 7

Found by J. Martin.
acs.44.8.txt and acsdk.44.8.txt contain a 44-qubit anticoherent state to order 8

Found by J. Martin.
acs.45.8.txt and acsdk.45.8.txt contain a 45-qubit anticoherent state to order 8

Found by J. Martin.
acs.46.8.txt and acsdk.46.8.txt contain a 46-qubit anticoherent state to order 8

Found by J. Martin.
acs.47.8.txt and acsdk.47.8.txt contain a 47-qubit anticoherent state to order 8

Found by J. Martin.
acs.48.9.txt and acsdk.48.9.txt contain a 48-qubit anticoherent state to order 9

Found by J. Martin.
acs.49.8.txt and acsdk.49.8.txt contain a 49-qubit anticoherent state to order 8

Found by J. Martin.
acs.50.9.txt and acsdk.50.9.txt contain a 50-qubit anticoherent state to order 9

Found by J. Martin.
acs.51.8.txt and acsdk.51.8.txt contain a 51-qubit anticoherent state to order 8

Found by J. Martin.
acs.52.9.txt and acsdk.52.9.txt contain a 52-qubit anticoherent state to order 9

Found by J. Martin.
acs.53.8.txt and acsdk.53.8.txt contain a 53-qubit anticoherent state to order 8

Found by J. Martin.
acs.54.9.txt and acsdk.54.9.txt contain a 54-qubit anticoherent state to order 9

Found by J. Martin.
acs.55.9.txt and acsdk.55.9.txt contain a 55-qubit anticoherent state to order 9

Found by J. Martin.
acs.56.9.txt and acsdk.56.9.txt contain a 56-qubit anticoherent state to order 9

Found by J. Martin.
acs.57.9.txt and acsdk.57.9.txt contain a 57-qubit anticoherent state to order 9

Found by J. Martin.
acs.58.9.txt and acsdk.58.9.txt contain a 58-qubit anticoherent state to order 9

Found by J. Martin.
acs.59.9.txt and acsdk.59.9.txt contain a 59-qubit anticoherent state to order 9

Found by J. Martin.
acs.60.9.txt and acsdk.60.9.txt contain a 60-qubit anticoherent state to order 9

Found by J. Martin.
acs.61.9.txt and acsdk.61.9.txt contain a 61-qubit anticoherent state to order 9

Found by J. Martin.
acs.62.10.txt and acsdk.62.10.txt contain a 62-qubit anticoherent state to order 10

Found by J. Martin.
acs.63.9.txt and acsdk.63.9.txt contain a 63-qubit anticoherent state to order 9

Found by J. Martin.
acs.64.10.txt and acsdk.64.10.txt contain a 64-qubit anticoherent state to order 10

Found by J. Martin.
acs.65.9.txt and acsdk.65.9.txt contain a 65-qubit anticoherent state to order 9

Found by J. Martin.
acs.66.10.txt and acsdk.66.10.txt contain a 66-qubit anticoherent state to order 10

Found by J. Martin.
acs.67.10.txt and acsdk.67.10.txt contain a 67-qubit anticoherent state to order 10

Found by J. Martin.
acs.68.10.txt and acsdk.68.10.txt contain a 68-qubit anticoherent state to order 10

Found by J. Martin.
acs.69.10.txt and acsdk.69.10.txt contain a 69-qubit anticoherent state to order 10

Found by J. Martin.
acs.70.10.txt and acsdk.70.10.txt contain a 70-qubit anticoherent state to order 10

Found by J. Martin.
acs.71.10.txt and acsdk.71.10.txt contain a 71-qubit anticoherent state to order 10

Found by J. Martin.
acs.72.10.txt and acsdk.72.10.txt contain a 72-qubit anticoherent state to order 10

Found by J. Martin.
acs.73.10.txt and acsdk.73.10.txt contain a 73-qubit anticoherent state to order 10

Found by J. Martin.
acs.74.11.txt and acsdk.74.11.txt contain a 74-qubit anticoherent state to order 11

Found by J. Martin.
acs.75.10.txt and acsdk.75.10.txt contain a 75-qubit anticoherent state to order 10

Found by J. Martin.
acs.76.11.txt and acsdk.76.11.txt contain a 76-qubit anticoherent state to order 11

Found by J. Martin.
acs.77.11.txt and acsdk.77.11.txt contain a 77-qubit anticoherent state to order 11

Found by J. Martin.
acs.78.11.txt and acsdk.78.11.txt contain a 78-qubit anticoherent state to order 11

Found by J. Martin.
acs.79.11.txt and acsdk.79.11.txt contain a 79-qubit anticoherent state to order 11

Found by J. Martin.
acs.80.11.txt and acsdk.80.11.txt contain a 80-qubit anticoherent state to order 11

Found by J. Martin.
acs.81.11.txt and acsdk.81.11.txt contain a 81-qubit anticoherent state to order 11

Found by J. Martin.
acs.82.11.txt and acsdk.82.11.txt contain a 82-qubit anticoherent state to order 11

Found by J. Martin.
acs.83.11.txt and acsdk.83.11.txt contain a 83-qubit anticoherent state to order 11

Found by J. Martin.
acs.84.12.txt and acsdk.84.12.txt contain a 84-qubit anticoherent state to order 12

Found by J. Martin.
acs.85.11.txt and acsdk.85.11.txt contain a 85-qubit anticoherent state to order 11

Found by J. Martin.
acs.86.12.txt and acsdk.86.12.txt contain a 86-qubit anticoherent state to order 12

Found by J. Martin.
acs.87.11.txt and acsdk.87.11.txt contain a 87-qubit anticoherent state to order 11 (*)

Found by J. Martin.
acs.88.12.txt and acsdk.88.12.txt contain a 88-qubit anticoherent state to order 12

Found by J. Martin.
acs.89.12.txt and acsdk.89.12.txt contain a 89-qubit anticoherent state to order 12

Found by J. Martin.
acs.90.12.txt and acsdk.90.12.txt contain a 90-qubit anticoherent state to order 12

Found by J. Martin.
acs.91.12.txt and acsdk.91.12.txt contain a 91-qubit anticoherent state to order 12

Found by J. Martin.
acs.92.12.txt and acsdk.92.12.txt contain a 92-qubit anticoherent state to order 12

Found by J. Martin.
acs.93.12.txt and acsdk.93.12.txt contain a 93-qubit anticoherent state to order 12

Found by J. Martin.
acs.94.12.txt and acsdk.94.12.txt contain a 94-qubit anticoherent state to order 12 (*)

Found by J. Martin.
acs.95.12.txt and acsdk.95.12.txt contain a 95-qubit anticoherent state to order 12

Found by J. Martin.
acs.96.13.txt and acsdk.96.13.txt contain a 96-qubit anticoherent state to order 13

Found by J. Martin.
acs.97.12.txt and acsdk.97.12.txt contain a 97-qubit anticoherent state to order 12

Found by J. Martin.
acs.98.13.txt and acsdk.98.13.txt contain a 98-qubit anticoherent state to order 13

Found by J. Martin.
acs.99.12.txt and acsdk.99.12.txt contain a 99-qubit anticoherent state to order 12

Found by J. Martin.
acs.100.13.txt and acsdk.100.13.txt contain a 100-qubit anticoherent state to order 13


© ULiege 2024 - John Martin