Mutually unbiased bases

Short description: Concept in quantum information theory

An example of three bases in a two-dimensional space, where bases B1 and B2 are mutually unbiased, whereas the vectors of basis B3 do not have equal overlap with the vectors of basis B1 (as well as B2) and so B3 is not mutually unbiased with B1 (and B2).

In quantum information theory, a set of bases in Hilbert space Cd are said to be mutually unbiased if when a system is prepared in an eigenstate of one of the bases, then all outcomes of the measurement with respect to the other basis are predicted to occur with an equal probability of 1/d.

## Contents

* 1 Overview
* 2 Definition and examples
* 2.1 Example for d = 2
* 2.2 Example for d = 4
* 3 Existence problem
* 4 Related problems
* 5 Search methods
* 5.1 Weyl group method
* 5.2 Unitary operators method using finite fields
* 5.3 Hadamard matrix method
* 6 Entropic uncertainty relations
* 7 Infinite dimension Hilbert spaces
* 8 See also
* 9 References

## Overview

The notion of mutually unbiased bases was first introduced by Julian Schwinger in 1960,[1] and the first person to consider applications of mutually unbiased bases was I. D. Ivanovic[2] in the problem of quantum state determination.

Mutually unbiased bases (MUBs) and their existence problem is now known to have several closely related problems and equivalent avatars in several other branches of mathematics and quantum sciences, such as SIC-POVMs, finite projective/affine planes, complex Hadamard matrices and more [see section: Related problems].

MUBs are important for quantum key distribution, more specifically in secure quantum key exchange.[3] MUBs are used in many protocols since the outcome is random when a measurement is made in a basis unbiased to that in which the state was prepared. When two remote parties share two non-orthogonal quantum states, attempts by an eavesdropper to distinguish between these by measurements will affect the system and this can be detected. While many quantum cryptography protocols have relied on 1-qubit technologies, employing higher-dimensional states, such as qutrits, allows for better security against eavesdropping.[3] This motivates the study of mutually unbiased bases in higher-dimensional spaces.

Other uses of mutually unbiased bases include quantum state reconstruction,[4] quantum error correction codes,[5][6] detection of quantum entanglement,[7][8] and the so-called "mean king's problem".[9][10]

## Definition and examples

A pair of orthonormal bases {|e1⟩,…,|ed⟩} and {|f1⟩,…,|fd⟩} in Hilbert space Cd are said to be mutually unbiased, if and only if the square of the magnitude of the inner product between any basis states |ej⟩ and |fk⟩ equals the inverse of the dimension d:[11]

|⟨ej|fk⟩|2=1d,∀j,k∈{1,…,d}.

These bases are unbiased in the following sense: if a system is prepared in a state belonging to one of the bases, then all outcomes of the measurement with respect to the other basis are predicted to occur with equal probability.

### Example for d = 2

The three bases

M0={|0⟩,|1⟩}
M1={|0⟩+|1⟩2,|0⟩−|1⟩2}
M2={|0⟩+i|1⟩2,|0⟩−i|1⟩2}

provide the simplest example of mutually unbiased bases in C2. The above bases are composed of the eigenvectors of the Pauli spin matrices σz,σx and their product σxσz, respectively.

### Example for d = 4

For d = 4, an example of d + 1 = 5 mutually unbiased bases where each basis is denoted by Mj, 0 ≤ j ≤ 4, is given as follows:[12]

M0={(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)}
M1={12(1,1,1,1),12(1,1,−1,−1),12(1,−1,−1,1),12(1,−1,1,−1)}
M2={12(1,−1,−i,−i),12(1,−1,i,i),12(1,1,i,−i),12(1,1,−i,i)}
M3={12(1,−i,−i,−1),12(1,−i,i,1),12(1,i,i,−1),12(1,i,−i,1)}
M4={12(1,−i,−1,−i),12(1,−i,1,i),12(1,i,−1,i),12(1,i,1,−i)}

## Existence problem

| Unsolved problem in mathematics:

What is the maximum number of MUBs in any given non-prime-power dimension d?

(more unsolved problems in mathematics)

|

Let 𝔐(d) denote the maximum number of mutually unbiased bases in the d-dimensional Hilbert space Cd. It is an open question[13] how many mutually unbiased bases, 𝔐(d), one can find in Cd, for arbitrary d.

In general, if

d=p1n1p2n2⋯pknk

is the prime-power factorization of d, where

p1n1<p2n2<⋯<pknk

then the maximum number of mutually unbiased bases which can be constructed satisfies[11]

p1n1+1≤𝔐(d)≤d+1.

It follows that if the dimension of a Hilbert space d is an integer power of a prime number, then it is possible to find d + 1 mutually unbiased bases. This can be seen in the previous equation, as the prime number decomposition of d simply is d=pn. Therefore,

𝔐(pn)=pn+1.

Thus, the maximum number of mutually unbiased bases is known when d is an integer power of a prime number, but it is not known for arbitrary d.

The smallest dimension that is not an integer power of a prime is d = 6\. This is also the smallest dimension for which the number of mutually unbiased bases is not known. The methods used to determine the number of mutually unbiased bases when d is an integer power of a prime number cannot be used in this case. Searches for a set of four mutually unbiased bases when d = 6, both by using Hadamard matrices[11] and numerical methods[14][15] have been unsuccessful. The general belief is that the maximum number of mutually unbiased bases for d = 6 is 𝔐(6)=3.[11]

## Related problems

| Unsolved problem in mathematics:

Do SIC-POVMs exist in all dimensions?

(more unsolved problems in mathematics)

|

The MUBs problem seems similar in nature to the symmetric property of SIC-POVMs. William Wootters points out that a complete set of d+1 unbiased bases yields a geometric structure known as a finite projective plane, while a SIC-POVM (in any dimension that is a prime power) yields a finite affine plane, a type of structure whose definition is identical to that of a finite projective plane with the roles of points and lines exchanged. In this sense, the problems of SIC-POVMs and of mutually unbiased bases are dual to one another.[16]

In dimension d=3, the analogy can be taken further: a complete set of mutually unbiased bases can be directly constructed from a SIC-POVM.[17] The 9 vectors of the SIC-POVM, together with the 12 vectors of the mutually unbiased bases, form a set that can be used in a Kochen–Specker proof.[18] However, in 6-dimensional Hilbert space, a SIC-POVM is known, but no complete set of mutually unbiased bases has yet been discovered, and it is widely believed that no such set exists.[19][20][21]

## Search methods

### Weyl group method

Let X^ and Z^ be two unitary operators in the Hilbert space Cd such that

Z^X^=ωX^Z^

for some phase factor ω. If ω is a primitive root of unity, for example ω≡e2πid then the eigenbases of X^ and Z^ are mutually unbiased.

By choosing the eigenbasis of Z^ to be the standard basis, we can generate another basis unbiased to it using a Fourier matrix. The elements of the Fourier matrix are given by

Fab=ωab,0≤a,b≤N−1

Other bases which are unbiased to both the standard basis and the basis generated by the Fourier matrix can be generated using Weyl groups.[11] The dimension of the Hilbert space is important when generating sets of mutually unbiased bases using Weyl groups. When d is a prime number, then the usual d + 1 mutually unbiased bases can be generated using Weyl groups. When d is not a prime number, then it is possible that the maximal number of mutually unbiased bases which can be generated using this method is 3.

### Unitary operators method using finite fields

When d = p is prime, we define the unitary operators X^ and Z^ by

X^=∑k=0d−1|k+1⟩⟨k|
Z^=∑k=0d−1ωk|k⟩⟨k|

where {|k⟩|0≤k≤d−1} is the standard basis and ω=e2πid is a root of unity.

Then the eigenbases of the following d + 1 operators are mutually unbiased:[22]

X^,Z^,X^Z^,X^Z^2,…,X^Z^d−1.

For odd d, the t-th eigenvector of the operator X^Z^k is given explicitly by[13]

|ψtk⟩=1d∑j=0d−1ωtj+kj2|j⟩.

When d=pr is a power of a prime, we make use of the finite field 𝔽d to construct a maximal set of d + 1 mutually unbiased bases. We label the elements of the computational basis of Cd using the finite field: {|a⟩|a∈𝔽d}.

We define the operators Xa^ and Zb^ in the following way

Xa^=∑c∈𝔽d|c+a⟩⟨c|
Zb^=∑c∈𝔽dχ(bc)|c⟩⟨c|

where

χ(θ)=exp⁡[2πip(θ+θp+θp2+⋯+θpr−1)],

is an additive character over the field and the addition and multiplication in the kets and χ(⋅) is that of 𝔽d.

Then we form d + 1 sets of commuting unitary operators:

{Zs^|s∈𝔽d} and {Xs^Zsr^|s∈𝔽d} for each r∈𝔽d

The joint eigenbases of the operators in one set are mutually unbiased to that of any other set.[22] We thus have d + 1 mutually unbiased bases.

### Hadamard matrix method

Given that one basis in a Hilbert space is the standard basis, then all bases which are unbiased with respect to this basis can be represented by the columns of a complex Hadamard matrix multiplied by a normalization factor. For d = 3 these matrices would have the form

U=1d[111eiϕ10eiϕ11eiϕ12eiϕ20eiϕ21eiϕ22]

The problem of finding a set of k+1 mutually unbiased bases therefore corresponds to finding k mutually unbiased complex Hadamard matrices.[11]

An example of a one parameter family of Hadamard matrices in a 4-dimensional Hilbert space is

H4(ϕ)=12[11111eiϕ−1−eiϕ1−11−11−eiϕ−1eiϕ]

## Entropic uncertainty relations

There is an alternative characterization of mutually unbiased bases that considers them in terms of uncertainty relations.[23]

Entropic uncertainty relations are analogous to the Heisenberg uncertainty principle, and Hans Maassen and J. B. M. Uffink[24] found that for any two bases B1={|ai⟩i=1d} and B2={|bj⟩j=1d}:

HB1+HB2≥−2log⁡c.

where c=max⁡|⟨aj|bk⟩| and HB1 and HB2 is the respective entropy of the bases B1 and B2, when measuring a given state.

Entropic uncertainty relations are often preferable[25] to the Heisenberg uncertainty principle, as they are not phrased in terms of the state to be measured, but in terms of c.

In scenarios such as quantum key distribution, we aim for measurement bases such that full knowledge of a state with respect to one basis implies minimal knowledge of the state with respect to the other bases. This implies a high entropy of measurement outcomes, and thus we call these strong entropic uncertainty relations.

For two bases, the lower bound of the uncertainty relation is maximized when the measurement bases are mutually unbiased, since mutually unbiased bases are maximally incompatible: the outcome of a measurement made in a basis unbiased to that in which the state is prepared in is completely random. In fact, for a d-dimensional space, we have:[26]

HB1+HB2≥log⁡(d)

for any pair of mutually unbiased bases B1 and B2. This bound is optimal:[27] If we measure a state from one of the bases then the outcome has entropy 0 in that basis and an entropy of log⁡(d) in the other.

If the dimension of the space is a prime power, we can construct d + 1 MUBs, and then it has been found that[28]

∑k=1d+1HBk≥(d+1)log⁡(d+12)

which is stronger than the relation we would get from pairing up the sets and then using the Maassen and Uffink equation. Thus we have a characterization of d + 1 mutually unbiased bases as those for which the uncertainty relations are strongest.

Although the case for two bases, and for d + 1 bases is well studied, very little is known about uncertainty relations for mutually unbiased bases in other circumstances.[28] [29]

When considering more than two, and less than d+1 bases it is known that large sets of mutually unbiased bases exist which exhibit very little uncertainty.[30] This means merely being mutually unbiased does not lead to high uncertainty, except when considering measurements in only two bases. Yet there do exist other measurements that are very uncertain.[28][31]

## Infinite dimension Hilbert spaces

While there has been investigation into mutually unbiased bases in infinite dimension Hilbert space, their existence remains an open question. It is conjectured that in a continuous Hilbert space, two orthonormal bases |ψsb⟩ and |ψs′b′⟩ are said to be mutually unbiased if[32]

|⟨ψsb|ψs′b′⟩|2=k>0,s,s′∈ℝ

For the generalized position and momentum eigenstates |q⟩,q∈ℝ and |p⟩,p∈ℝ, the value of k is

|⟨q|p⟩|2=12πℏ

The existence of mutually unbiased bases in a continuous Hilbert space remains open for debate, as further research in their existence is required before any conclusions can be reached.

Position states |q⟩ and momentum states |p⟩ are eigenvectors of Hermitian operators x^ and −i∂∂x, respectively. Weigert and Wilkinson[32] were first to notice that also a linear combination of these operators have eigenbases, which have some features typical for the mutually unbiased bases. An operator αx^−iβ∂∂x has eigenfunctions proportional to exp⁡(i(ax2+bx)) with α+2βa=0 and the corresponding eigenvalues bβ. If we parametrize α and β as cos⁡θ and sin⁡θ, the overlap between any eigenstate of the linear combination and any eigenstate of the position operator (both states normalized to the Dirac delta) is constant, but dependent on β:

|⟨xθ|x⟩|2=12π|sin⁡θ|,

where |x⟩ and |xθ⟩ stand for eigenfunctions of x^ and cos⁡θx^−isin⁡θ∂∂x.

## See also

* Measurement in quantum mechanics
* POVM
* SIC-POVM
* QBism

## References

1. ↑ Schwinger, J. (1960). "Unitary Operator Bases, Harvard University". Proc. Natl. Acad. Sci. U.S.A. 46 (4): 570–9. doi:10.1073/pnas.46.4.570. PMID 16590645. Bibcode: 1960PNAS...46..570S.
2. ↑ Ivanovic, I. D. (1981). "Geometrical description of quantal state determination". J. Phys. A 14 (12): 3241–3245. doi:10.1088/0305-4470/14/12/019. Bibcode: 1981JPhA...14.3241I.
3. ↑ 3.0 3.1 Planat, M. (14 November 2006). "A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements". Foundations of Physics 36 (11): 1662–1680. doi:10.1007/s10701-006-9079-3. Bibcode: 2006FoPh...36.1662P. http://hal.ccsd.cnrs.fr/docs/00/07/99/18/PDF/MUB_FP.pdf.
4. ↑ Wootters, W. K.; Fields, B. D. (1989). "Optimal State-Determination by Mutually Unbiased Measurements". Ann. Phys. 191 (2): 363–381. doi:10.1016/0003-4916(89)90322-9. Bibcode: 1989AnPhy.191..363W.
5. ↑ Gottesman, D. (1996). "Class of quantum error-correcting codes saturating the quantum Hamming bound". Phys. Rev. A 54 (3): 1862–1868. doi:10.1103/physreva.54.1862. PMID 9913672. Bibcode: 1996PhRvA..54.1862G.
6. ↑ Calderbank, A. R. (1997). "Quantum Error Correction and Orthogonal Geometry". Phys. Rev. Lett. 78 (3): 405–408. doi:10.1103/physrevlett.78.405. Bibcode: 1997PhRvL..78..405C.
7. ↑ Huang, Yichen (29 July 2010). "Entanglement criteria via concave-function uncertainty relations". Physical Review A 82 (1). doi:10.1103/PhysRevA.82.012335. Bibcode: 2010PhRvA..82a2335H.
8. ↑ Spengler, C.; Huber, M.; Brierley, S.; Adaktylos, T.; Hiesmayr, B. C. (2012). "Entanglement detection via mutually unbiased bases". Phys. Rev. A 86 (2). doi:10.1103/physreva.86.022311. Bibcode: 2012PhRvA..86b2311S.
9. ↑ Vaidman, L. (1987). "How to ascertain the values of σx,σy, and σz of a spin-1/2 particle". Phys. Rev. Lett. 58 (14): 1385–1387. doi:10.1103/PhysRevLett.58.1385. PMID 10034422. Bibcode: 1987PhRvL..58.1385V.
10. ↑ Englert, B.-G.; Aharonov, Y. (2001). "The mean king's problem: prime degrees of freedom". Phys. Lett. A 284 (1): 1–5. doi:10.1016/s0375-9601(01)00271-7. Bibcode: 2001PhLA..284....1E.
11. ↑ 11.0 11.1 11.2 11.3 11.4 11.5 Bengtsson, Ingemar (2007). "Three Ways to Look at Mutually Unbiased Bases". AIP Conference Proceedings. 889. pp. 40–51. doi:10.1063/1.2713445.
12. ↑ Klappenecker, Andreas; Roetteler, Martin (2004). "Constructions of Mutually Unbiased Bases". Finite Fields and Applications. Lecture Notes in Computer Science. 2948. Springer. pp. 137–144. Bibcode: 2003quant.ph..9120K.
13. ↑ 13.0 13.1 Durt, T.; Englert, B.-G.; Bengtsson, I.; Życzkowski, K. (2010). "On mutually unbiased bases". International Journal of Quantum Information 8 (4): 535–640. doi:10.1142/s0219749910006502.
14. ↑ Butterley, Paul; Hall, William (September 2007). "Numerical evidence for the maximum number of mutually unbiased bases in dimension six" (in en). Physics Letters A 369 (1–2): 5–8. doi:10.1016/j.physleta.2007.04.059. Bibcode: 2007PhLA..369....5B.
15. ↑ Brierley, S.; Weigert, S. (2008). "Maximal sets of mutually unbiased quantum states in dimension six". Phys. Rev. A 78 (4). doi:10.1103/physreva.78.042312. Bibcode: 2008PhRvA..78d2312B.
16. ↑ Wootters, William K. (8 February 2006). "Quantum measurements and finite geometry". Foundations of Physics 36 (1, Festschrift for Asher Peres): 112–126. doi:10.1007/s10701-005-9008-x. Bibcode: 2006FoPh...36..112W.
17. ↑ Stacey, Blake C. (2016). "SIC-POVMs and Compatibility among Quantum States". Mathematics 4 (2): 36. doi:10.3390/math4020036.
18. ↑ Bengtsson, Ingemar; Blanchfield, Kate; Cabello, Adán (2012). "A Kochen–Specker inequality from a SIC". Physics Letters A 376 (4): 374-376. doi:10.1016/j.physleta.2011.12.011. Bibcode: 2012PhLA..376..374B.
19. ↑ Grassl, Markus (2004). "On SIC-POVMs and MUBs in Dimension 6". arXiv:quant-ph/0406175.
20. ↑ Bengtsson, Ingemar; Życzkowski, Karol (2017). Geometry of quantum states: an introduction to quantum entanglement (Second ed.). Cambridge, United Kingdom: Cambridge University Press. pp. 313-354. ISBN 978-1-107-02625-4. OCLC 967938939.
21. ↑ Horodecki, Paweł; Rudnicki, Łukasz; Życzkowski, Karol (2022-03-03). "Five Open Problems in Quantum Information Theory" (in en). PRX Quantum 3 (1). doi:10.1103/PRXQuantum.3.010101. Bibcode: 2022PRXQ....3a0101H.
22. ↑ 22.0 22.1 Bandyopadhyay, Somshubhro; Oscar Boykin, P.; Roychowdhury, Vwani; Vatan, Farrokh (2002). "A new proof for the existence of mutually unbiased bases". Algorithmica 32 (4): 512–528. doi:10.1007/s00453-002-0980-7. Bibcode: 2001quant.ph..3162B.
23. ↑ Hirschman, I. I. Jr. (1957). "A note on entropy". American Journal of Mathematics 1957 (1): 152–156. doi:10.2307/2372390.
24. ↑ Maassen, Hans; Uffink, J. B. M. (1988-03-21). "Generalized entropic uncertainty relations" (in en). Physical Review Letters 60 (12): 1103–1106. doi:10.1103/PhysRevLett.60.1103. Bibcode: 1988PhRvL..60.1103M.
25. ↑ Damgård, Ivan B.; Fehr, Serge; Renner, Renato; Salvail, Louis; Schaffner, Christian (2007). "A Tight High-Order Entropic Quantum Uncertainty Relation with Applications". in Menezes, A.. Advances in Cryptology - CRYPTO 2007. Lecture Notes in Computer Science. 4622. Springer. doi:10.1007/978-3-540-74143-5_20. ISBN 978-3-540-74143-5.
26. ↑ Deutsch, D. (1982). "Uncertainty in Quantum Measurements". Physical Review Letters 50 (9): 631–633. doi:10.1103/physrevlett.50.631. Bibcode: 1983PhRvL..50..631D.
27. ↑ Ambainis, Andris (2010-09-01). "Limits on entropic uncertainty relations for 3 and more MUBs". Quantum Info. Comput. 10 (9): 848–858. https://dl.acm.org/doi/abs/10.5555/2011464.2011474.
28. ↑ 28.0 28.1 28.2 Wehner, Stephanie; Winter, Andreas (2010-02-26). "Entropic uncertainty relations—a survey". New Journal of Physics 12 (2). doi:10.1088/1367-2630/12/2/025009. Bibcode: 2010NJPh...12b5009W.
29. ↑ Wu, S.; Yu, S.; Mølmer, K. (2009). "Entropic uncertainty relation for mutually unbiased bases". Phys. Rev. A 79 (2). doi:10.1103/physreva.79.022104. Bibcode: 2009PhRvA..79b2104W.
30. ↑ Ballester, M.; S. Wehner (2007). "Entropic uncertainty relations and locking: tight bounds for mutually unbiased bases". Physical Review A 75 (1): 022319. doi:10.1103/PhysRevA.75.012319. Bibcode: 2007PhRvA..75a2319C. http://ddd.uab.cat/pub/artpub/2007/115650/phyreva_a2007m1v75n1p012319.pdf.
31. ↑ Wehner, S.; A. Winter (2008). "Higher entropic uncertainty relations for anti-commuting observables". Journal of Mathematical Physics 49 (6): 062105. doi:10.1063/1.2943685. Bibcode: 2008JMP....49f2105W.
32. ↑ 32.0 32.1 Weigert, Stefan; Wilkinson, Michael (2008). "Mutually unbiased bases for continuous variables". Physical Review A 78 (2). doi:10.1103/PhysRevA.78.020303. Bibcode: 2008PhRvA..78b0303W.

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