Five Open Problems in Quantum Information Theory

Horodecki, Paweł; Rudnicki, Łukasz; Życzkowski, Karol

doi:10.1103/prxquantum.3.010101

Cited by 45 publications

(20 citation statements)

References 148 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…With N bosons and N outputs in the Bell multiport, there are in total |S| := 2N − 1 N different signatures possible. Then, we find that apart from some specific N, the total number of signatures in S Incidentally, N = 6, 10, 12 are the first three elements in the sequence of numbers which are not integer power of a prime number, and it is a well known open problem in mathematical physics and quantum information whether there exists N + 1 MUB for Hilbert space with dimension of such numbers [68,95]. Given that one of the important tools in studying MUB is the complex Hadamard matrices for which discrete Fourier transform is one of such [68,94], it is then perhaps not surprising to have such a correspondence between the non-unity ratio of |S 1 |/|S| and the MUB problem.…”

Section: Discussionmentioning

confidence: 97%

Multiparameter estimation for qubit states with collective measurements: a case study

Len

2022

New J. Phys.

View full text Add to dashboard Cite

Quantum estimation involving multiple parameters remains an important problem of both theoretical and practical interest. In this work, we study the problem of simultaneous estimation of two parameters that are respectively associate with the length and direction of the Bloch vector for identically prepared qubit states that is confined to a plane, where in order to obtain the optimal estimation precision for both parameters, collective measurements on multiple qubits are necessary. Upon treating N qubits as an ensemble of spin-1/2 systems, we show that simultaneous optimal estimation for both parameters can be attained asymptotically with a simple collective measurement strategy---First, we estimate the length parameter by measuring the populations in spaces corresponding to different total angular momentum values j, then we estimate the direction parameter by performing a spin projection onto an optimal basis. Furthermore, we show that when the state is nearly pure, for sufficiently but not arbitrarily large N, most information will be captured in the largest three j-subspaces. Then, we study how the total angular-momentum measurement can be realized by observing output signatures from a Bell multiport setup, either exactly for N=2,3, or approximately when the qubits are nearly pure for other N values. We also obtain numerical results that suggest that using a Bell multiport setup, one can distinguish between projection onto the j=N/2 and j=N/2-1 subspaces from their respective interference signatures at the output.

show abstract

Section: Discussionmentioning

confidence: 97%

Multiparameter estimation for qubit states with collective measurements: a case study

Len

2022

New J. Phys.

View full text Add to dashboard Cite

show abstract

“…Two permutation matrices that bring U Γ to P 1 U Γ P 2 = U Γ block , shown in Fig. D2, can be written as two vectors consisting of 36 integers, [6,2,36,24,13,29,22,10,32,27,1,17,31,26,3,19,23,9,18,5,12,33,28,16,11,34,25,15,20,7,14,30,8,4,35,21], 3,4,9,10,7,8,1,2,27,28,33,34,16,15,…”

Section: Appendix D Block Diagonal Form Of Ame and Bell Basesmentioning

confidence: 99%

“…Since the times of Euler [27] and Tarry [28] it is known that two orthogonal Latin squares of order d = 6 do not exist. Therefore, it was not known whether AME states for four subsystems do exist for d = 6 [29]. Our very recent result [30] can thus be considered as a quantum solution to a classically impossible problem -we constructed an explicit analytic example of a pair of quantum orthogonal Latin squares of size six.…”

Section: Introductionmentioning

confidence: 99%

9 $\times$ 4 = 6 $\times$ 6: Understanding the quantum solution to the Euler's problem of 36 officers

Życzkowski¹,

Bruzda²,

Rajchel-Mieldzioć³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

The famous combinatorial problem of Euler concerns an arrangement of 36 officers from six different regiments in a 6 × 6 square array. Each regiment consists of six officers each belonging to one of six ranks. The problem, originating from Saint Petersburg, requires that each row and each column of the array contains only one officer of a given rank and given regiment. Euler observed that such a configuration does not exist. In recent work, we constructed a solution to a quantum version of this problem assuming that the officers correspond to quantum states and can be entangled. In this paper, we explain the solution which is based on a partition of 36 officers into nine groups, each with four elements. The corresponding quantum states are locally equivalent to maximally entangled two-qubit states, hence each officer is entangled with at most three out of his 35 colleagues. The entire quantum combinatorial design involves 9 Bell bases in nine complementary 4-dimensional subspaces.

show abstract

“…In subsections III A and III B, we have shown that proving M satisfies Conjecture 1 is equivalent to proving N p ∼ N in (54) satisfies Conjecture 1. Further, it has been proved by Lemma 5 that N in (25) satisfies Conjecture 1 if k 1 = K in (27). So from ( 26) and ( 27), we assume that…”

Section: Proof Of Conjecturementioning

confidence: 99%

“…The PPT entanglement represents quantum resources which cannot be distillable into pure entangled states under local operations and classical communications (LOCC). What's more, bipartite non-PPT states of rank at most four turn out to be distillable [20][21][22], and some non-PPT states are conjectured to be non distillable [23][24][25][26][27]. On the other hand, the Schmidt rank is a basic parameter of characterizing bipartite pure states, and has been extended to multipartite pure states as an entanglement monotone [3,28].…”

Section: Introductionmentioning

confidence: 99%

A complete picture of the four-party linear inequalities in terms of the 0-entropy

Song,

Chen,

Sun

et al. 2021

Preprint

View full text Add to dashboard Cite

Multipartite quantum system is complex. Characterizing the relations among the three bipartite reduced density operators ρAB, ρAC and ρBC of a tripartite state ρABC has been an open problem in quantum information. One of such relations has been reduced by [Cadney et al, LAA. 452, 153, 2014] to a conjectured inequality in terms of matrix rank, namely r(ρAB) • r(ρAC ) ≥ r(ρBC ) for any ρABC . It is denoted as open problem 41 in the website "Open quantum problems-IQOQI Vienna". We prove the inequality, and thus establish a complete picture of the four-party linear inequalities in terms of the 0-entropy. Our proof is based on the construction of a novel canonical form of bipartite matrices under local equivalence. We apply our result to the marginal problem and the extension of inequalities in the multipartite systems, as well as the condition when the inequality is saturated.

show abstract

Five Open Problems in Quantum Information Theory

Cited by 45 publications

References 148 publications

Multiparameter estimation for qubit states with collective measurements: a case study

Multiparameter estimation for qubit states with collective measurements: a case study

9 $\times$ 4 = 6 $\times$ 6: Understanding the quantum solution to the Euler's problem of 36 officers

A complete picture of the four-party linear inequalities in terms of the 0-entropy

Contact Info

Product

Resources

About