CC BY
85
Discrete & Continuous Models
#& Applied Computational Science 2021, 29 (4) 361-386
ISSN 2658-7149 (online), 2658-4670 (print) http://journals-rudn-ru/miph
Research article
UDC 519.872:519.217
PACS 07.05.Tp, 02.60.Pn, 02.70.Bf
DOI: 10.22363/2658-4670-2021-29-4-361-386
Parameterizing qudit states
Arsen Khvedelidze1,2'3, Dimitar Mladenov4, Astghik Torosyan3
1 A. Razmadze Mathematical Institute Iv. Javakhishvili Tbilisi State University 1, Ilia Chavchavadze Avenue, Tbilisi, 0179, Georgia 2 Institute of Quantum Physics and Engineering Technologies Georgian Technical University 77, Kostava St., Tbilisi, 0175, Georgia 3 Meshcheryakov Laboratory of Information Technologies Joint Institute for Nuclear Research 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation
4 Faculty of Physics Sofia University St. Kliment Ohridski" 15, Tsar Osvoboditel Boulevard, Sofia, 1164, Bulgaria
(received: August 23, 2021; accepted: September 22, 2021)
Quantum systems with a finite number of states at all times have been a primary element of many physical models in nuclear and elementary particle physics, as well as in condensed matter physics. Today, however, due to a practical demand in the area of developing quantum technologies, a whole set of novel tasks for improving our understanding of the structure of finite-dimensional quantum systems has appeared.
In the present article we will concentrate on one aspect of such studies related to the problem of explicit parameterization of state space of an AT-level quantum system. More precisely, we will discuss the problem of a practical description of the unitary S'^(W)-invariant counterpart of the AT-level state space i.e., the unitary orbit space /SU(N). It will be demonstrated that the combination of well-known methods of the polynomial invariant theory and convex geometry provides useful parameterization for the elements of /SU(N). To illustrate the general situation, a detailed description of /SU(N) for low-level systems: qubit (N = 2), qutrit (N = 3), quatrit (N = 4) — will be given.
Key words and phrases: density matrix parameterization, quantum system, qubit, qutrit, quatrit, qudit, polynomial invariant theory, convex geometry
1. Introduction
Quantum mechanics is a unitary invariant probabilistic theory of finite-dimensional systems. Both basic features, the invariance and the randomness, strongly impose on the mathematical structure associated with the state
© Khvedelidze A., MladenovD., TorosyanA., 2021
This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/
space ty of a quantum system. In particular, the geometrical concept of the convexity of the state space originates from the physical assumption of an ignorance about the quantum states. Furthermore, the convex structure of the state space, according to the Wigner [1] and Kadison [2] theorems about quantum symmetry realization, leads to unitary or anti-unitary invariance of the probability measures (short exposition of the interplay between these two theorems see e.g. in [3]). In turn of the action of unitary/anti-unitary transformations q —> q' = UgU^ sets the equivalence relation q ^ q' between the states q,q' G ty and defines the factor space ty/U. This space is a fundamental object containing all physically relevant information about a quantum system. An efficacious way to describe ] •= /SU(N) for an A^-level quantum system is a primary motivation of the present article. The properties of 0[tyN], as a semi-algebraic variety, are reflected in the structure of the center of the enveloping algebra *d(su(N)). Hence, it is pertinently to describe 0[tyN] using the algebra of real SV(^)-invariant polynomials defined over the state space . Following this observation in a series of our previous publications [4]-[8], we develop description of 0[tyN] using the classical invariant theory [9].
It is worth noting that within this description of the state space the entanglement properties of binary composite systems can be analyzed as well. In [5], [6] qubit-qubit and qubit-qutrit pairs were studied from this standpoint. In particular, the optimal integrity basis for the polynomial SU(2) x SU(2) invariant ring of a two-qubit system was proposed and the separability criterion was formulated via polynomial inequalities in three SU(4) Casimir invariants and two determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices, which are the SU(2) x SU(2) polynomial scalars.
On the other hand, /SU(N) is related to the co-adjoint orbits space su* (N)/SU(N) and hence it is natural to describe /SU(N) directly in terms of non-polynomial variables — the spectrum of density matrices. Below we will describe a scheme which combines these points of view and provides description of the orbit space tyN/SU(N) in terms of one second order polynomial invariant, the Bloch radius of a state and additional non-polynomial invariants, the angles corresponding to the projections of a unit (N — 2)-dimensional vector on the weight vectors of the fundamental representation of SU(N).
The article is organised as follows. The next section is devoted to brief statements of general results about the state space of ^-dimensional quantum systems, including discussion of its convexity (Section 2.1) and semi-algebraic structure (Section 2.2). Particularly, the set of polynomial inequalities in an (N2 — 1)-dimensional Bloch vector and the equivalent set of inequalities in N — 1 polynomial SU(^)-invariants will be presented for arbitrary A^-level quantum systems. Section 3 contains information on the orbit space ] — the factor space of the state space under equivalence relation against the unitary group adjoint action. In Section 3.3.1 we introduce a new type of parameterization of a qubit, a qutrit and a quatrit based on the representation of the orbit space of a qudit as a spherical polyhedron on SN_2. This parameterization allows us to give a simple formulation of the conception of the hierarchy of subsystems inside one another. In Section 3.3.2 we present
formal elements of the suggested scheme for an arbitrary final-dimensional system. Section 4 contains a few remarks on possible applications of the introduced version of the qudit parameterization.
2. The state space
The state space of a quantum system comes in many faces. One can discuss its mathematical structure from several points of view: as a topological set, as a measurable space, as a convex body, as a Riemannian manifold.1 Below we concentrate mainly on a brief description of as a convex body
realized as a semi-algebraic variety in Rn 2-1 following in general the publications [4]-[8].
2.1. The state space as a convex body
According to the Hilbert space formulation of the quantum theory, a possible state of a quantum system is associated to a self-adjoint, positive semi-definite "density operator" acting on a Hilbert space. Considering a non-relativistic ^-dimensional system whose Hilbert space ft is CN, the density operator can be identified with the Hermitian, unit trace, positive semi-definite N x N density matrix [14], [15].
The set of all possible density matrices forms the state space of an ^-dimensional quantum system. It is a subset of the space of complex N x N matrices:
= {gGMN(C)\g = gl g^0, Tr^=1}.
A generic non-minimal rank matrix g describes the mixed state, while the singular matrices with rank (g) = 1 are associated to pure states. Since the set of N-th order Hermitian matrices has a real dimension N2, and due to the finite trace condition Tr (g) = 1, the dimension of the state space is dim(^w) = N2 — 1. The semi-positivity condition g > 0 restricts it further to a certain (N2 — 1)-dimensional convex body. The convexity of is the fundamental property of the state space. The next propositions summarize results on a general pattern of the state space as a convex set with an interior Int ) and a boundary [10].
Proposition 1. Given two states g1, g2 G Int(^w) and a "probability" p G [0,1], consider the convex combination
QP ■ = (1—P)Qi +P82,
then gp G Int ).
Proposition 2. The boundary consists of non-invertible matrices of all possible non-maximal ranks:
= {gG^N \ det(e) = 0}.
xHere is a short and extremely subjective list of publications on these issues [10]-[13].
The subset of pure states C = [g G | rank(^) = 1},
contains N extreme boundary points Pi (g) which generate the whole by taking the convex combination:
N N
Q = ^rip(Q), = 1, ri (1)
i=0 i=0
In (1) every extreme component Pi(g) can be related to the standard rank-one projector by a common unitary transformation U G SU(N) and transposition Pi(1) interchanging the first and i-th position:
Pz(g) = UPz(1) diag(1,°,...,°)PiW U t.
For any dimension of the quantum system the subset of extreme states provides important information about the properties of all possible states, even the pure states comprise a manifold of a real dimension dim(£w) = 2N — 2, smaller than that dimension of the whole state space boundary dim(<9^w ) = N2 — 2.
2.2. The state space as a semi-algebraic variety
According to the decomposition (1), the neighbourhood of a generic point of (RN2-1) is locally homeomorphic to (U(N)/U(1)n) x DN-1, where the component DN-1 is an (N — 1)-dimensional disc (cf. [10], [13]). Following this result, below we will describe how the state space can be realized
as a convex body in RN2-1 defined via a finite set of polynomial inequalities involving the Bloch vector of a state. In order to formalize the description of the state space, we consider the universal enveloping algebra K(su(N)) of the Lie algebra su(N). Choosing the orthonormal basis X1, X2,..., XN2-1 for
su(N),
N2-1
su(N) = Y, ZiK, (2)
i=1
the density matrix will be identified with the element from K(su(N)) of the form:
i IN — 1 n2-1
= + (3)
i=1
The analysis (see e.g. consideration in [4], [6]) shows the possibility of description of the state space via polynomial constraints on the Bloch vector of an A^-level quantum system.
Proposition 3. If a real (N2 — 1)-dimensional vector f = (^,... ,^N2-1) in (3) satisfies the following set of polynomial inequalities:
Sk k = 1,2,...N,
(4)
where Sk (£) are coefficients of the characteristic equation of the density matrix q:
det \\x — — xN — S1 xN-1 + S2xN-2 — ■■■ + (—1)N SN — 0, (5) then the equation (3) defines the states q G .
The inequalities (4), which guarantee the semi-positivity of the density matrix, remain unaffected by unitary changes of the basis of the Lie algebra and thus the semi-algebraic set (4) can be equivalently rewritten in terms of the elements of the SV(^)-invariant polynomial ring [R[^W]SU(N). This ring can be equivalently represented by the integrity basis in the form of homogeneous polynomials P — (t1 ,t2,... ,tN),
,s2,..-,sn 2-1 ]SU(N) = ^[h ,..-,tN].
The useful, from a computational point of view, polynomial basis P is given by the trace invariants of the density matrix:
tk ■ — Tr(gk). (6)
The coefficients Sk, being S^^-invariant polynomial functions of the density matrix elements, are expressible in terms of the trace invariants via the well-known determinant formulae:
0 ^
1
k — 1
tk-1 tk-2 ■ t1 /
Aiming at more economic description of , we pass from N2 — 1 Bloch variables to N — 1 independent trace variables tk. Pay for such a simplification is necessity to take into account additional constraints on tk which reflect the Hermicity of the density matrix. Below we give the explicit form of these constraints in terms of P = (t1 ,t2,... ,tN).
In accordance with the classical results, the Bezoutian, the matrix B = ATA, constructed from the Vandermonde matrix A, accommodates information on the number of distinct roots (via its rank), numbers of real roots (via its signature), as well as the Hermicity condition. A real characteristic polynomial has all its roots real and distinct if and only if the Bezoutian is positive definite. For generic invertible density matrices — matrices with all eigenvalues different, the positivity of the Bezoutian reduces to the requirement
det \\B\\ > 0. (7)
Noting that the entries of the Bezoutian are simply the trace invariants:
Sl. —
1
h 1 0
h h 2
det h t2 h
Bij — ti+j-2,
(8)
one can get convinced that the determinant of the Bezoutian is nothing else than the discriminant of the characteristic equation of the density matrix,
Disc = ni>:/ (ri — rj)2, rewritten in terms of the trace polynomials2
Disc(i1 ,t2,...,tN) •= det ||B||. (9)
Hence, we arrive at the following result.
Proposition 4. The following set of inequalities in terms of the trace SU (^)-invariants,
Disc (t1 ,t2 ,...,tN )>°, Sk (t1 ,t2 ,...,tN)^°, t1 = 1, (10)
define the same semi-algebraic variety as the inequalities (4) in N2 — 1 Bloch coordinates do.
3. Orbit space ty^/SU(N)
3.1. Parameterizing tyN/SU(N) via polynomial invariants
Proposition 4 is a useful starting point for establishing a stratification of the under the adjoint action of the SU(N) group. It turns out that, due to the unitary invariant character of the inequalities (10), they accommodate all nontrivial information on possible strata of unitary orbits on the state space . Indeed, it is easy to find the link between the description of given in the previous section and the well-known method developed by Abud-Sartori-Procesi-Schwarz (ASPS) for construction of the orbit space of compact Lie group [16]-[18]. The basic ingredients of this approach can be very shortly formulated as follows.
Consider a compact Lie group G acting linearly on a real d-dimensional vector space V. Let R[V]G be the corresponding ring of the G-invariant polynomials on V. Assume P = (t1 ,t2,... ,tq) is a set of homogeneous polynomials that form the integrity basis, R[|1,|2,... ,Cd]G = R^,t2, . ,tq]. Elements of the integrity basis define the polynomial mapping:
t• V^Ri; (£1 ,$2,..-,Sd)^(t1 ,.,tq). (11)
Since the map t is constant on the orbits of G, it induces a homeomorphism of the orbit space V/G and the image X of ¿-mapping; V/G ^ X [19]. In order to describe X in terms of P uniquely, it is necessary to take into account the syzygy ideal of P, i.e.,
= {hGR [V1 ,V2,...,yq] • h(p1 ,P2,..-,Pq) = °, in R[V]}.
Let Z C Rq denote the locus of common zeros of all elements of 1then Z is an algebraic subset of Rq such that X C Z. Denoting by R[Z] the restriction
2The dependence of the discriminant on trace invariants only up to order N pointed in the left side of (9) assumes that all higher trace invariants tk with k > N in (9) are expressed via polynomials in t1 ,t2, ■■■, tN (the Cayley-Hamilton Theorem).
of R[y1,y2,..., y ] to Z, one can easily verify that R[Z] is isomorphic to the quotient [[^ ,y2,..., yq]/Ip and thus R[Z] ^ R[V]G. Therefore, the subset Z essentially is determined by R[V]G, but to describe X the further steps are required. According to [17], [18], the necessary information on X is encoded in the structure of the q x q matrix with elements given by the inner products of gradients, grad(i^) ■
WGrad^ — (grad (t,), grad (t) . (12)
Hence, applying the ASPS method to the construction of the orbit space /SU(N), one can prove the following proposition.
Proposition 5. The orbit space /SU(N) can be identified with the semi-algebraic variety, defined as points satisfying two conditions:
a) The integrity basis for SU(^)-invariant ring contains only N independent polynomials, i.e., the syzygy ideal is trivial and the integrity basis elements of R[^W]SU(W) are subject to only semi-positivity inequalities
,t2, . ) ^ 0;
b) ASPS inequality Grad(z) ^ 0 is equivalent to the semi-positivity of the Bezoutian, provided by existence of the d-tuple where x — (1,2,... ,d) ■
Grad (t1 ,t2,.,td) = XB (t1, tr2 ,.,td)XT. (13)
3.2. /SU(N) — as a AN-1 -simplex of eigenvalues
The decomposition of the density matrix (1) over the extreme states explicitly displays the equivalence relation between states,
SU(N)
Q ^ q' if g' — UqUt, UGSU(N).
Matrices with the same spectrum are unitary equivalent. Furthermore, since the eigenvalues of the density matrix r — (r1 ,r2,..., rN) in (1) can be always disposed in a decreasing order, the orbit space /SU(N) can be identified with the following ordered (N — 1)-simplex:
rz — 1,1^rx >r2 ^0 } . (14)
3.3. /SU(N) — as a spherical polyhedron on SN-2
We are now ready to combine the above stated methods of the description of the state space , the polynomial invariant theory and convex geometry for writing down certain parameterization of density matrices. Based on the extreme decomposition of states (1), the parameterization of the elements of reduces to fixing the coordinates on the flag manifolds of SU(N) and the simplex AN of eigenvalues of density matrices. In the remaining
AN-1 — {r G R
N
N £
i=1
part of the article, we will describe tyn/SU(N) in terms of the second order polynomial invariant, which is determined uniquely by the Euclidean length r of the Bloch vector, and N — 2 angles on the sphere SN_2, whose radius in
its turn is given as J —jj1 r.
3.3.1. Qubit, qutrit and quatrit
In order to demonstrate the main idea of the parameterization, we start with its exemplification by considering three the lowest-level systems, qubit, qutrit and quatrit and afterwards the general case of an A^-level system will be briefly outlined.
Qubit. A two-level system, the qubit, is described by a three-dimensional Bloch vector \ = fa,^,£3}:
Q(2) = \(h + 6 ^). (15)
The qubit state with the spectrum r = [r1, r2} G A1 is characterized by the only one independent second order SU(2)-invariant polynomial t2 = rf + . Introducing the length of the qubit Bloch vector, r = , we see
that3 1 2 3
Hence, the eigenvalues of the qubit density matrix (15) can be parameterized as
ri = 2 + r»i. (16)
It will be explained later that the coincidence of the constants ^1 = 1/2 and ^2 = —1/2 in (16) with the standard weights of the fundamental SU(2) representation, when the diagonal Pauli matrix a3 is used for the Cartan element of su(2) algebra, is not accidental. Below we will give a generalization of (16) for the qudit, an arbitrary A^-level system. With this aim it is sapiential to start with considering the N = 3 and N = 4 cases.
Qutrit. We assume that a generic qutrit state (N = 3) has the spectrum r = [r1 ,r2,r3} from the simplex A2 and thus is an eight-dimensional object. According to the normalization chosen in (3), it is characterized by the
8-dimensional Bloch vector f = fa,£2,... ,£g),
1 1 8
Q(3) = ^3 + -73^^ \. (17)
3The semi-positivity of state (15) dictates the constraint, — 1/2(1 - t2) > 0, which
restricts the value of the Bloch vector length: 0 ^ r ^ 1.
A qutrit has two independent SU(3) trace invariant polynomials, the first
one, t2 = rl + r"2 + r2, is expressible via the Euclidean length of the Bloch
8
vector, r2 = Y 0,
i=1
12 = l + 3r2, (18)
and the third order polynomial invariant, t3 = rf + r'3 + , which rewritten in terms of eight components of the Bloch vectors reads:
h = l + lr2 + /fii & & + ) +
+ ^2 — ) + //3 + — — ) +
+ (6 ($2 + & + $) - 3 (a + n + % + #) - 2d). (19)
Now we want to rewrite (19) in terms of the Bloch vector of a length r and an additional SU(3) invariant. Having this in mind, it is convenient to pass to new coordinates linked to the structure of the Cartan subalgebra of su(3). Choosing the latter as the span of the diagonal SU(3) Gell-Mann matrices and noting that the state (17) is S^(3)-equivalent to the diagonal state:
SU(3) 1 1
e(3) * ^3 + /J3)(J3 ^3 + ?8 \), (20)
one can consider two coordinates (J3, J8) in the Cartan subalgebra of su(3) as independent coordinates in ty3/SU(3). Taking into account that for the given values of the second trace invariant (18) the coefficients obey relation J2 + J2 = r2, we pass to the polar coordinates on the (J3, J8)-plane,
J3 = r cos (I), J8 =r sin (|). (21)
In terms of new variables (r, <p) the expression (19) for the SU(3)-polynomial invariant t3 simplifies,
h = 9 + lr2 + \r3 sin <P, (22)
and the image of the ordered simplex A2 in (J3, J8)-plane under the mapping (21) is given by the triangle AABC:
A2 ^ {°<J3 <J8
depicted in the figure 1.
1 m V-
2
•'1 V3
Figure 1. The image of the ordered simplex A2 in (J3, J8)-plane under the mapping (21)
In the figure 1 the A2-simplex of the qutrit eigenvalues is mapped to the triangle AABC inscribed in a unit-radius circle J3 + Jg — 1. Its inner part AABC comprises the points of the maximal rank-3 states 3 with 1 > r1 > r2 > r3 >0. All these points generate the regular SU(3) orbits 0123 of dimension dim(0123) = 6. The points on the line AB also generate regular orbits 0123, however the corresponding states have rank(^) = 2. In contrast to the above case, the line AC/{A} and line BC/{B} correspond to the subspace of ^33, but now the eigenvalues of the states are degenerate, either r1 = r2 > r3, or r1 > r2 = r3, hence representing the degenerate orbits 01\23 and 0 12\3, respectively. The dimensions of both types of orbits are the same, dim(0 1\23) — dim(012\3) — 4. Finally, the single point C(0,0) represents a maximally mixed state which belongs also to the set of rank-3 states.
The polar form of the invariants (21) prompts us to introduce a unit 2-vector n — (cos (p/3), sin (p/3)) and represent the qutrit eigenvalues as
1 2 ^ ^ r% = 3+v3r^% 'n,
with the aid of the weights of the fundamental SU(3) representation:
* = fei^ 112 = (-l1,2^73), — (°,-jil- (24)
Gathering all together, we convinced that the representation (23) is nothing else than the well-known trigonometric form of the roots of the 3-rd order characteristic equation of the qutrit density matrix:
1 2 . fp + 4n\ 1 2 . fi —---r sin ( - I , r0 =---r sin
1 - — - — I sin ( - ) , / o - — - — I sin ( - I ,
1 33 \ 3 J , 2 3 3 v 3 J
1 2 f^
r3 —---r sin ( —
3 3 3 V3
It is in order to present a 3-dimensional geometric picture associated to the parameterization (23). The three drawings in the figure 2 with different values of r show that (23) are parametric form of the arc of the red circle which is the intersection A2 f Si (^2/3r).
The picture in the figure 2 illustrates a geometrical meaning of the parameterization of qutrit eigenvalues (25) in terms of the Bloch radius r and the angle p G [0,^]. Consider an intersection of a qutrit simplex A2 with 2-sphere r\ + r2 + r^ = 3/3 + (2/3)r2. The intersection depends on a value of a qutrit Bloch vector. For r = 0 the sphere and the simplex A2 intersect at point C = (3/3, 3/3, 3/3), while for 0 < r < 3 the intersection is an arc Gr of a circle on the plane r1 + r2 + r3 = 3 of the radius s/2/3r centered at the point C(3/3, 3/3, 3/3). The intersection for r = 3 takes place at B(3,0,0). The ordering of eigenvalues 3 ^ ri ^ r2 ^ T3 ^ 0 determines the length of the arc Gr. For any r, the arc Gr is described by (25), the depicted curve in the figure corresponds to the fixed value r = 3/4. Furthermore, varying r within the interval r G [0, 3], provides the slices covering the whole simplex A2 = [0,n]xer.
Figure 2. The geometrical meaning of the parameterization of qutrit eigenvalues (25) in terms of the Bloch radius r and the angle ^ E [0, w]
Qutrit Boundary. The introduced parameterization is very useful for analyzing the structure of a qutrit boundary states. The qutrit space admits decomposition
=y3>3 uy3>2 (26)
into 8d-component of maximal rank-3, 7d-component of rank-2 and extreme pure states. Every component of (26) can be associated with the corresponding domains in the orbit space dO[ty3]. Particularly, the boundary dO[$3] consists of two components and is described as follows:
— Qubit inside Qutrit. For a chosen decreasing order of the qutrit eigenvalues, r\ ^ v2 ^ r3, the rank-2 states belong to the edge A3, given by equation ^ = 0, which in the parameterization (25) reads:
rank-2 states : {r — ^ gin^3^ for p £ [0, (27)
Considering (27) as a polar equation for a plane curve, we find that the rank-2 states 03 2 can be associated to the part of a 3-order plane curve. Indeed, rewriting (27) in Cartesian coordinates x — r cos p, y — r sin p,
(x2 + y2)(y-3a) + 4a3 — 0,
we identify this curve with the famous Maclaurin trisectrix with a special choice of a — 1/2. For the boundary states (27), the equations (25) reduce to
— 2(1 + r2c3), ^2 — \(1-r*2c3), (28)
where
^ — (29)
These expressions for non-vanishing eigenvalues of a qutrit indicate the existence of a "qubit inside qutrit" whose effective radius is r2c3. Since the radius of the Bloch vector of rank-2 states associated to a qubit in qutrit lies in the interval 1/2 < r < 1, the length of its Bloch vector, r2c3, takes the same values as a single isolated qubit, 0 < r2c3 < 1. Orbit space of pure states of qutrit. The boundary dO[#3 1 ] corresponding to all pure states 1 is attainable by SU(3) transformation from the point, r — 1 for p — n.
Quatrit. Now, following the qutrit case, consider a 4-level system, the qua-trit, whose mixed state is described by the Bloch vector ^ — {^, £2,... ,£15},
1 3 15
The integrity basis for a quatrit ring of SU(4)-invariant polynomials R[^1,...,^15]su(4) consists of three polynomials \R[t2,t3,t4]. Using the compact notations (see details in Appendix 5.1), they can be represented in terms of the Casimir invariants of su(4) algebra in the following form:
13 2 1 9 2 3 - - -
4 4 16 16 16
1 9 2 9 4 ( )
^ — 64 + 32 r2 + 16 ^ 64r +
From the expressions (30) one can see that apart from the length r of the Bloch vector, there are two independent parameters required to unambiguously
characterize the quatrit eigenvalues. To find them, let us proceed as in the qutrit case. Consider the diagonal form corresponding to a quatrit state:
SU(4) - 3
e(4) - + 2^6{J3x3 + J8\ + K*)■ (31)
The coefficients J3 ,J8 and J15 in (31) are invariants under the adjoint SU(4) transformations of q. By equivalence relation (31), the quatrit state space is projected to the following convex body:
The 2-dimensional slice J15 = 1/3 of this body corresponds to rank-3 states, see the figure 3. In terms of new invariants, all states with a given length of Bloch vector r belong to a 2-sphere: J" + J" + J\5 = r2. Hence, the corresponding spherical angles p and 9 of these invariants,
m m
J3 =r sin 9 cos —, J8 =r sin 9 sin—, J15 = r cos 9, (33)
33
can be used as two additional parameters needed for the parameterization of a quatrit eigenvalues.
2 VT © Jl5=1/3
3
/ \Î2 L
3
IT
V 3
Figure 3. Slice of the convex body (32) as a result of cutting by the plane J15 = 1/3
Let us now, in accordance with (33), introduce the unit 3-vector n = (sin 9 cos(p/3), sin 9 sin(^/3), cos 9) and parameterize 4-tuple of the eigenvalues of the density matrix r = (r1 ,r2, r3, r4) via the following projections:
ri = \+^3r™^V'г, (34)
where 3-vectors ,fi,3 and denote the weights of the fundamental
SU(4). Explicitly the weights read:
11 1 A (11 1
(37)
^ *2 V 2' 2\f3 2\[6/
_ ( 1 1\ ^ ( 3 ^ (35)
= {0,-7=з,2v=6), 114 = {0,0,-2vE
Note that the weights are normalised in a way leading to a unit norm of the simple roots of algebra su(4) and obey relations:
4 4 1
= 0 and ,4 = -5^. (36)
1=1 i=i 2
Using these expressions, we arrive at the following parameterization of a quatrit eigenvalues:
1 1 (. a . 4ft 1
r\ =---( sin 0 sin---—
1 4 V 3 2V2
1 1 (■ a ■ V + 2* 1
r2 =---^r ( sin 0 sin---—
2 4 V 3 2\[2
1 1 ( ■ a • V 1
r3 =---^r ( sin 0 sin---= coi
3 4 v^ v 3 2^2 1 3
r4 =---r cos V.
4 4 4
To ensure the chosen ordering of the eigenvalues ri G A3, the Bloch radius should vary in the interval r G [0,1] and angles d be defined over the domains:
ft & ft 1 (,
6<1<2, cot sin ^ (38)
A geometric interpretation of (37), in full analogy with the qutrit case, is described in figure 4.
In the figure 4 the 3-sphere ^4 rf = 1/4+(3/4)r2 intersects the hyperplane
^4 ri = 1 in the positive quadrant. The intersection occurs iff 1/4 <
1/4 + (3/4)r2 < 1, and represents the 2-sphere S2(^ r) centered at the point D = (1/4,1/4,1/4,1/4). The intersection with the ordered simplex A3 is given by a spherical polyhedron with 3 or 4 vertices, depending on the Bloch radius r.
The boundary of a quatrit orbit space dO[$4] can be decomposed into 2d-component of rank-3, ld-component of rank-2 and extreme zero-dimensional component of rank-1, corresponding to pure states:
00^4] = d(D№«3] U d0[^42] u dom
Figure 4. A geometric illustration of (37)
Qutrit inside Quatrit. The boundary component ^[^43] of rank-3 states is determined by the intersection of 3D simplex A3 with the hyperplane:
r4 =0.
(39)
Parameterizing quatrit eigenvalues in terms of angles, the solution to the equation (39) is
cos Q = —, if rG 3r
1 ' 3-1
(40)
Hence, the parametric form of the 2-dimensional surface 3] is given in terms of the remaining three non-vanishing eigenvalues:
(41)
rn =---= r(r) sin I-) , r9 =---= f(r) sin ( -)
1 3 V2 ) v 3 ) , 2 3 y/2JK ) \ 3 )
1 1 f/ \ •
f3 =---= r(r) sin ( — ) ,
3 3 s/2 \?J)
where /(r) = yV2 — 1.
Consequences of the above derived formulae deserve few comments.
1. According to the formula (41) for the eigenvalues of boundary rank-3 states, their expressions are similar to the qutrit eigenvalues given in (25). This observation prompts us to introduce the conception of the "effective qutrit inside quatritwhose Bloch radius value is determined by the Bloch radius of a quatrit:
tytT -
r 3c4 —
2V2
f2 9.
Note that since the admissible range of the Bloch radius of rank-3 quatrit states is r G [1/3,1], then the effective radius r3c4 takes values
in the interval 0 ^ rjjC < 1
2. The idea to identify qutrit inside quatrit is based on the establishing correspondence on the level of orbit spaces 3 and 3. The generic qutrit state in (26) is 8-dimensional, while dim(^4 3) = 14. Thus, one can speak about the correspondence between quatrit rank-3 states and qutrit states only modulo unitary transformations.
3. In favour of the idea considering "effective qutrit inside quatrit" is a relation between the polynomial invariants for states on bulk and boundary. Particularly, using expressions for trace polynomials given in Appendix 5.2, we get:
Qubit inside Qutrit inside Quatrit. In A3 the rank-2 boundary component 0[$4 2] is comprised from points on a line given by its intersection with two hypersurfaces:
r4 = 0, r:i = 0.
Following in complete analogy with the rank-3 states, we arrive at "ma-tryoshka" structure with "effective qubit inside qutrit which in turn is inside quatrit". The Bloch radius of this effective qubit is given by the Bloch radius of a quatrit:
Note that for rank-2 states r G [1/\f3,1] and hence 0 < r^2c3c4 < 1.
Finally, the rank-1 boundary component 0[$4 1] is generated by one point r = (1, 0, 0, 0) which represents all pure states in A3.
3.3.2. Generalization to A^-level system
Now after examining main features of the introduced parameterization for a qutrit and quatrit, we are ready to give a straightforward generalization to the case of an arbitrary A^-level system. With this aim, we will use the Cartan subalgebra of SU(N) as span of the following diagonal N x N Gell-Mann matrices:
H-i = diag (1,-1,0,^,0), H2 = -j= diag (1,1,-2,^,0),
k times
Hk = diag i1,1,-,1,-4,0,-,0!,
'(N-1) times
=^N;N_1)diagi ïx—,-^-1
The corresponding weights of the fundamental SV(N) representation are 11 1 1
=
^2 =
2' 2V3'"" //2fc(fc+1)'"'' //2^(^-1)/ '
11 1 1 \ '2'2V3'^' //2fc(fc + 1), "', - 1) J '
2 1 1 ^ = (0, -273' , +1), "', //2N(w -1)J '
'(fc-2) times
^fc = I 0,0, ... ,0 , -
fc -1 1 1
2fc '"''+1)'"''/2^(^-1) / '
(W-2) times
= I 0,0,...,0 ,...,-
N - 1 2N
It is easy to verify that the following relations are true:
N N
a P
II, II.. =
=0, and ^f ^^.
¿=i ¿=i
Taking into account these observations, one can write down the following parameterization for the roots r of the Hermitian N x N matrix:
1 /2(^-1) ^ ^ r, = (42)
where n G Sw_2 (1) and parameter r provides the fulfillment of the correspondence with a value of the second order invariant,
1 N- 1 2
Writing the traceless part of the density matrix as the expansion over the Cartan subalgebra H of su(N),
1 Stf(W) /(^-1)
" VHF"1,
AE^
we see that N - 2 angles of the unit norm vector n (42) are related to
2,
the invariants , ^, ... , , whose values are constrained by the Bloch
radius r :
N
= r2. (43)
s=2
Finally, it is worth to give the geometric arguments which are emphasizing the introduced parameterization (42) of qudit eigenvalues. With this goal consider the intersection SN-1 (R) f of (N — 1)-sphere of radius R
and hyperplane T,N-1 ■ ^^ ri = 1 in RN. Let us describe the hyperplane in parametric form, with parameters ^,s2, ■■■, sN-1:
r = d + e(1)s- + e(2) S2 + - + e(N-1) sN--, (44)
where ^-vector d fixes the point P G and the basis vectors (Darboux
frame) obey conditions:
d • e(a) =0, e(a) • e^ = , a, ft = 1, 2,..., N — 1.
Using this parameterization, the equation for (N — 1)-sphere reduces to the constraint
d2 + s2 + s2 + - + s2N-1 = R2 for all points of intersection SN-1 (R) f T.N-1. Hence, the intersection is
nothing else as the (N — 2)-sphere of radius R^-2 = \/R2 — d2 centered at a point associated to the vector d G -1. Now if we fix the point P such that d = (1/N,... ,1/N), express the parameters in (44) in terms of the Bloch radius and the components of the unit vector by relation sa = ^2(N — 1)/Nrna and define the frame vectors e(a), so that4
e^ = , i = 1,2,..., N, while a = 1,2,..., N — 1,
we arrive at the representation (42) with the radius of intersection sphere RN-2 = ^(N — 1)/Nr.
Passing from hyperplane -1 to its subset, the simplex AN-1, we note that SN-1 (R) f AN-1 will be determined uniquely for every chosen order of the eigenvalues and the value of r. For an arbitrary N, a special analysis is required to write down explicitly SN-1 (R) f AN-1. Here we only note that the intersection is given by one out of all possible tillings of SN-2 by the spherical polyhedra. For N = 3 such polyhedron degenerates to an arc of a circle, whereas for N = 4 the intersection will be given by two types of polyhedra, either a spherical triangle, or a spherical quadrilateral, depending on the value of the Bloch radius r.
4. Concluding remarks
Since the introduction of the concept of mixed quantum states, the problem of an efficient parameterization of density matrices in terms of independent variables became one of the important tasks of numerous studies. Starting with the famous Bloch vector parameterization [20], several alternative types of "coordinates" for points of quantum states have been suggested [21]-[30]. According to the generalization of Bloch vector parameterization, initially introduced for a 2-level system, the Bloch vector for an A^-level system is a real
4Here a component of i-th weights determines i-th component of basis vector e(a).
(^2 — 1)-dimensional vector. However, owing to the unitary symmetry of an isolated quantum system, those — 1 parameters can be divided into two special subsets. The first subset is given by N—1 unitary invariant parameters, and the second one is compiled from the coordinates on a certain flag manifold constructed from the SV(N) group. Introduction of the coordinates on both subsets has a long history. A description of the former set of SV(^)-invariant parameters is related to the classical problem of determination of roots of a polynomial equation, while the latter corresponds to a description of the homogeneous spaces of SV(^) group 5.
In the present article we have discussed the first part of the problem of parameterization of N x N density matrices and proposed a general form of parameterization of A^-tuple of its eigenvalues in terms of a length r of the Bloch vector and N — 2 angles on sphere
-2(V(^— 1)/^>). We expect that this parameterization will be useful from a computational point of view in many physical applications including the models of elementary particles. Particularly, in forthcoming publications it will be used for the evaluation of very recently introduced indicators of quantumness/classicality of quantum states which are based on the potential of the Wigner quasidistributions to attain negative values [35]-[37].
5. Appendix 5.1. Constructing Casimir invariants for 5u(^) algebra
In this Appendix we collect few notions and formulae explaining the construction of the polynomial Casimir invariants on the Lie algebra g = su(^) of the group G =
Consider algebra g = ^^ -1 ^A^, spanned by the orthonormal basis (A^} with the multiplication rule
2
= + + ^Zijfc) ^fc, (45)
defined via the symmetric and anti-symmetric structure constants. Let (uj1} be the dual basis in i.e., uj%(A^) = , and introduce the G-invariant symmetric tensor S of order r:
# = UJZ1 . (46)
The G-invariance of tensor S means that
r
^ fm S.....=0 (47)
s=1
5 Among the important contributions to the problem of parameterizing we would
like to mention the following publications that influenced the present work: [31]-[34].
Using the tensor S, one can construct the elements of the enveloping algebra U(o):
cr = ^v-v\K2. , (48)
which turns to belong to the center of U(g), i.e., [Cr, Xi] = 0, for all generators Xi. Having in mind the solution to the invariance equations (47), one can build the polynomials in N2 — 1 real variables £ = (C1,C2, — Cn2-1 ) :
^r ) ^ y — ,
i
which are invariant under the adjoint SU(^)-transformations:
p(Adg (O)-P(t).
It can be proved that the symmetric tensors k(r) defined in the given basis of algebra as kiir\ i = Tr (A^ Xi — Xi }), satisfy invariance equation (47)
'1'2'"V V {i1 ^2"' V} /
and form the basis for the polynomial ring of G-invariants. The tensors kir) admit decomposition with the aid of the lowest symmetric invariants tensors, and dijk. Particularly, the following combinations are valid candidates for the basis:
h(4) — rl rl
lb A A A A Hi f A A \ Q Hi
i1i2i3iA u{iii2}su{i3i4}s,
U(5) = A f] f]
ili2i'ii4i5 U,{iii2}SU,si3tU,{i4i5}t,
U(6) = r] r] r] r]
rviii2i3i4i5i6 U-{iii2}^si3tU-t,i4,UU-{i5i6}U.
As an example, for A^-level system the G-invariant polynomials up to order six read:
£2 = (N — 1)f, £3 = (N — 1)i^vi £4 = (N — 1)tvbivi (49)
£5 = (N—1)tV£V£V^i
£6 = (N — 1)(ivivi)2.
In the equation (49) the Casimir invariants are represented in a dense vectorial notation using the auxiliary (N2 — 1)-dimensional vector defined via the symmetrical structure constants dirjk of the algebra su(N):
(m )k ■— \IN(N0 l) dl3k Uj.
5.2. Polynomial SU(N)-invariants on ^
N
In this section the explicit formulae for polynomial invariants for quatrit will be given in terms of the suggested parameterization of density matrices.
Since the traceless part of the density matrices, q — ^IN = jJ8, belongs
to the algebra su(N), all trace polynomials can be expanded over the su(W) Casimir invariants. The corresponding decomposition of independent polynomials for the quatrit (^ = 4) read:
= 4(! + 3^2 ),
is = 42 (1 + 3^2 + ^3 ),
Î4 = 413 (1+6^2 + 4£s + ^2 + ^4).
In order to derive the explicit form of polynomials £2 and £3, the knowledge of components of the symmetric structure tensor d is needed. It is convenient at first to express the invariants for diagonal states, characterized by 73, 78 and J15, and afterwards rewrite them for generic states using parameterization (33). With this aim, we collect (up to permutations) in the table 1 all non-zero coefficients for the Cartan subalgebra of su(3) and su(4).
Table 1
Symmetric structure constants for the Cartan subalgebra of su(3) and su(4)
i.j.k 3.3.8 3.3.15 8.8.8 8.8.15 15.15.15
^SU(4) 1 V3 1 V6 1 V3 1 V6
WSU(3) 1 V3 1
Taking into account the values for structure constant d from the table 1, the Casimir invariants of the third and fourth order of a quatrit read:
£3 = 9^5 (.72 + 78) + 9^2 (72 - 1 78) - 6 , (50)
£4 = 9 (72 + 72) 2 + 36^2 .8 7i5 (73 - 1 72) + 12 745. (51)
Finally, plugging expressions (33) into (50) and (51), we arrive at the representation of the su(4) Casimir invariants in terms of quatrit Bloch radius r and two angles (0,
£3 = 3 r3 [W2 sin3 (0) sin(p) - 3 cos(0) - 5 cos(30)] ,
£4 = 3r4 [3^v/2 sin3(0) cos(0) sin(p) + 4 cos(20) + 7 cos(40) + 2l] , 8 L J
as well as directly for the trace polynomial invariants,
=
1 3 2
2 4 4
19 3
— + —r2 + —r3 (4^ sin3 0 sin -3 cos 0-5 cos(30)) ,
1 fi 1 fi nd \ '
1
i4 = — + —r2 + —r3 (W2 sin3 0 sin W - 3 cos 0-5 cos(30)) + 4 64 32 64 v ^
+ ^r4 (3^v/2sin3 0 cos 0sin ^ + 4 cos(20) + 7 cos(40) + 45)
Acknowledgments
The work is supported in part by the Bulgaria-JINR Program of Collaboration. One of the authors (AK) acknowledges the financial support of the Shota Rustaveli National Science Foundation of Georgia, Grant FR-19-034. DM has been supported in part by the Bulgarian National Science Fund research grant DN 18/3.
References
[1] E. P. Wigner, Group theory. New York: Academic Press, 1959.
[2] R. V. Kadison, "Transformation of states in operator theory and dynamics," in Topology, ser. 2. 1965, vol. 3, pp. 177-198.
[3] W. Hunziker, "A note on symmetry operations in quantum mechanics," Helvetica Physica Acta, vol. 45, pp. 233-236, 1972. DOI: 10.5169/seals-114380.
[4] V. Gerdt, A. Khvedelidze, and Y. Palii, "On the ring of local polynomial invariants for a pair of entangled qubits," Journal of Mathematical Sciences, vol. 168, pp. 368-378, 2010. DOI: 10.1007/s10958-010-9988-8.
[5] V. Gerdt, A. Khvedelidze, and Y. Palii, "Constraints on SW(2) x 5^(2) invariant polynomials for a pair of entangled qubits," Adernaa fizika, vol. 74, pp. 919-925, 2011.
[6] V. Gerdt, D. Mladenov, A. Khvedelidze, and Y. Palii, Casimir invariants and SV(2) ® 5^(3) scalars for a mixed qubit-qutrit state," Journal of Mathematical Sciences, vol. 179, pp. 690-701, 2011. DOI: 10.1007/s10958-011-0619-9.
[7] V. Gerdt, A. Khvedelidze, and Y. Palii, "Constructing the 5^(2) x orbit space for qutrit mixed states," Journal of Mathematical Sciences, vol. 209, pp. 878-889, 2015. DOI: 10.1007/s10958-015-2535-x.
[8] V. Gerdt, A. Khvedelidze, and Y. Palii, "On the ring of local unitary invariants for mixed X-states of two qubits," Journal of Mathematical Sciences, vol. 224, pp. 238-249, 2017. DOI: 10.1007/s10958-017-34-09-1.
[9] V. L. Popov and E. B. Vinberg, "Invariant theory," in Encylopaedia of Mathematical Sciences, ser. Algebraic Geometry IV, vol. 55, Berlin: Springer-Verlag, 1994, pp. 123-273.
[10] M. Adelman, J. V. Corbett, and C. A. Hurst, "The geometry of state space," Foundations of Physics, vol. 23, pp. 211-223, 1993. DOI: 10. 1007/BF01883625.
[11] M. Kus and K. Zyczkowski, "Geometry of entangled states," Physical Review A, vol. 63, p. 032 307, 2001. DOI: 10.1103/PhysRevA.63.032307.
[12] J. Grabowski, M. Kus, and G. Marmo, "Geometry of quantum systems: density states and entanglement," Journal of Physics A: Mathematical and General, vol. 38, no. 7, pp. 10 217-44, 2005. DOI: 10.1088/03054470/38/47/011.
[13] I. Bengtsson and K. Zyczkowski, Geometry of quantum states: an introduction to quantum entanglement. Cambridge: Cambridge University Press, 2006. DOI: 10.1017/CB09780511535048.
[14] J. von Neumann, "Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik," German, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pp. 245-272, 1927.
[15] L. D. Landau, "Das Dampfungsproblem in der Wellenmechanik," German, Z. Physik, vol. 45, pp. 430-441, 1927.
[16] M. Abud and G. Sartori, "The geometry of spontaneous symmetry breaking," Annals of Physics, vol. 150, no. 2, pp. 307-372, 1983. DOI: 10.1016/0003-4916(83)90017-9.
[17] C. Procesi and G. Schwarz, "The geometry of orbit spaces and gauge symmetry breaking in supersymmetric gauge theories," Physics Letters B, vol. 161, pp. 117-121, 1985. DOI: 10.1016/0370-2693(85)90620-3.
[18] C. Procesi and G. Schwarz, "Inequalities defining orbit spaces," In-ventiones mathematicae, vol. 81, pp. 539-554, 1985. DOI: 10.1007/ BF01388587.
[19] D. Cox, J. Little, and D. O'Shea, Ideals, varieties, and algorithms, 3rd ed. Springer, 2007.
[20] F. Bloch, "Nuclear induction," Physical Review, vol. 70, no. 7, pp. 464474, 1946. DOI: 10.1103/PhysRev.70.460.
[21] U. Fano, "Description of states in quantum mechanics by density matrix and operator techniques," Reviews of Modern Physics, vol. 29, pp. 7493, 1957. DOI: 10.1103/RevModPhys.29.74.
[22] S. M. Deen, P. K. Kabir, and G. Karl, "Positivity constraints on density matrices," Physical Review D, vol. 4, p. 1662, 1971. DOI: 10. 1103/ PhysRevD.4.1662.
[23] F. J. Bloore, "Geometrical description of the convex sets of states for spin-1/2 and spin-1," Journal of Physics A: Mathematical and General, vol. 9, no. 12, pp. 2059-67, 1976. DOI: 10.1088/0305-4470/9/12/011.
[24] F. T. Hioe and J. H. Eberly, "N-level coherence vector and higher conservation laws in quantum optics and quantum mechanics," Physical Review Letters, vol. 47, p. 838, 1981. DOI: 10.1103/PhysRevLett.47. 838.
[25] P. Dita, "Finite-level systems, Hermitian operators, isometries and novel parametrization of Stiefel and Grassmann manifolds," Journal of Physics A: Mathematical and General, vol. 38, no. 12, pp. 2657-68, 2005. DOI: 10.1088/0305-4470/38/12/008.
[26] S. Akhtarshenas, "Coset parameterization of density matrices," Optics and Spectroscopy, vol. 103, pp. 411-415, 2007. DOI: 10. 1134/ S0030400X0709010X.
[27] E. Brüning, D. Chruscinski, and F. Petruccione, "Parametrizing density matrices for composite quantum systems," Open Systems & Information Dynamics, vol. 15, no. 4, pp. 397-408, 2008. DOI: 10. 1142/ S1230161208000274.
[28] C. Spengler, M. Huber, and B. Hiesmayr, "A composite parameterization of unitary groups, density matrices and subspaces," Journal of Physics A: Mathematical and Theoretical, vol. 43, no. 38, p. 385 306, 2010. DOI: 10.1088/1751-8113/43/38/385306.
[29] E. Brüning, H. Mäkelä, A. Messina, and F. Petruccione, "Parametriza-tions of density matrices," Journal of Modern Optics, vol. 59, no. 1, pp. 1-20, 2012. DOI: 10.1080/09500340.2011.632097.
[30] C. Eltschka, M. Huber, S. Morelli, and J. Siewert, "The shape of higher-dimensional state space: Bloch-ball analog for a qutrit," Quantum, vol. 5, p. 485, 2021. DOI: 10.22331/q-2021-06-29-485.
[31] L. Michel and L. A. Radicati, "The geometry of the octet," Ann. Inst. Henri Poincare, Section A, vol. 18, no. 3, pp. 185-214, 1973.
[32] A. J. MacFarline, A. Sudbery, and P. H. Weisz, "On Gell-Mann's A-matrices, d- and /-tensors, octets, and parametrizations of SU(3)," Communications in Mathematical Physics, vol. 11, pp. 77-90, 1968. DOI: 10.1007/BF01654302.
[33] D. Kusnezov, "Exact matrix expansions for group elements of SU(N)," Journal of Mathematical Physics, vol. 36, pp. 898-906, 1995. DOI: 10. 1063/1.531165.
[34] T. S. Kortryk, "Matrix exponentsial, SU(N) group elements, and real polynomial roots," Journal of Mathematical Physics, vol. 57, no. 2, p. 021701, 2016. DOI: 10.1063/1.4938418.
[35] V. Abgaryan and A. Khvedelidze, "On families of Wigner functions for A^-level quantum systems," Symmetry, vol. 13, no. 6, p. 1013, 2021. DOI: 10.3390/sym13061013.
[36] V. Abgaryan, A. Khvedelidze, and A. Torosyan, "The global indicator of classicality of an arbitrary A^-level quantum system," Journal of Mathematical Sciences, vol. 251, pp. 301-314, 2020. DOI: 10.1007/ s10958-020-05092-6.
[37] V. Abgaryan, A. Khvedelidze, and A. Torosyan, "Kenfack-Zyczkowski indicator of nonclassicality for two non-equivalent representations of Wigner function of qutrit," Physics Letters A, vol. 412, no. 7, p. 127591, 2021. DOI: 10.1016/j.physleta.2021.127591.
For citation:
A. Khvedelidze, D. Mladenov, A. Torosyan, Parameterizing qudit states, Discrete and Continuous Models and Applied Computational Science 29 (4) (2021) 361-386. DOI: 10.22363/2658-4670-2021-29-4-361-386.
Information about the authors:
Khvedelidze, Arsen — PhD in physics and mathematics, Head of Group of Algebraic and Quantum Computations of Meshcheryakov Laboratory of Information Technologies, Joint Institute for Nuclear Research; Director of Institute of Quantum Physics and Engineering Technologies, Georgian Technical University; Researcher in A. Razmadze Mathematical Institute, Iv. Javakhishvili Tbilisi State University (e-mail: akhved@jinr.ru, phone: +7(496)2164338, ORCID: https://orcid.org/0000-0002-5953-0140)
Mladenov, Dimitar — PhD in Physics and Mathematics, Associate professor of department of Theoretical Physics of Faculty of Physics, Sofia University "St. Kliment Ohridski" (e-mail: mladim2002@gmail.com, ORCID: https://orcid.org/0000-0003-3817-5976)
Torosyan, Astghik — Junior Researcher in Meshcheryakov Laboratory of Information Technologies, Joint Institute for Nuclear Research (e-mail: astghik@jinr.ru, phone: +7(496)2164800, ORCID: https://orcid.org/0000-0002-4514-2884)
УДК 519.872:519.217
РАСЯ 07.05.Tp, 02.60.Pn, 02.70.Bf
DOI: 10.22363/2658-4670-2021-29-4-361-386
Параметризация состояний кудита
А. Хведелидзе1,2'3, Д. Младенов4, А. Торосян'
.3
3
1 Математический институт им. А. Размадзе Тбилисский государственный университет им. И. Джавахишвили проспект Ильи Чавчавадзе, д. 1, Тбилиси, 0179, Грузия 2 Институт квантовой физики и инженерных технологий Грузинский технический университет ул. Костава, д. 77, Тбилиси, 0175, Грузия 3 Лаборатория информационных технологий им. М. Г. Мещерякова
Объединённый институт ядерных исследований ул. Жолио-Кюри, д. 6, Дубна, Московская область, 141980, Россия
4 Факультет физики Софийский университет им. св. Климента Охридского ул. «Царь-Освободитель», д. 15, София, 1164, Болгария
2
4
Квантовые системы с конечным числом состояний всегда были основным элементом многих физических моделей в ядерной физике, физике элементарных частиц, а также в физике конденсированного состояния. Однако сегодня, в связи с практической потребностью в области развития квантовых технологий, возник целый ряд новых задач, решение которых будет способствовать улучшению нашего понимания структуры конечномерных квантовых систем.
В статье мы сфокусируемся на одном из аспектов исследований, связанных с проблемой явной параметризации пространства состояний Л?-уровневой квантовой системы. Говоря точнее, мы обсудим вопрос практического описания унитарного пространства орбит — (Ж)-инвариантного аналога Л^-уровневого пространства состояний фдг. В работе будет показано, что сочетание хорошо известных методов теории полиномиальных инвариантов и выпуклой геометрии позволяет получить удобную параметризацию для элементов фдг/<§^(Ж). Общая схема параметризации фдгбудет детально проиллюстрирована на примере низкоуровневых систем: кубита (Ж = 2), кутрита (Ж = 3), куатрита
Ключевые слова: параметризация матрицы плотности, квантовая система, кубит, кутрит, куатрит, кудит, теория полиномиальных инвариантов, выпуклая геометрия
(Ж = 4).
