Finding maximal quantum resources Jonathan Steinberg1 2and Otfried G uhne1 1Naturwissenschaftlich-Technische Fakult at Universit at Siegen Walter-Flex-Straße 3 57068 Siegen Germany

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Finding maximal quantum resources

Jonathan Steinberg1, 2, ∗and Otfried G¨uhne1, †

1Naturwissenschaftlich-Technische Fakult¨at, Universit¨at Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany

2State Key Laboratory for Mesoscopic Physics, School of Physics and Frontiers

Science Center for Nano-Optoelectronics, Peking University, Beijing 100871, China

(Dated: January 3, 2025)

For many applications the presence of a quantum advantage crucially depends on the availability

of resourceful states. Although the resource typically depends on the particular task, in the context

of multipartite systems entangled quantum states are often regarded as resourceful. We propose an

algorithmic method to ﬁnd maximally resourceful states of several particles for various applications

and quantiﬁers. We discuss in detail the case of the geometric measure, identifying physically

interesting states and also deliver insights to the problem of absolutely maximally entangled states.

Moreover, we demonstrate the universality of our approach by applying it to maximally entangled

subspaces, the Schmidt-rank, the stabilizer rank as well as the preparability in triangle networks.

I. INTRODUCTION

The access to multipartite quantum states is an indis-

pensable prerequisite for many applications in quantum

information, turning them into a powerful resource which

potentially outperform their classical counterparts [1–3].

Indeed, magic states turn out to be a resource for fault-

tolerant quantum computation [4,5] while cluster states

are resourceful for measurement-based quantum compu-

tation [6,7]. Furthermore, the power of quantum metrol-

ogy heavily relies on the ability to prepare multipartite

quantum states. However, for a particular given task it is

in general very challenging to identify those multipartite

states which yield the largest advantage.

For many important applications entanglement has

been proven to be a powerful resource. An example of

resourceful states are the absolutely maximally entangled

(AME) states which maximized the entanglement in the

bipartitions, but are notoriously diﬃcult to character-

ize [8–14]. Still, the analysis of AME states is important

for understanding quantum error correction and regarded

as one of the central problems in the ﬁeld [15,16]. How-

ever, multiparticle entanglement oﬀers a complex and

rich structure resulting in the impossibility of quantiﬁ-

cation by means of a single number. Consequently, there

is a variety of quantiﬁers, each emphasizing a diﬀerent

property that makes a state a valuable resource [17–19].

The geometric measure of entanglement [20–23] quan-

tiﬁes the proximity of a quantum state to the set of prod-

uct states, has an intuitive meaning and also oﬀers mul-

tiple operational interpretations. For instance, it relates

to multipartite state discrimination using LOCC [24],

the additivity of channel capacities [25], quantum state

estimation [26] and was also used to describe quantum

phase transitions [27–30]. In complexity theory, identify-

ing maximally entangled states and computing their geo-

metric measure allows for the identiﬁcation of cases where

∗steinberg@physik.uni-siegen.de

†otfried.guehne@uni-siegen.de

FIG. 1. Schematic illustration of the iteration step of the

algorithm. The set of all states is represented by the half

sphere and the set of product states by the lower dimensional

manifold P. If the algorithm is initialized in state |φ⟩(red

arrow), we ﬁrst compute the best approximation within P,

denoted by |π⟩(blue arrow). Then, we compute the projector

into the orthocomplement of |π⟩, which is here given by the

xy-plane. The portion of |φ⟩within the xy-plane is given by

|η⟩(gray arrow). The new state |˜φ⟩∼|φ⟩+ϵ|η⟩is then the

normalized version of |φ⟩shifted by a small amount ϵ > 0 into

the direction |η⟩.

the MAX-N-local Hamiltonian problem and its product

state approximation deviate maximally [31,32]. So, al-

though high geometric entanglement does not guarantee

that a quantum state is useful for all tasks [33–35], ﬁnding

maximally entangled states has been recognised as a nat-

ural and important problem [32]. So far, however, max-

imally entangled states have only been identiﬁed within

the low-dimensional family of symmetric qubit states,

where their computation is related to the problem of

distributing charges on the unit sphere [19,36,37], or

within the family of graph states that stem from bipar-

arXiv:2210.13475v3 [quant-ph] 1 Jan 2025

tite graphs [38].

Mathematically, the complexity of the task reﬂects the

fact that pure multiparticle states are described by ten-

sors. In contrast to the matrix case, notions like ranks

and eigenvalues are for tensors much less understood

and their computation turns out to be a hard prob-

lem [39,40]. Interestingly, the geometric measure is

closely related to the recently introduced concept of ten-

sor eigenvalues [22,41–44], oﬀering a much more complex

structure as the matrix case [45] as well as to the no-

tion of injective tensor norms [46,47] and matrix perma-

nents [48]. Here, maximally entangled states oﬀer max-

imal tensor eigenvalues [49] and it was conjectured that

the overlap of a multipartite qubit state with the set of

product states decreases exponentially in the number of

particles [47]. So, the identiﬁcation of maximally en-

tangled states provides valuable intuition to decide this

conjecture.

In this paper we design an iterative method for ﬁnd-

ing maximally resourceful multipartite quantum states.

Choosing initially a generic quantum state, we show

that in each step of the algorithm the resourcefulness

increases. We illustrate the universality of our method

by applying it to various diﬀerent resource quantiﬁers

and present a detailed analysis for the geometric mea-

sure. Here we identify for moderate sizes the correspond-

ing states, revealing an interesting connection to AME

states. Further, we introduce a novel quantiﬁer for max-

imally entangled subspaces for which we provide a full

characterization for the case of three qubits.

II. THE GEOMETRIC MEASURE

This measure quantiﬁes how well a given multiparti-

cle quantum state can be approximated by pure prod-

uct states. More formally [20–23], given |φ⟩one deﬁnes

G(|φ⟩)=1−λ2(|φ⟩) with

λ2(|φ⟩) = max

|π⟩|⟨π|φ⟩|2,(1)

where the maximization runs over all product states |π⟩

of the corresponding system. This quantity for pure

states can be extended to mixed states via the convex roof

construction and is then a proper entanglement mono-

tone [22].

While computing λ2for a generic pure state is, in

principle, diﬃcult [50], there is a simple see-saw iter-

ation that can be used [51–53]. For a three-partite

state |φ⟩, the algorithm starts with a random product

state |a0b0c0⟩. From this we can compute the non-

normalized state |˜a⟩=⟨b0c0|φ⟩, and make the update

|a0⟩ 7→ |a1⟩=|˜a⟩/p⟨˜a|˜a⟩. The procedure is repeated

for the second qubit |b0⟩, starting in the product state

|a1b0c0⟩. This is then iterated until one reaches a ﬁxed

point. Of course, this ﬁxed point is not guaranteed to be

the global optimum, in practice, however, this method

works very well.

III. IDEA OF THE ALGORITHM

We present the algorithm for the case of three qubits.

The generalization to arbitrary multiparticle systems is

straightforward and is discussed in Appendix A. As ini-

tial state |φ⟩we choose a random pure three qubit state.

Then, we compute its closest product state |π⟩via the

see-saw algorithm described above. We can assume

without loss of generality that |π⟩=|000⟩. We write

λ=|⟨φ|π⟩| for the maximal overlap of |φ⟩with the set

of all product states. Note that for a generic quantum

state the closest product state is unique.

The key idea is now to perturb the state |φ⟩in a way

that the overlap with |π⟩decreases. If |π⟩is the unique

closest product state and the perturbation is small, one

can then expect that the overlap with all product states

decreases. So, we consider the orthocomplement of

|π⟩=|000⟩, that is, the complex subspace spanned by

|001⟩,|010⟩,|100⟩,|011⟩,|101⟩,|110⟩,|111⟩. This subspace

gives rise to a projection operator Π = 11−|π⟩⟨π|and we

compute the best approximation of the state |φ⟩within

this subspace, given by |η⟩= Π|φ⟩/M, where Mdenotes

the normalization. Then, we shift the state |φ⟩in the di-

rection of |η⟩by some small amount θ > 0. Hence the

state update rule is given by

|φ⟩ 7→ |˜φ⟩:= 1

N(|φ⟩+θ|η⟩) (2)

where Nis a normalization factor.

In the next step, we calculate the best rank one ap-

proximation to |˜φ⟩. This process is iterated until the

geometric measure is not increasing under the update

rule (2). In this case, one can reduce the step size or the

algorithm terminates.

One can directly check that the overlap with |π⟩is

smaller for |˜φ⟩than for |φ⟩. Indeed one has

|⟨π|˜φ⟩|2=1

N2|⟨π|φ⟩+θ⟨π|η⟩|2=λ2

N2< λ2(3)

since N>1 if θ > 0. In fact, a much stronger statement

holds:

Observation 1. For a generic quantum state |ψ⟩there

always exists a Θ >0 such that the updated state |˜

ψ⟩

according to Eq. (2) with step size θ < Θ fulﬁlls G(|ψ⟩)<

G(|˜

ψ⟩).

Observation 1provides the guarantee that the pro-

posed algorithm yields a sequence of states with increas-

ing geometric measure. The proof is given in Appendix A

and comes with an interesting feature. It turns out that

the proof does not rely on the particular product state

structure of |π⟩, so any ﬁgure of merit based on maximiz-

ing the overlap with pure states from some subset can be

optimized with our method. This turns the algorithm

into a powerful and tool with a universal applicability.

Indeed, we adapt it to the dimensionality of entangle-

ment (see Appendix E), the stabilizer rank (Appendix

D), matrix product states (Appendix E) as well as to

the preparability in quantum networks (Appendix F).

Remarkably, here novel states are found which are more

distant to any network state as those known so far.

IV. IMPLEMENTATION

Thanks to the update rule in Eq. (2), we can make

use of advanced descent optimization algorithms in or-

der to obtain faster convergence and higher robustness

against local optima [54,55]. We have implemented a

descent algorithm with momentum as well as the Nes-

terov accelerated gradient (NAG) [56]. The idea behind

the momentum version is to keep track of the direction of

the updates. More precisely, the update direction |ηn⟩in

the n-th iteration will be a running average of the previ-

ously encountered updates |η1⟩, ..., |ηn−1⟩. For the NAG

method, the update vector is, contrary to Eq. (2), eval-

uated at a point estimated from previous accumulated

updates, and not at |φ⟩. A more detailed discussion and

a comparison of the diﬀerent methods can be found in

Appendices Gand B. The convergence of the algorithm

in the case of qubits is also shown in Fig. 1.

After the algorithm has terminated, one obtains a ten-

sor presented in a random basis. In order to identify the

state and to obtain a concise form, we have to ﬁnd ap-

propriate local basis for each party, such that the state

becomes as simple as possible. This can be done by a

suitable local unitary transformation, see Appendix H

for details.

V. RESULTS FOR QUBITS

For two qubits, the maximally entangled state is the

Bell state and our algorithm directly converges to this

maximum. For the case of three qubits there are two

diﬀerent equivalence classes of genuine tripartite en-

tangled states with respect to stochastic local opera-

tions and classical communication (SLOCC) [57], namely

|W⟩= (|001⟩+|010⟩+|100⟩)/√3 and |GHZ⟩= (|000⟩+

|111⟩)/√2. While G(|GHZ⟩) = 1/2 one has G(|W⟩) =

5/9, what turns out to be the maximizer among all tri-

partite states [58]. Indeed, after 150 iterations, the algo-

rithm yields the W state. It should be noted, that in this

case the maximizer belongs to the family of symmetric

states. Operationally, the W state is the state with the

maximal possible bipartite entanglement in the reduced

two-qubit states [57].

For four qubits, the algorithm yields after 300 itera-

tions the state

|˜

M⟩=1

√3(|GHZ⟩+e2πi/3|GHZ34⟩+e4πi/3|GHZ24⟩)

(4)

where |GHZij ⟩means a four-qubit GHZ state where a

bit ﬂip is applied at party iand j. Note that the phases

n Gsymm

max Gmax |φ⟩max

2 1/2 1/2|ψ−⟩

3 0.5555 ≈5/9 0.5555 ≈5/9|W⟩

4 0.6666 ≈2/3 0.7777 ≈7/9|M⟩

5≈0.7006 0.8686 ≈(1/36)(33 −√3) |G5⟩

6 0.7777 ≈7/9 0.9166 ≈11/12 |G6⟩

7≈0.7967 ≥0.941 MMS(7,2)

TABLE I. Maximally entangled states found by the algorithm

for systems between two and seven qubits. Here |φ⟩max refers

to the state found by algorithm and Gmax denotes the ge-

ometric measure of the corresponding state. Gsymm

max denotes

the maximal entanglement among symmetric states, as shown

in [36].

form a trine in the complex plane and that the state is

a phased Dicke state [59]. This state can be shown to

be LU-equivalent to the so called Higuchi-Sudbery or M

state [11,60], which appears as maximizer of the Tsallis

α-entropy in the reduced two-particle states for 0 < α <

2. Note that this state is not symmetric with respect to

permutations of the parties. Similar to the W state, the

entanglement of the state in Eq. (4) appears to be robust,

that is, uncontrolled decoherence of one qubit does not

completely destroy the entanglement of the remaining

qubits [60].

For ﬁve qubits the algorithm converges to a state

|G5⟩, that can be identiﬁed with the ring cluster state

(a 5-cycle graph state) yielding a geometric measure of

0.86855 ≈1

36 (33 −√3). The state |G5⟩appears in the

context of the ﬁve-qubit error correcting code [61]. Some

basic facts concerning cluster and graph states are given

in Appendix C. Similarly, for six qubits we obtain a

graph state |G6⟩with a measure of 0.9166 ≈11

12 . This

is again connected to quantum error correction, indeed,

both states |G5⟩and |G6⟩are AME states, see also be-

low. For seven qubits, we ﬁnd a numerical state with

maximally mixed two-body marginals, where the spec-

tra of the three-body marginals are all the same. This

motivates to introduce the class of maximally marginal

symmetric states (MMS), as a natural extension of notion

of AME states.

VI. RELATION TO AME STATES

One calls a multiparticle state AME, if it is a maxi-

mally entangled state for any bipartition. For bipartite

entanglement measures, it is well known that pure states

with maximally mixed marginals are the maximally en-

tangled states [60,62,63]. Consequently, an n-partite

pure multi-qudit state is AME if all reductions to ⌊n

2⌋

parties are maximally mixed, such a state is then de-

noted by AME(n, d). Interestingly, not for all values of

nand dAME states exist and quest for AME states is

a central problem in entanglement theory [15]. The non-

existence of AME states was ﬁrst encountered for four

qubits [60], but interestingly, the M state in Eq. (4) that

maximizes the geometric measure can be viewed as the

best possible replacement, since the one-body marginals

are maximally mixed and all 2-body marginals, albeit

not being maximally mixed have the same spectrum [11].

For ﬁve and six qubits the states found by our algorithm

are just the known AME states, see also Appendix C.

For n≥7 no AME(n, 2) state exist [8,12]. A poten-

tial approximation |F⟩to the AME state of seven qubits

has been identiﬁed [12], which is a graph state corre-

sponding to the Fano plane. This state has a measure of

G(|F⟩) = 15/16 = 0.9375, thus smaller as the measure of

the MMS state found by the algorithm.

While for certain values of nand dno AME state ex-

ists, it appears that multiple AME states can exist for

other choices. In fact, it has been shown that those states

can be even SLOCC inequivalent [64,65]. It is in general

a diﬃcult problem to decide whether two states belong to

the same SLOCC class, but for the case of AME states it

can be drastically simpliﬁed using our algorithm in com-

bination with the Kempf-Ness theorem [66]. First, notice

that states with maximally mixed 1-body marginals be-

long to the so-called class of critical states [67]. The the-

orem then assures that two critical states belong to the

same SLOCC class if and only if they are LU equivalent.

Hence, if the geometric measure of those states diﬀers, it

already implies SLOCC inequivalence.

VII. SYSTEMS OF HIGHER DIMENSIONS

For the bipartite case the generalized Bell states |ϕd⟩=

Pd−1

j=0 |jj⟩/√dare maximally entangled with G(|ϕd⟩) =

1−1/d. We ﬁnd that for 2 ≤d≤10 the algorithm yields

the corresponding state |ψd⟩with high ﬁdelity, and that

the number of iterations needed until convergence ap-

pears to be only weakly dependent on d, see also Ap-

pendix G.

In the three-qutrit case we obtain the total antisym-

metric state |Ψ3⟩, given by

|Ψ3⟩=1

√6(|012⟩+|201⟩+|120⟩−|210⟩−|102⟩−|021⟩)

(5)

In general, antisymmetric states |Ψn⟩can be constructed

for all n-partite n-level systems and their geometric mea-

sure can be easily computed analytically as n!−1

n![68]. In

the particular case n= 3 we obtain G=5

6≈0.8333.

Note that |Ψ3⟩is an AME state.

More generally, in the tripartite case a procedure is

known to construct AME states for arbitrary d[69]. This

leads to

AME(3, d)∼

d−1

i,j=0 |i⟩|j⟩|i+j⟩,(6)

where i+jis computed modulo d. Note that the state

AME(3,3) constructed according to Eq. (6) only has a

measure of 2/3.

For the case of three ququads our algorithm gives in-

sights into the AME problem. First, the AME(3,4) state

corresponding to Eq. (6) has a geometric measure of

G= 0.75. However, our algorithm yields a state given

|ϕ3,4⟩=1

2√2(|022⟩+|033⟩+|120⟩+|131⟩

+|212⟩+|203⟩+|310⟩+|301⟩)

(7)

with G(|ϕ3,4⟩) = 7/8 = 0.875. After applying local uni-

taries, this state can be seen as arising from three Bell

pairs distributed between three parties in a triangle like

conﬁguration. In addition, |ϕ3,4⟩is an AME state, i.e., all

one-party marginals are maximally mixed. As the geo-

metric measure of the states |ϕ3,4⟩and AME(3,4) diﬀers,

they belong to diﬀerent SLOCC classes.

In the case of four qutrits the algorithm converges to a

state with a geometric measure of 0.888 ≈8

9. This state

can be identiﬁed to be the AME(4,3) state given by [69]

AME(4,3) =1

3(|0000⟩+|0112⟩+|0221⟩+|1011⟩+|1120⟩

+|1202⟩+|2022⟩+|2101⟩+|2210⟩)

(8)

For four ququads, the algorithm converges to the anti-

symmetric state |Ψ4⟩, yielding a measure of 23

24 ≈0.9583.

Interestingly, while being 1-uniform, this state is not

AME. The so far only known AME(4,4), a graph state

[65,70], yields a measure of 15

16 = 0.9375. Finally, the re-

cently found AME(4,6) [16] is not maximally entangled

with respect to the geometric measure, i.e., the algorithm

ﬁnds states yielding a higher geometric measure.

Similarly, there exists a general procedure to construct

AME(5, d) states given by [69,71]

AME(5, d)∼

d−1

i,j,l=0

ωil|i⟩|j⟩|i+j⟩|l+j⟩|l⟩,(9)

where ω=e2πi/d. In the case of a three-dimensional

system, the algorithm converges to the AME(5,3) state

yielding a measure of approximately 0.96122. Finally, we

have G(AME(5,4)) = 31

32 = 0.96875. Here the algorithm

yields a state |ϕ5,4⟩with a larger geometric measure, in

particular G(|ϕ5,4⟩)>0.975. However, here we cannot

identify a closed expression of the state. The numerical

result suggests that the maximizer is again an AME state.

VIII. MAXIMALLY ENTANGLED SUBSPACES

One can extend our method such that it also applies

to subspaces. More precisely, we want to construct an

orthonormal basis for a subspace Vsuch that the least

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FindingmaximalquantumresourcesJonathanSteinberg1,2,∗andOtfriedG¨uhne1,†1Naturwissenschaftlich-TechnischeFakult¨at,Universit¨atSiegen,Walter-Flex-Straße3,57068Siegen,Germany2StateKeyLaboratoryforMesoscopicPhysics,SchoolofPhysicsandFrontiersScienceCenterforNano-Optoelectronics,PekingUniversity,Beijing...

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Finding maximal quantum resources Jonathan Steinberg1 2and Otfried G uhne1 1Naturwissenschaftlich-Technische Fakult at Universit at Siegen Walter-Flex-Straße 3 57068 Siegen Germany

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