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arXiv:1106.2947v1 [quant-ph] 15 Jun 2011

Disorder overtakes Order in Information Concentration over Quantum Networks

R. Prabhu, Saurabh Pradhan, Aditi Sen(De), and Ujjwal Sen

Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India

We consider different classes of quenched disordered quantum XY spin chains, including quantum

XY spin glass and quantum XY model with a random transverse field, and investigate the behavior

of genuine multiparty entanglement in the ground states of these models. We find that there are

distinct ranges of the disorder parameter that gives rise to a higher genuine multiparty entanglement

than in the corresponding systems without disorder – an order-from-disorder in genuine multiparty

entanglement. Moreover, we show that such a disorder-induced advantage in the genuine multiparty

entanglement is useful – it is almost certainly accompanied by a order-from-disorder for a multiport

quantum dense coding capacity with the same ground state used as a multiport quantum network.

I.INTRODUCTION

Quantum mechanical laws can be utilized to enhance

the performance of a large spectrum of information trans-

mission protocols – remarkable discoveries in the last

two decades – which can potentially revolutionize com-

munication networks. Examples include classical infor-

mation transfer through a quantum channel (quantum

dense coding) [1], arbitrary quantum state transmission

with the use of two bits of classical communication and

a shared quantum state (quantum teleportation) [2], and

secret transmission of classical information (quantum

cryptography or quantum key distribution) [3]. These

protocols form the main pillars in the development of

quantum communication [4], and at the same time, are

the basic ingredients for quantum computational tasks

[5]. That the protocols have already been experimen-

tally realized in a variety of substrates, makes the field all

the more exciting. These include the realizations in pho-

tonic systems, by which quantum communication proto-

cols can now be implemented between nearby cities and

islands (see e.g. [6, 7], and references therein). And in

ion trap systems, where quantum communication proto-

cols can potentially be useful in a future quantum com-

puter realized in that system (see [8–13], and references

therein).

However, while quantum communication protocols be-

tween two parties have received a lot of attention, both

on the experimental [6–8, 11, 14–17] and theoretical [18]

fronts, the same is rather limited in the multiparty sce-

nario. This is despite the fact that multi-access quantum

communication networks involving several senders and

receivers have immense applications and form one of the

ultimate goals of such studies.

The multiparty scenario presents a very rich struc-

ture indeed. See e.g. [19–24], and references therein.

In particular, while capacities for classical as well as

quantum information transmission has a one-to-one cor-

respondence with the shared entanglement between the

sender and the receiver in a bipartite situation (single

sender and a single receiver), the multiparty case does

not lend itself to such a direct correspondence in every

setting. Certain multi-access quantum capacities of mul-

tiparty quantum states can not be related to its multi-

party entanglement content, irrespective of the mode of

quantification of the multiparty entanglement [25]. Here

we show that in physically realizable quantum spin mod-

els used as multi-access quantum networks, the capacity

of transmitting classical information is related to the gen-

uine multiparty entanglement content of that network.

The genuine multiparty entanglement content is quan-

tified by the recently introduced generalized geometric

measure [25, 26], while the quantum spin models consid-

ered are the one-dimensional anisotropic quantum XY

models, with nearest-neighbor interactions, and with a

transverse field [27]. We consider both ordered as well as

quenched disordered XY spin systems. The disordered

models that we consider are (i) the quantum XY spin

glass – the randomness appearing in the coupling con-

stant, (ii) the quantum XY model with a random trans-

verse field, and (iii) the quantum XY spin glass with a

random transverse field. Disordered systems form one of

the centerstages of studies in many-body physics [28–30].

In particular, a lot of attention is recently being given to

quantum spin glass systems (disordered couplings, with

or without a disordered transverse field) (see [31–36], and

references therein).

Disordered systems appear due to certain environmen-

tal conditions and other parameters in the system that

are not possible to control experimentally. One may ex-

pect that disorder would reduce properties like magneti-

zation, conductivity, etc., of a given system, and is indeed

true for a large variety of systems. However, phenomena

where disorder enhances magnetization and other prop-

erties of a system are also known to occur in both clas-

sical and quantum systems [30, 37]. Studies in quan-

tum transport have shown that disordered systems are

always worse than ordered ones for carrying quantum

information [38–44]. However, we find that the ground

states of a class of quantum spin models with disorder,

for certain values of the impurity parameters, can give

a clear advantage for carrying classical information en-

coded in a quantum state, over the corresponding or-

dered system. Moreover, we observe that the disorder

can enhance the amount of genuine multipartite entan-

glement of the ground state in the quantum spin model,

and this is found in almost the same range in which the

disorder-induced amplification of the multiport capacity

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is obtained.

Technological limitations currently restricts the physi-

cally realizable genuinely multiparty entangled quantum

states to about 10 qubits [7, 8, 11, 12, 45, 46]. More-

over, photonic systems are widely envisaged as the sub-

strates for quantum communication protocols, and in

that case, currently available technology limitations are

even stricter on the number of qubits [47, 48]. We have

therefore focussed our attention on quantum communi-

cation protocols involving 7 or 8 parties, constituting the

sending and receiving ports of the communication pro-

tocol. However, our investigations with a few more par-

ties have shown that the qualitative behavior remains the

same.

The quantum XY spin chain can be solved exactly by

using the Jordan-Wigner transformation [49]. Since we

will be interested in finding global properties of the mul-

tiparty quantum states generated from the correspond-

ing disordered spin models, and in particular will be en-

gaged in finding out a genuine multiparty entanglement

measure, we will have to carry out the computations by

exact diagonalization.

The paper is organized as follows. In Sec. II, we briefly

describe the one dimensional quantum XY model, with-

out disorder, as well as with disorder in the couplings

and/or the fields. The disorders that we consider in this

paper are of the “quenched” type, requiring a “quenched

average” of the physical quantities considered in such

systems. The meanings of these terms are also briefly

explained in Sec. II. In Sec. III, we introduce the gen-

uine multiparty entanglement measure called generalized

geometric measure (GGM) which is used to investigate

the multiparty entanglement in the models considered in

this paper. In Sec. IV, we describe the classical infor-

mation transmission protocols that we consider – along

with their capacities – for the case of a single sender and

a single receiver and as well as for multiple senders and

a single receiver. The results are presented in Sec. V.

We begin there with the comparison between the XY

spin glass and the ordered XY systems (Sec. VA). We

find that there is large range of parameters for which the

quenched averaged genuine multiparty entanglement (as

quantified by GGM) in the disordered system is better

than the same in the ordered system – an order-from-

disorder phenomenon for a genuine multiparty entangle-

ment. And often there exists a critical coupling constant

below which the system shows this behavior. A similar

picture appears for the multiport channel capacity that

we consider. We observe that an order-from-disorder for

GGM is a prerequisite for the same feature to appear for

the multiport capacity. Disorder in the spin glass system

is introduced through randomness in the interactions. A

disordered field term (while maintaining an ordered in-

teraction) is considered in Sec. VB. Again, order-from-

disorder is observed for both GGM and the capacity, but

there appeared qualitative differences with the spin glass

case. It is aproiri unclear as to whether the two types

of disorder, if appearing in the same system, would also

allow for the order-from-disorder phenomenon. But we

find that such a situation is indeed allowed, in Sec. VC.

We conclude with a discussion in Sec. VI.

II.ORDERED AND DISORDERED QUANTUM

XY MODELS

The one dimensional quantum XY model with nearest

neighbor interactions in a transverse field is described by

the Hamiltonian [27]

H =J

4

?

?ij?

[(1 + γ)σx

iσx

j+ (1 − γ)σy

iσy

j] −h

2

?

i

σz

i, (1)

where J is the coupling constant and will be positive for

antiferromagnetic systems and negative for ferromagnetic

systems. γ ?= 0 is the anisotropy constant. σx, σy, and

σzare the Pauli spin matrices at the corresponding sites,

and ?ij? (i,j = 1,2,3,...,N) indicates that the interac-

tions are between all nearest-neighbor spins. N is the

length of the spin chain. Periodic boundary conditions,

i.e., ? σN+1= ? σNis considered here. Moreover, we assume

that J,h > 0. Such systems can be realized in different

physical system, including in ultracold gases [27, 50].

Disorder can be introduced in the quantum XY Hamil-

tonian through several routes. Here we consider the fol-

lowing three options: (i) the one-dimensional quantum

XY spin glass, where the coupling constant at each site is

an independent but identically distributed random vari-

able, while the field strength is a constant throughout the

chain, (ii) the one-dimensional quantum XY model with

a random transverse magnetic field, where the random-

ness behavior of the coupling constant and field are inter-

changed with respect the spin glass system, and (iii) the

one-dimensional quantum XY spin glass with a random

transverse magnetic field, where the coupling constant

and the field are both random.

A.Quenched disorder and quenched average

We assume that the disorder in all the different systems

considered here are “quenched”. That is, the time scales

in which the dynamics of the system takes place is much

shorter in comparison to the equilibrating times of the

disorder. On time scales over which the dynamics of the

system takes place, a particular realization of the random

disorder parameters remains static (“quenched”).

This has the implication that to find the average of a

particular physical quantity, the averaging over the dis-

order has to be performed after the calculation of the

physical parameters for particular realizations of the dis-

order.

In the opposite extreme, where the time scales of the

dynamics are much longer than the disorder equilibrat-

ing times, the disorder is said to be “annealed”, and the

corresponding annealed averaging is typically easier to

handle.

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B.Gaussian random variable

Along with being quenched, we also assume that the

disorder in the models considered, arise from Gaussian

distributed random variables. A random variable X is

said to follow the Gaussian distribution N(M,S), with

mean M and standard deviation S, if the probability

density P[X = x] is given by

P[X = x] =

1

S√2πexp

?

−1

2

?x − M

S

?2?

. (2)

Typically, the qualitative behavior of an averaged

physical quantity in a disordered system is independent

of the particular form of the distribution function of the

random variables involved, and essentially depends on a

few first moments of the distribution.

C.The disordered systems and their Hamiltonians

We now explicitly write down the Hamiltonians of the

disordered systems that we consider in this paper.

1. Quantum XY spin glass

The Hamiltonian of this model is given by [27] (see

Ref. [51] for experiments)

HSG=

N

?

i

Ji

4[(1+γ)σx

iσx

i+1+(1−γ)σy

iσy

i+1]−h

2

N

?

i

σz

i,

(3)

where the Jiare independent and identically distributed

(i.i.d.) Gaussian quenched random variables, each fol-

lowing the Gaussian (normal) distribution N(J,1). We

assume that h > 0.

2.Quantum XY model with random transverse magnetic

field

Reversing the randomness behavior of the coupling

constant and the field with respect to the spin glass

Hamiltonian leads to the quantum XY model with a ran-

dom transverse field (see [27, 52] and references therein),

with the Hamiltonian

HRF=J

4

N

?

i

[(1+γ)σx

iσx

i+1+(1−γ)σy

iσy

i+1]−

N

?

i

hi

2σz

i.

(4)

Here hiare i.i.d. Gaussian quenched random variables,

each following the Gaussian distribution N(h,1), and

J > 0.

3.Quantum XY spin glass with random transverse

magnetic field

Finally, we consider the case of the one-dimensional

quantum XY spin glass with random transverse mag-

netic fields hiapplied in the z-direction (see [27, 29, 53–

56] and references therein). The Hamiltonian for this

model is given by

HRF

SG=

N

?

i

Ji

4[(1+γ)σx

iσx

i+1+(1−γ)σy

iσy

i+1]−

N

?

i

hi

2σz

i,

(5)

where the Jiand hiare all i.i.d. quenched random vari-

ables, with the Jibeing Gaussian distributed as N(J,1),

and the hias N(h,1).

III.

ENTANGLEMENT MEASURE

GENUINE MULTIPARTITE

In this section, we introduce the genuine multipartite

entanglement measure that will be used in this paper to

investigate the behavior of multiparty entanglement in

the systems under study. The multipartite entanglement

measure that we consider here was introduced in Refs.

[25, 26], and is called the generalized geometric measure.

For a given N-party pure quantum state |ψN?, the GGM

is defined as

E(ψN) = 1 − max|?φN|ψN?|,

where the maximization is taken over all N-party pure

quantum states |φN? that are not genuinely multipartite

entangled. An N-party quantum state is said to be gen-

uinely multiparty entangled if it is not a product state

across any partition of the N parties that constitute the

whole system.

It turns out that above expression for GGM simplifies

to [25, 26]

(6)

E(ψN) = 1−max{λ2

A:B|A∪B = {1,....N},A∩B = ∅},

(7)

where λA:Bis the maximal Schmidt coefficient of |ψN? in

the bipartite split A : B.

The measure possesses the usual properties of a gen-

uine multiparty entanglement measure. In particular, it

does not increase under local quantum operations and

classical communication. Moreover, it is possible to com-

pute GGM for an arbitrary multiparty pure quantum

state of an arbitrary number of parties in arbitrary di-

mensions [25, 26] (cf. [57]).

IV.CLASSICAL CAPACITY OF A QUANTUM

COMMUNICATION CHANNEL

The paradigmatic classical information transmission

protocol over a quantum channel is quantum dense cod-

ing [1]. In this paper, we will consider the dense coding

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protocol, by using the multiparty ground states of phys-

ically realizable quantum spin models, between several

senders and a single receiver. For completeness, and to

set the terminology, we begin with the case of a single

sender and a single receiver.

A.Dense Coding: A Single Sender and a Single

Receiver

Let us suppose that an observer Alice (A), wants to

send some classical information to another observer, Bob

(B), who is in a distant laboratory, and with whom Alice

shares a quantum state ρAB. Suppose that dA(dB) is the

dimension of the Hilbert space on which Alice’s (Bob’s)

part of the state ρABis defined.

Consider the situation where Alice wants to send the

classical information i, which is known to happen with

probability pi, to Bob. The dense coding protocol now

runs as follows. Depending on the information i that is

to be sent, Alice performs the unitary transformation Ui

(cf. [58]) on her side of the state ρAB, and sends her part

of the quantum state to Bob over a noiseless quantum

channel. Therefore, Bob now has the two-party ensemble

{pi,ρAB

is the identity operator on Bob’s Hilbert space. Bob’s

aim is to gather as much information as possible, about

the index i, by performing quantum mechanically allowed

measurements on the two-particle ensemble that is in his

possession now. This involves an optimization process,

and it is possible to perform the optimization to obtain

the dense coding capacity, of the state ρAB, as [59]

i

}, where ρAB

i

= Ui⊗ IBρABU†

i⊗ IB, where IB

C(ρAB) = log2dA+ S(ρB) − S(ρAB),

where S(σ) = −tr[σlog2σ] is the von Neumann entropy

of the quantum state σ, and ρB= trA[ρAB] is a reduced

density matrix of the shared state ρAB. Here the capacity

is measured in “bits”.

To make the capacity values bounded by unity, inde-

pendent of the dimensions in which we are working, we

divide the actual capacity by the maximum (quantum

mechanically) achievable capacity, log2dA+ log2dBbits,

and we call this form of the capacity as normalized ca-

(8)

pacity, which is then given by

C(ρAB) =log2dA+ S(ρB) − S(ρAB)

log2dA+ log2dB

.(9)

Note that the normalized capacity is dimensionless.

B.Dense Coding between Multiple Senders and a

Single Receiver

Let us now consider a dense coding protocol in which

there are many senders, say Alice1, Alice2, ..., AliceM

(denoted as A1,A2,...,AM), and a single receiver, say

Bob (denoted as B). The M Alices and Bob share the

quantum state ρA1,A2,...,AM,B. Suppose that Alicek(k =

1,2,...,M) wants to send the classical information ikto

Bob, where it is previously known that ik appears with

probability pik.

The dense coding protocol then runs as follows. To

encode the message ik, Alicek applies the unitary Uik

on her part of the multiparty quantum state. All the

Alices then send their parts of the quantum state to Bob,

over noiseless quantum channels, and so now Bob is in

possession of the multiparty ensemble

?M

k=1

?

pik,⊗M

k=1Uik⊗ IBρA1,A2,...,AM,B⊗M

k=1U†

ik⊗ IB

?

(10)

,

and his job is to find an optimal measurement strategy to

obtain as much information as possible about the indices

ik.

It is possible to find an optimal strategy, and the re-

sulting dense coding capacity is given by [60]

C(ρA1,A2,...,AM,B) = log2dA1+ log2dA2+ ... + log2dAM

+S(ρB) − S(ρA1,A2,...,AM,B) (11)

bits, where dAkis the dimension of the Hilbert space of

Alicek’s subsystem. Again, to normalize the capacity,

we divide the actual capacity by log2dA1+ log2dA2+

...+log2dAMbits, to obtain the dimensionless normalized

capacity as

C?ρA1,A2,...,AM,B?=log2dA1+ log2dA2+ ... + log2dAM+ S?ρB?− S?ρA1,A2,...,AM,B?

log2dA1+ log2dA2+ ... + log2dAM+ log2dB

,(12)

where ρB= trA1,A2,...,AM

dimension of the Hilbert space of Bob’s subsystem.

?ρA1,A2,...,AM,B?, and dBis the

V.ORDER-FROM-DISORDER IN GENUINE

MULTIPARTITE ENTANGLEMENT AND

CHANNEL CAPACITY

We will now investigate the behavior of genuine mul-

tipartite entanglement (as quantified by generalized geo-

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metric measure) and capacity of dense coding in multi-

port scenarios, in the ground states of different quantum

XY spin models. In particular, we will be interested in

the effect of the different types of disorder (as provided by

the systems described by the Hamiltonians HSG, HRF,

and HRF

SG) on the multiparty entanglement measure and

the multiport classical capacity, and compare them with

the same quantities in the system described by the quan-

tum XY Hamiltonian H in Eq. (1).

We consider the ground states of the models for the

comparisons. This is dictated by the fact that while the

GGM can be obtained efficiently for arbitrary pure mul-

tiparty quantum states (of arbitrary number of parties

and dimensions), it is much harder to find it for mixed

quantum states. This feature of GGM is similar to that

of most other measures of entanglement, where calcula-

tions for pure states is much easier than for mixed states.

Exceptions include entanglement of formation (for two

qubits) [61] and logarithmic negativity [62]. However,

both these examples are for two-party situations. The

multiport channel capacity that we consider can how-

ever be efficiently calculated for mixed states also. But

since one of our main interests lie in the comparison of

the behavior of GGM with that of the multiport capac-

ity, with the introduction of disorder, we only work with

the ground states of the models, and in regions in which

there is no degeneracy in the ground state. We will have

occasion to discuss more on the issue of degeneracy, later

in the paper.

A. Quantum XY spin glass vs. quantum XY

In this subsection, we compare the values of GGM and

the multiport channel capacity in the quantum XY spin

glass with those of the quantum XY model. The Hamil-

tonian for the (ordered) quantum XY model is given in

Eq. (1) and that for the (disordered) quantum XY spin

glass model Hamiltonian is given in Eq. (3).

Let us first consider the calculations for GGM. For the

ordered system, we find the GGM in the ground state of

the system. For the disordered system, we find the GGM

in the ground state for a particular realization of the

disorder, and then average over the disorder. We perform

calculations for upto 8 spins in a periodic chain. The

results for odd and even number of spins are significantly

different, and we display the plots here for N = 7 and

for N = 8. See Fig. 1. The plots for other odd and even

number of spins are qualitatively similar, respectively to

the presented plots. For the plots, we have chosen the

anisotropy parameter (γ) in the Hamiltonians as 0.7. The

plots for the other anisotropies are qualitatively similar.

For the comparison, we find the GGM for the ordered

system for a particular value of J = J′and h = h′.

We then find the quenched averaged GGM in the disor-

dered system, where the transverse field is fixed at h′, and

where the Ji’s are all i.i.d. quenched random variables,

with each Ji/h′following the Gaussian distribution with

mean J′/h′and unit standard deviation.

For an odd number of spins, the quenched averaged

GGM in the disordered system is always higher than the

GGM in the corresponding ordered system. For an even

number of spins, there is a “cross-over value”, λGGM

of λ = J/h, before which the quenched averaged GGM

in the disordered system is larger than the GGM in the

ordered system.

Therefore, there are distinct regimes where introduc-

tion of disorder enhances the amount of genuine mul-

tiparty entanglement present in the quantum system

[64, 65]. The existence of entanglement in a quantum sys-

tem, consisting of several subsystems, indicates the ther-

modynamic signature of high global order in the whole

system while the local order in the subsystems is low

[66]. This disparity between global and local order is

particularly pronounced in a maximally entangled state,

e.g. the singlet state, where the global order is com-

plete (the global state is pure) while the local order is

completely absent (the local states are completely depo-

larized). What we show is that this disparity between

global and local order, in the sense of increasing genuine

multipartite entanglement, can be enhanced by introduc-

ing disorder into the system. The interplay of global and

local order to create entanglement is intrinsic to the mul-

tiparty quantum state under consideration. It is intrigu-

ing to see that agents of disorder that are external (in

the sense that they can be externally manipulated on

the system) can interact with the intrinsic order-disorder

mechanism, to produce an increase of entanglement by

increasing the disorder.

We now move over to the comparison of the informa-

tion carrying capacities of the same systems. The channel

capacity that we consider will have multiple senders and

a single receiver. For the ground state of a system of N

spins, we assume that N − 1 spins are in possession of

N − 1 Alices (who act as senders) while the remaining

spin is in possession of Bob (who will act as the receiver

of the information). Due to symmetry of the system, it

is immaterial as to which spin is in possession of Bob, in

the sense that the capacity will be independent of that

choice.

Similarly as for the GGM, the multiport capacity be-

haves differently for odd and even number of spins. In

Fig. 2, we plot the normalized capacities for N = 7 and

N = 8. The situation is qualitatively similar for other

odd and even spins, respectively. Just like for GGM, the

quenched averaged capacity in the disordered case is al-

ways better than the capacity in the ordered situation,

if the total number of spins is odd, while for even N,

there is a cross-over λcap

c

, until which the capacity of the

disordered case is better than that of the ordered Hamil-

tonian.

Therefore, again there arises situations where introduc-

tion of disorder into the system increases the information

carrying capacity of a quantum system.

Additionally, we find that it is necessary to have an

order-from-disorder feature for multiparticle entangle-

c

,

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0.05

0.10

0.15

0.20

0.25

Λ

GGM

? Disorder

? Order

N?7

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0.0

?

?

?

?

?

?????????????????

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0.51.01.52.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Λ

GGM

? Disorder

? Order

N?8

FIG. 1: (Color online.) Order-from-disorder for a genuine

multiparty entanglement in quantum XY spin glass.

upper plot is for 7 spins, while the lower one is for 8. In

both the plots, GGM is plotted on the vertical axes. The

curves with black circles are for the ordered systems (with

the Hamiltonian H), and in these cases, the GGM are plot-

ted against λ = J/h on the horizontal axes. The curves with

red squares are for the disordered systems (with the Hamilto-

nian HSG), and in these cases, the GGM are plotted against

the mean values λ = J/h of the Gaussian distributed Ji/h

(with mean J/h and unit standard deviation) on the hori-

zontal axes. Both axes represent dimensionless quantities in

both the plots. The whole range of parameters considered of-

fers disorder-induced enhancement of GGM for the case of an

odd number of spins. For an even number of spins, there is a

cross-over λ, before which the disorder-induced enhancement

occurs. See [63] for the strategy assumed for the very small

number degenerate ground states appearing.

The

ment, to have the same in the multiport capacity. In

other words, we observe that λGGM

cases considered.

We term the disorder-induced enhancements for the

genuine multiparty entanglement as well as the multi-

port capacity as “order-from-disorder”, since a higher

efficiency (in the sense of increased genuine multiparty

entanglement, which results in an increased multiport ca-

pacity, and possibly increased capacities for other quan-

tum communication tasks) in the generated ground state

is obtained by introducing disorder into the physical sys-

tem under consideration.

c

≥ λcap

c

in all the

B.Quantum XY : Random vs Non-random

transverse fields

For the ordered Hamiltonian, scanning over different

values of the coupling constant is physically equivalent

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1.52.0

0.86

0.88

0.90

0.92

0.94

0.96

Λ

Capacity

?

Classical

?

Disorder

?

Order

N?7

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????????????????????????????????????????

?

?

?

?

?

???????????????????

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0.00.51.01.52.0

0.88

0.90

0.92

0.94

0.96

0.98

Λ

Capacity

?

Classical

?

Disorder

?

Order

N?8

FIG. 2: (Color online.) Order-from-disorder for a multiport

classical capacity of a quantum channel in quantum XY spin

glass. The upper and lower plots are for 7 and 8 spins respec-

tively. In both the plots, the normalized capacity (dimension-

less) are plotted on the vertical axes. All other notations are

the same as in Fig. 1. Again, the whole range of parame-

ters considered offers disorder-induced enhancement for the

capacity for an odd number of spins, while there is a cross-

over λ before which the disorder-induced enhancement occurs

for an even number of spins. The cross-over λ for the capacity

is lower than that for the GGM. The horizontal lines (with

blue diamonds) correspond to the classical capacities that can

be attained if there are no previously shared quantum states

between the senders and the receivers.

to scanning over the transverse field. It may seem plau-

sible that similarly, sweeping over different mean values

of the coupling constant in the disordered Hamiltonian

is equivalent to sweeping over mean values of a disor-

dered transverse field. However, we show below that in-

troduction of a Gaussian disorder in the transverse field

produces a qualitatively different behavior for the GGM

as well as the multiport capacity, as compared to their

behavior after introduction of a Gaussian disorder in the

coupling constant.

As in the preceding subsection, we consider the ground

state of the ordered Hamiltonian (Eq.

J = J′and h = h′, and find the GGM and the

multiport classical capacity.

with those obtained from the ground state of the dis-

ordered Hamiltonian in Eq.

i.i.d. quenched random variables hi, with each hi/J dis-

tributed as N(h′/J′,1). And again, in the disordered

case, the averaging over the disorder is performed after

the physical quantity has been calculated for randomly

obtained values of the disorder parameter. See Figs. 3

and 4.

In contrast to the situation when disorder is introduced

(1)) for given

We then compare them

(4), for J = J′and for

Page 7

7

?

?

?

?

?

?

?

?

??????????????????????

??????????????????????????????

0.60.81.01.21.41.61.82.0

0.00

0.05

0.10

0.15

0.20

Μ

GGM

? Disorder

? Order

N?7

????????????????

????????????????????????????????????????

?

?

?

?????????????????????

0.00.51.01.5 2.0

0.0

0.1

0.2

0.3

0.4

Μ

GGM

? Disorder

? Order

N?8

FIG. 3: (Color online.)

quantum XY system with random transverse field. The up-

per plot is for N = 7, while the lower one is for N = 8.

The vertical axes in both the plots represents the GGM. The

curves with black circles are for the systems without disor-

der (represented by the Hamiltonian H), in which cases, the

GGM are plotted against µ = h/J on the horizontal axes.

The curves with red squares are for the systems with disorder,

and described by the Hamiltonian HRF, and in these cases,

the horizontal axes represent the mean values µ = h/J of the

Gaussian distributed random variables hi/J (with mean h/J

and unit standard deviation). All the axes represent dimen-

sionless quantities. In contrast to the situation in Fig. 1, the

cross-overs now appear for both odd and even N.

Order-from-disorder for GGM in

in the coupling constants (as discussed in the preceding

subsection), there are now cross-overs for both odd and

even total number of spins. That is, for both odd and

even total number of spins, there are cross-over values,

µGGM

c

, of µ = h/J, before which the quenched aver-

aged physical parameters are lower than the correspond-

ing physical parameter in the ordered Hamiltonian, and

after which the situation is the opposite.

There does however appear a difference between the

odd and even cases, but it is of a separate kind. Precisely,

the ordered Hamiltonian for an odd number of spins has a

degenerate ground state for low µ, and as said before, we

will therefore disregard this range of µ. The ground state

is however nondegenerate for the whole range of µ for an

even number of spins. This feature for odd N was also

obviously present in the considerations of the preceding

subsection, for high λ, and were also disregarded there.

Just like when we introduced disorder in the coupling

constants, again we have obtained distinct regimes where

introduction of disorder in the transverse field enhances

the genuine multipartite entanglement as well as the mul-

tiport channel capacity of the quantum system.And

again, a disorder-enhancement in GGM is necessary for

a similar feature in the capacity.

??????????????????????????????

??????????????????????????????

0.88

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

0.86

0.6 0.81.0 1.21.41.6 1.8 2.0

0.90

0.92

0.94

0.96

Μ

Capacity

?

Classical

?

Disorder

?

Order

N?7

??????????????????

????????????????????????????????????????

?

?

?

???????????????????

????????????????????????????????????????

0.00.5 1.01.52.0

0.88

0.90

0.92

0.94

0.96

0.98

1.00

Μ

Capacity

?

Classical

?

Disorder

?

Order

N?8

FIG. 4: (Color online.) Order-from-disorder for the multi-

port capacity in quantum XY system with random transverse

field. The vertical axes in both the plots represent the normal-

ized capacity (dimensionless). All other notations remain the

same as in Fig. 3. There are cross-overs again, and the cross-

over µ’s for the capacity are higher than those for GGM. The

horizontal lines (with blue diamonds) have the same meaning

as in Fig. 2.

C.Quantum XY spin glass with random transverse

field vs. quantum XY

We have already investigated the situations where dis-

order is introduced into the quantum system through sep-

arate channels – via coupling constants and via transverse

magnetic fields. Introduced separately, both have given

rise to regimes where disorder enhances the amount of a

genuine multiparty entanglement as well as the amount

of a multiport channel capacity. The question that we

want to raise now is whether the two types of disor-

der, if introduced together, will still allow for regimes of

disorder-enhanced physical quantities. This is not apri-

ori obvious as the effects of disorder-enhancement due

to the two types of disorder may not work in tandem,

and may, in principle, completely wash out the disorder-

enhancement phenomena. This however is not the case,

as we show now.

Consider a particular realization of the ordered Hamil-

tonian in Eq. (1) for J = J′and h = h′. Correspond-

ingly, let us consider the Hamiltonian HRF

quenched random variables Jiand hi, where the Ji/κ are

distributed as N(J′/κ,1) and the hi/κ are distributed

as N(h′/κ,1), for a positive κ with the unit of energy.

We compare the GGM and the capacity in the ground

SG, for i.i.d.

Page 8

8

-0.1

-0.05

0

0.05

0.1

0.15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

FIG. 5: (Color online.) Order-from-disorder for genuine mul-

tiparty entanglement in quantum XY spin glass with random

transverse field. The upper plot is for N = 7 and the lower

one is for N = 8.The vertical axes represent J/κ, while

the horizontal axes represent h/κ, with κ > 0. The plotted

quantity is the difference (disorder case − order case) be-

tween the quenched averaged GGM for the system described

by HRF

Ji/κ are distributed as N(J/κ,1) and the hi/κ are distributed

as N(h/κ,1)) and the GGM for the system described by H.

See text for further details. The plotted quantity as well as

the quantities represented by the axes are dimensionless.

SG(with i.i.d. random variables Ji and hi, where the

states of the two Hamiltonians. The results are plotted

in Figs. 5 and 6. For the plots, we find the difference

(disorder case − order case) of the values of a particular

physical quantity (GGM or capacity) as obtained in the

ordered case and in the corresponding disordered case

(after quenching). As before, the situations are different

for odd and even N. The results are stated below sepa-

rately for the two cases. For specificity, we consider the

cases N = 7 and N = 8, and γ = 0.7, in the plots. The

situation is similar for other odd and even total number

of spins, and for other anisotropies.

We begin with the case where the total number of spins

is odd. There are three qualitatively different regions.

See the upper plots in Figs. 5 and 6.

• Regions I. In the upper plot in Fig. 5, this is

the whole (J,h) region shown, except for roughly a

quadrant of an ellipse with corners approximately

at (0.2,0), (2,1), and (2,0). [Online, this region is

blue in color.] In this region, the quenched averaged

GGM in the disordered Hamiltonian is higher than

the GGM in the corresponding ordered system. A

similar, but slightly smaller, region is in the upper

plot of Fig. 6, where the quenched averaged capac-

-0.03-0.03

-0.02-0.02

-0.01-0.01

0 0

0.01 0.01

0.02 0.02

0.03 0.03

0.04 0.04

0.05 0.05

0.06 0.06

0.07 0.07

0.08 0.08

0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 1.2 1.2 1.4 1.4 1.6 1.6 1.8 1.8 2 2

0 0

0.2 0.2

0.4 0.4

0.6 0.6

0.8 0.8

1 1

1.2 1.2

1.4 1.4

1.6 1.6

1.8 1.8

2 2

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

FIG. 6: (Color online.) Order-from-disorder for the multiport

capacity in quantum XY spin glass with random transverse

field. All notations are the same as those in Fig. 5, except that

the plotted quantity is the difference between the multiport

capacities in the two situations considered. Also see text for

further details.

ity is greater than that in the ordered Hamiltonian.

[Online, this region is also blue in color.] This lat-

ter region is contained in the former region, again

implying that a disorder-enhancement in GGM is

a pre-requisite for disorder-enhancement in the ca-

pacity.

• Regions II. These are tiny portions of the (J,h)

region of both the plots, just below Region I, in

the bottom left corners of the plots. [Online, these

regions are black to grey in color.] In these regions,

there is no disorder-enhancement.

• Regions III. Finally, in the remaining portions

of the (J,h) regions plotted, the ordered Hamilto-

nians have degenerate ground states, and and are

thereby disregarded. These are the white regions

in the bottom right of the plots. They should not

be confused with the white boundary between Re-

gions I and Regions II, which indicates an equality

between the quenched averaged physical quantity

and that obtained from the ordered Hamiltonian.

Going over to the case where the total number of spins

is even, we find that the the picture is simpler here, as

there is no degeneracy observed in the region under study,

and consequently there are no Regions III here. See the

lower plots in Figs. 5 and 6. The other two regions are

however distinctly present, and placed approximately on

two sides of the antidiagonals.

Page 9

9

• Regions I. In case of the GGM, which is con-

sidered in the lower plot of Fig.

above the antidiagonal J = h. In this region, the

quenched averaged GGM is higher than the ordi-

nary variety. [Online, this region is blue in color.]

The status of the multiport capacity is considered

in the lower plot of Fig. 6, where the Region I is

similar, but slightly smaller, to the Region I in the

lower plot of Fig. 5. Again, the latter region is

contained in the former region.

5, Region I is

• Regions II. This is the complement of the Regions

I, and indicates that the quenched averaged values

are lower than the ordinary ones.

VI.DISCUSSION

0.00.51.0 1.52.0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Λ

GGM

FIG. 7: (Color online.) Behavior of quenched averaged GGM

for different anisotropy parameters in the spin glass system.

We consider a system of 8 spins. The vertical axis represents

the quenched averaged GGM. The different curves are for

different values of the anisotropic parameter γ. From bottom

to top, they are respectively for γ = 0.1 (black), 0.3 (red), 0.5

(blue), 0.7 (green), and 1.0 (magenta). The GGM is plotted

for the ground state of the spin glass system described by

the Hamiltonian HSG, and is plotted against the mean values

λ = J/h of the Gaussian quenched random variables Ji/h.

The standard deviations of the random variables are all unity.

Both the axes represent dimensionless quantities.

Summarizing, we have found that the ground states

of several quenched disordered quantum spin systems,

in the presence of certain ranges of the disorder parame-

ters, have higher genuine multipartite entanglement than

in their counterparts in systems without disorder. In al-

most the same range of parameters, the ground states of

disordered models turn out to be better carriers of classi-

cal information than their equivalents in systems without

disorder. The models considered include the quantum

anisotropic XY spin glass with and without a random

transverse magnetic field.

The calculations are performed by exact diagonaliza-

tions, and subsequent quenched averaging, whenever re-

quired. The results shown in the plots displayed in the

paper are given, for definiteness, for a specific value of the

anisotropy parameter γ. However, we have performed

the calculations for a large range of γ, and the general

qualitative behavior is similar for all γ. To exemplify the

genericity of the qualitative behavior obtained from the

previous plots in the paper, let us present a figure (Fig.

7) where we plot the quenched averaged genuine multi-

partite entanglement for different values of γ, in the spin

glass Hamiltonian HSG.

We term the phenomenon of increased genuine mul-

tiparty entanglement and increased multiport capacity

subsequent upon the introduction of disorder into the

system as order-from-disorder. Experimental and theo-

retical work in quantum information processing in the

last two decades or so have gradually convinced us that

the phenomenon of entanglement, observed in a variety

of physical systems, is not a fragile quantity. We hope

that this work will add to that increasing belief.

Acknowledgments

We acknowledge computations performed at the clus-

ter computing facility in HRI (http://cluster.hri.res.in/).

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