Subsystem Trace-Distances of Two Random States Joaquim Telles de Miranda1and Tobias Micklitz1 1Centro Brasileiro de Pesquisas F ısicas Rua Xavier Sigaud 150 22290-180 Rio de Janeiro Brazil

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Subsystem Trace-Distances of Two Random States

Joaquim Telles de Miranda1and Tobias Micklitz1

1Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil

(Dated: May 30, 2023)

We study two-state discrimination in chaotic quantum systems. Assuming that one of two N-qubit

pure states has been randomly selected, the probability to correctly identify the selected state from

an optimally chosen experiment involving a subset of N−NBqubits is given by the trace-distance

of the states, with NBqubits partially traced out. In the thermodynamic limit N→ ∞, the average

subsystem trace-distance for random pure states makes a sharp, ﬁrst order transition from unity

to zero at f= 1/2, as the fraction f=NB/N of unmeasured qubits is increased. We analytically

calculate the corresponding crossover for ﬁnite numbers Nof qubits, study how it is aﬀected by the

presence of local conservation laws, and test our predictions against exact diagonalization of models

for many-body chaos.

I. INTRODUCTION

The capability for storing and processing quantum in-

formation relies on the ability to discriminate between

quantum states. Quantifying distinguishability of quan-

tum states, speciﬁcally when only access to subregions is

available, is thus a problem of fundamental and practical

interest. In chaotic systems initially localized informa-

tion is ‘scrambled’ into the many degrees of freedom of

the system, and subsystem density matrices of generic

pure states are nearly indistinguishable from fully ther-

mal states. The ‘nearly’ was speciﬁed by Page for pure

random states, which also serve as proxies for eigenstates

of chaotic systems, in his classic paper [1]. There he

showed that for these states the information stored in the

smaller subsystem, deﬁned as the deﬁcit of the subsystem

entanglement entropy SAfrom the maximum entropy of

a fully mixed state, is on average IA≡ln DA−SA=

A/2D. Here DA, and Dare the Hilbert space dimen-

sions of the smaller subsystem, respectively, the joint sys-

tem, and corrections small in 1/D have been neglected.

For reasonably large systems the entanglement entropy

is self averaging and the average SAis also typical [2, 3].

Conservation laws generally reduce entanglement, in-

creasing thus the capability to store information [2, 4–11].

If multiple charges are conserved, entanglement is (on av-

erage) promoted if the charges fail to commute with each

other. That is, Page curves for non-commuting charges

lie above that of commuting charges, as recently pointed

out in Ref. [12]. Morevover, in the presence of locally con-

served charges, the entire charge distributions of initial

states are conserved. The largest amount of information

can then be stored in states with the broadest charge

distribution [13].

While Page’s formula provides information on subsys-

tems, it does not give an answer to state-discrimination in

fully information scrambling systems. Speciﬁcally, imag-

ine one of two known random pure states ρ,σ, both com-

posed of Nqubits, has been randomly selected. What

is the (average) probability Pρσ that performing an opti-

mally chosen experiment on NAof the qubits we correctly

identify the selected state, and how is Pρσ aﬀected by

conservation laws? In this paper we want to investigate

these questions, and the outline is as follows: We start

brieﬂy reviewing in Section II the concepts of the trace-

distance D1, its generalization the Schatten n-distances

Dn, Page states, and the calculation of D1from Dnvia

the replica trick. We then discuss in Section III the com-

binatorics involved in the calculation of average subsys-

tem Schatten n-distances of random pure states. In Sec-

tion IV we analyze the average subsystem trace-distances

of random pure states and consequences of local conser-

vation laws. We conclude in Section V with a summary

and discussion, and give further technical details in the

Appendices.

II. SCHATTEN-DISTANCES, PAGE STATES,

AND REPLICA TRICK

Optimal quantum state discrimination is generally

challenging, and the only completely analyzed case is

for two states, see e.g. Ref. [14] for a review. The

trace-distance provides a natural metric for two-state

discrimination. According to the Holevo–Helstrom the-

orem the best success probability for the latter is en-

coded in the 1-Schatten- or trace-distance as Pρσ =

2(1 + D1(ρ, σ)) [15]. General Schatten n-distances here

are deﬁned as Dn(ρ, σ)≡1

21/n ||ρ−σ||n, with n-norm

of a matrix Λ determined by its singular values λias

||Λ||n= (Piλn

i)1/n [16]. Notice that all Schatten dis-

tances are symmetric in the inputs, positive semi-deﬁnite,

equal to zero if and only if inputs are identical, and obey

the triangular inequality. That is, they all satisfy the

properties of a metric, with normalization here chosen

such that 0 ≤Dn≤1. Our focus here is on two-state

discrimination and we thus concentrate on the 1-distance.

Consider then a D-dimensional Hilbert space with

entanglement-cut bi-partitioning the total system into

subsystems A,Bof dimensions DAand DB, respectively,

with DADB=D. Without much loss of generality, we

focus here on Nqubit systems parametrized by (a, b),

where the NA-bit vector alabels the DA= 2NAstates

of subsystem Aand bthe DB= 2NBstates of B, with

arXiv:2210.03213v3 [quant-ph] 26 May 2023

NB=N−NA. Following Page, we then consider two

random pure states |ψα⟩=Pa,b ψα

ab|a, b⟩,α=ρ, σ, with

Gaussian distributed complex amplitudes ψα

ab, chosen to

have zero mean and variances

⟨ψα

ab ¯

ψβ

cd⟩=1

Dδacδbdδαβ.(1)

|ψα⟩describe inﬁnite temperature thermal states of

generic chaotic systems, and using Eq. (1) we employ that

correlations induced by the normalization constraint are

negligible for reasonable large systems D≫1. Tracing

out subsystem B, information is lost and mixedness of

the reduced density matrices,

ρA= trB(|ψρ⟩⟨ψρ|), σA= trB(|ψσ⟩⟨ψσ|),(2)

increases with the number of partially traced qubits NB.

To ﬁnd trace-distances of Eq. (2) we employ the replica

trick recently discussed in Ref. [17]. We ﬁrst calculate

Schatten-distances for general even integer n, analyti-

cally continue to real n, and ﬁnally take the limit nto

unity,

⟨D1(ρA, σA)⟩=1

2lim

n→1⟨tr (ρA−σA)n⟩.(3)

Restricting to even integers here is important, since cor-

responding expression for odd integers vanish (see also

below), and a replica limit for the latter is thus trivially

zero [17]. Expanding powers in Eq. (3), we are confronted

with the averages

⟨trA(ρA−σA)n⟩

= sgn(σ)⟨ψα1

a1b1¯

ψα1

a2b1ψα2

a2b2¯

ψα2

a3b2···ψαn

anbn¯

ψαn

a1bn⟩,(4)

where sums over repeated indices αi=ρ, σ,ai=

1, .., DA,bi= 1, ..., DBare implicit, and the sign-factor

is sgn(σ) = ±1 if an even/odd number of density ma-

trices σAis involved in the product. Following previous

works [18] the bookkeeping of index conﬁgurations enter-

ing the products is conveniently done in a tensor network

representation shown in Fig. 1. The solid lines here in-

dicate how the indices of matrices ψ¯

ψαβ

ab;a′b′=ψα

ab ¯

ψβ

a′b′

are constrained due to matrix multiplication in subspace

A, subsystem-traces over B, and state indices α=ρ, σ,

respectively (see also ﬁgure caption). Further constraints

then arise from Gaussian averages Eq. (1). These are in-

dicated by the red lines, keeping track of Hilbert space

and state indices after contractions. For Page states, each

of the n! contributions resulting from the Gaussian aver-

ages of the 2ncomplex amplitudes in Eq. (4) is weighted

by an overall factor 1/Dn, and terms can be organized

according to the numbers of free subspace summations,

or ‘cycles’, as we discuss next.

III. COMBINATORICS OF AVERAGES

It is instructive to ﬁrst focus on the contribution in-

volving only a single state density matrix, say ρA,

⟨trA(ρn

A)⟩=⟨ψρ

a1b1¯

ψρ

a2b1ψρ

a2b2. . . ψρ

anbn¯

ψρ

a1bn⟩.(5)

tr(ρA−σA)4

p=id.

p= (4,3,2,1,6,5)

FIG. 1: Tensor network representation of averages Eq. (4).

Top left: Representation of a pair of amplitudes from the ex-

pansion αA=Pa,a′Pbψα

ab ¯

ψα

a′b|a⟩⟨a′|, with α=ρ, σ. Each

dot represents an index to be contracted, and contractions

must be between right- and left-side indices. Top right: Struc-

ture of tr(ρA−σA)4, with black lines representing index con-

tractions resulting from matrix multiplication in subspace A

(top line), traces in subspace B(middle line), and state in-

dices α=ρ, σ (bottom line). Notice that the index structure

of states follows that of subspace B. When averaging the trace

we only consider leading order contributions in 1/D resulting

from non-crossing permutations. Gaussian averages then im-

pose further contractions upon the structural ones, indicated

by the red lines. Middle: Resulting index structure for n= 6

and identity permuation p=id. This establishes six B-cycles

each consisting of one element, i.e. in the notation of main

text Λ6= (16,20,30,40,50,60). Bottom: Another example

of a non-crossing permutation for n= 6, p= (4,3,2,1,6,5).

This permutation establishes three B-cycles each consisting

of two elements, Λ6= (10,23,30,40,50,60). In the middle

diagram, contributions from states ρ(positive sign) and σ

(negative sign) sum to zero in each of the one-element cycles.

In the bottom diagram contributions from ρand σboth come

with positive sign and sum to two, i.e. contributions from the

three cycles add up to 23= 8.

These averages have been recently discussed in the con-

text of the entanglement entropy [19], and the main ob-

servations are [20, 21]: (i) the n! contributions result-

ing from the average of 2nGaussian distributed com-

plex variables can be organized as a sum over the per-

mutation group ⟨trA(ρn

A)⟩=1

DnPp∈SnDC(π−1◦p)

ADC(p)

where C(p) is the number of cycles in the permutation p

and πdeﬁned by π(i) = (i+ 1)mod(n), (ii) the maximal

number of cycles is C(π−1◦p) + C(p) = n+ 1 and is re-

alized by the non-crossing permutations, (iii) their com-

binatorics is encoded in the Narayana numbers N(n, k)

where kthe number of cycles in B, and (iv) contribu-

tions of crossing permutations are suppressed in pow-

ers of 1/D. Neglecting the latter, one thus arrives at

⟨trA(ρn

A)⟩=1

DnPn

k=1 N(n, k)Dn−k+1

ADk

B, which can be

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SubsystemTrace-DistancesofTwoRandomStatesJoaquimTellesdeMiranda1andTobiasMicklitz11CentroBrasileirodePesquisasF´ısicas,RuaXavierSigaud150,22290-180,RiodeJaneiro,Brazil(Dated:May30,2023)Westudytwo-statediscriminationinchaoticquantumsystems.AssumingthatoneoftwoN-qubitpurestateshasbeenrandomlyselected,...

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Subsystem Trace-Distances of Two Random States Joaquim Telles de Miranda1and Tobias Micklitz1 1Centro Brasileiro de Pesquisas F ısicas Rua Xavier Sigaud 150 22290-180 Rio de Janeiro Brazil

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