Randomized channel-state duality Bin Yan1 2and Nikolai A. Sinitsyn1 1Theoretical Division Los Alamos National Laboratory Los Alamos New Mexico 87545 USA

2025-04-29 0 0 631.62KB 15 页 10玖币

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Randomized channel-state duality

Bin Yan1, 2 and Nikolai A. Sinitsyn1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

2Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

Channel-state duality is a central result in quantum information science. It refers to the cor-

respondence between a dynamical process (quantum channel) and a static quantum state in an

enlarged Hilbert space. Since the corresponding dual state is generally mixed, it is described by a

Hermitian matrix. In this article, we present a randomized channel-state duality. In other words, a

quantum channel is represented by a collection of Npure quantum states that are produced from

a random source. The accuracy of this randomized duality relation is given by 1/N , with regard to

an appropriate distance measure. For large systems, Nis much smaller than the dimension of the

exact dual matrix of the quantum channel. This provides a highly accurate low-rank approximation

of any quantum channel, and, as a consequence of the duality relation, an eﬃcient data compres-

sion scheme for mixed quantum states. We demonstrate these two immediate applications of the

randomized channel-state duality with a chaotic 1-dimensional spin system.

Quantum channels are the most general framework for

describing dynamical quantum processes, from the time

evolution of closed or open quantum systems to quantum

communications between distant parties, and error cor-

rections on quantum computers. One of the most pow-

erful methods for investigating quantum channels is the

so-called channel-state duality [1–5]: For every quantum

channel, there exists a quantum state corresponding to

it. As a result, the dynamical information of the for-

mer can be fully encoded into the kinematic information

of the latter [6]. Hitherto, channel-state duality has be-

come a classic textbook result in quantum information

science. It not only oﬀers an elegant mathematical char-

acterization of the structure of quantum channels [7,8],

but also has a profusion of implications and applications

in various research areas, e.g., quantum process tomog-

raphy [9,10], non-local quantum correlations [11,12], or

non-Markovian quantum dynamics [13].

The dual state of a quantum channel “lives” in an en-

larged bipartite Hilbert space. In other words, for a chan-

nel that accepts an input state from a Hilbert space of

dimension da, and outputs a state with dimension db,

its corresponding dual state is a bipartite quantum state

with a Hilbert space dimension d=da×db. Additionally,

the dual state is in general a mixed quantum state, and

is therefore described by a density matrix—a Hermitian

matrix of dimension d×ddubbed the Choi matrix. The

rank of this matrix is also referenced as the rank of the

corresponding channel.

Although a quantum channel has a precise Choi matrix

representation, eﬃciently ﬁnding its low-rank approxi-

mate [14–16] still remains a challenging problem. Such

an approximation is highly desirable, because it can sig-

niﬁcantly reduce the complexity of describing and assess-

ing the channel’s properties. On the other hand, as a

consequence of the channel-state duality, this problem is

equivalent to ﬁnding a low-rank matrix approximation

of the channel’s Choi matrix. The latter problem is of

importance on its own [17–20], which has relevance in

areas even outside physics, such as engineering and data

sciences .

In this article, we introduce a randomized channel-state

duality. Instead of a single density matrix, we convert

the channel to a set of Npure states in the Hilbert space

of the same dimension d(Figure 1). These pure states

are all produced from a random source of input. Here,

the ﬁrst moment of these pure states—averaged with re-

spect to the probability distribution of the initial random

input—creates an exact dual state (density matrix) of the

quantum channel. Given that we employ Nrandom pure

state realizations, the average of these pure states serves

as a good approximation of the exact density matrix,

with a precision (quantiﬁed by the variance of a proper

distance measure) given by a factor of 1/N.

As a result, using N d-dimensional vectors, we can

approximate the precise dual state with a high degree of

accuracy. Nis set to meet the desired precision. It is

independent of, and much smaller than, the dimension d

for large systems.

0.35 0.14 ··· 0.23

0.280.13 0.29

0.33

···

0.18 0.15

···

....

da×db

0.15

0.21

0.13

0.23

0.12

0.10

· ·

FIG. 1. Channel-state duality. A quantum channel in-

duced by the unitary evolution of an interacting spin chain

system. The channel input is the state of the entire spin

chain (blue), whose Hilbert space dimension is da. The out-

put is the reduced state of a subsystem (red), with dimension

db. Through channel-state duality, this channel can be rep-

resented by a (generally mixed) quantum state in a da×db-

dimensional Hilbert space, known as the Choi matrix. We

show that the same channel can be described by a set of N

pure quantum states (hence vectors) of the same dimension,

generated from random sources. Here Ndetermines the pre-

cision of the representation.

arXiv:2210.03723v1 [quant-ph] 7 Oct 2022

a: Channel-state duality b: Randomized duality c: Concentration of measure

HbH

σX

U†

|Ψk⟩

|ψk⟩

f(ψ)

⟨f⟩]ϵ

Pk|ψk⟩⟨ψk| ∼ I

FIG. 2. Randomized channel-state duality. (a) A unitary induced channel (red portion) can be represented by a quantum

state σXin an extended Hilbert space, via the standard Jamio lkowshi-Choi isomorphism (2). (b) Randomized channel-state

duality maps the same channel to a set of quantum states |Ψki,k= 1,··· , N , from a set of random input states |ψki, as

deﬁned in equation. (5). The mixture of a few random states |ψkican approximate with very high accuracy the maximally

mixed state. This is in analog to the concentration of measure (c) in high dimensional spaces, where typical values of a smooth

function are close to the averaged value. Therefore, the random input |ψkican be replaced with a system that is maximally

entangled with an ancillary system (see the gray shaded region). The diagram in (b) together with the maximally entangled

input state is equivalent to the Jamio lkowshi-Choi representation, up to a local basis rotation U†⊗U†on the initial bipartite

canonical maximally entangled state.

Randomized dual states

Let us start by formulating the conventional channel-

state duality. A quantum channel is formally deﬁned as a

linear map X:L(Ha)→L(Hb) that transfers linear op-

erators on Hilbert space Hato Hb, whose dimensions are

respectively daand db.Xis demanded to be completely

positive and trace-preserving. These properties guaran-

tee the existence of the operator sum representation [21]

(Kraus representation) of channel X, i.e.,

X(ρ) =

k=1

MkρM†

k,X

M†

kMk=I. (1)

Here, Iis the identity operator. The minimal value of r

is the Kraus rank (or Choi rank) of X. To get the dual

state of X, consider a maximally entangled state |φ+iin

the composed Hilbert space Ha⊗Ha, and apply Xto one

of its subspace. This is also known as the Jamio lkowshi-

Choi isomorphism [22–24]:

X→σX≡I⊗X|φ+ihφ+|.(2)

Here, Iis the identity map. The canonical maximally

entangled state |φ+i ≡ Pi|iii/√dais represented in the

computational basis. This correspondence is illustrated

in Fig. 2a.

The dual state σX—known as the Choi matrix—is of

dimension d=da×db. It fully characterizes quantum

channel X, in the sense that any dynamical information

of the channel can be extracted from the dual state alone.

More precisely, for any Hermitian operators Aand Bthat

apply on Haand Hb, respectively, we have [1,6]

tr [X(A)B] = da·tr σXAt⊗B,(3)

where Atdenotes the matrix transpose of Ain the com-

putational basis. Note that the rank of the Choi matrix

is identical to the Kraus rank of the corresponding chan-

nel. Therefore, a low-rank (approximate) representation

of the Choi matrix directly gives rise to a low-rank rep-

resentation of the channel, and vice versa.

For the sake of transparency, let us present the ran-

domized channel-state duality for a special type of chan-

nel. We will generalize it later to generic channel. Con-

sider a channel Xinduced by a unitary evolution U. The

input Hilbert space Hamatches the full dimension of the

unitary, while the output Hilbert space Hbis a subspace

of Ha(Figure 1). Xcan be formally deﬁne as

X(ρ) = tr¯

bUρU†,(4)

where tr¯

bis the partial trace over the complement of Hb.

We then map Xto a pure state, i.e.,

X→ |Ψi ≡ I⊗U†|φ+i⊗|ψi.(5)

This transformation is illustrated in Fig. 2b (blue shaded

area). Here, |φ+iis the canonical maximally entangled

state in the bipartite Hilbert space Hb⊗Hb—tensor prod-

uct between the output Hilbert space and an ancillary

Hilbert space of the same dimension. |ψiis a random

state on the complement of Hb. Note that the identity

map applies on one subsystem of |φ+i.U†applies to the

other subsystem of |φ+itogether with |ψiTherefore, the

resulting pure state has the same dimension das the Choi

matrix. We also assume that the initial state |ψiis drawn

from an ensemble that forms a quantum state 2-design

[25]. With respect to the probability distribution of the

input ensemble, the ﬁrst moment of the output state |Ψi

is a density matrix

ρX≡Zdψ |ΨihΨ|.(6)

Here, the integral is performed with respect to the prob-

ability measure of the initial random input state |ψi.

⟨σz⟩:⟨σy⟩:

⟨O⟩

0 0.5

0246

⟨O⟩

−0.4 0 0.2

⟨O⟩

100101102

10−1

100

||ρN−ρX||2

nnanb

9 1 4 32

10 4 3 128

11 7 2 512

rank

n−na

a: Weak and strong thermalization b: Low rank approximation of density matrix

FIG. 3. (a) Weak and strong thermalization for a chaotic spin chain system with 12 spins. Left: Time evolution of the

expectation values of single spin observables. For weak (top) and strong (bottom) thermalization, the initial states of the spins

are polarized in the Zand Ydirection, respectively. Solid and dashed curves are direct numerical simulations of the evolution.

Markers correspond to the averaged value evaluated with N= 200 randomized dual states. Error bars show the conﬁdence

intervals of 3-sigma [3σNas deﬁned in (16)] of the data point. Right: Scatter plot of the standard deviation σdeﬁned in (12),

which is below the predicted upper bound. (b) Trace distance between ρXin (6) and its rank Napproximation ρest

X(8), for

various rank of ρXand N. The channel is generated by a unitary evolution of a n-site spin chain (17). The channel’s input

(output) space is the space of the ﬁrst na(nb) spins. The dotted curve is the predicted averaged scaling 1/√N. The dashed

curve is away from the average value by the predicted upper bound of the standard deviation.

This density matrix provides an exact characterization

of channel X, similar to the Choi matrix, through,

tr [X(A)B] = da·tr ρXA⊗Bt.(7)

Therefore, we get a new channel-state duality with the

exact dual state ρX.

Rigorous proof of the above duality relation is del-

egated to supplemental information. We now oﬀer a

more heuristic explanation: If one applies the standard

Jamio lkowshi-Choi isomorphism (2) not to |φ+i, but to a

maximally entangled state in a rotated basis other than

the computational basis, i.e., U†⊗U†|φ+i, one gets a new

transformation represented by a circuit diagram shown in

Fig. 2b including the grey shaded area. For an observer

who only has access to the space of the ﬁnal output states

|Ψi, the reduced state of one subsystem of the bipartite

maximally entangled state is indistinguishable from the

maximally mixed state. Hence, one can replace the max-

imally entangled input state (grey area in Fig. 2b) with

the maximally mixed state, which can be further approx-

imated by a collection of random pure states |ψki.

From this point of view, the exact density matrix ρX

is not special compared to the standard Choi matrix σX.

In fact, as evidenced by their duality relations (3) and

(7), they are connected by a global transpose, which is

an anti-unitary operation. However, the crucial point is

that one can approximate ρXwith Nrealizations of the

output pure state |Ψi, whose average serves as a good

estimator of ρX:

ρest

X≡1

k=1 |ΨkihΨk|.(8)

As will be seen, we can achieve a high accurate approxi-

mation with only a relatively small number N.

Bounding the variance

The idea underlying the above low-rank approxima-

tion is the typicality of quantum states among a ran-

dom ensemble. In our case, expectation values of observ-

ables evaluated on a single random dual state realization

|Ψiare highly likely to be around the averaged values

of many realizations (Figure 2c). Quantitatively, the

averaged distance between the exact dual state ρXand

the estimator ρest

Xwith Npure states can be bounded as

(supplemental information)

Zdψ||ρest

X−ρX||2≤r1

N.(9)

Here ||X||2≡√trXX†is the Hilbert-Schmidt norm, and

dψ≡QkdψkMoreover, the variance of the distance is

suppressed by Nas well, i.e.,

Zdψ||ρest

X−ρX||2

2≤1

N.(10)

This gives an upper bound 1/√Nfor the standard devi-

ation of the distance. Since ρest

X, as the ﬁrst moment of

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Randomizedchannel-statedualityBinYan1,2andNikolaiA.Sinitsyn11TheoreticalDivision,LosAlamosNationalLaboratory,LosAlamos,NewMexico87545,USA2CenterforNonlinearStudies,LosAlamosNationalLaboratory,LosAlamos,NewMexico87545,USAChannel-statedualityisacentralresultinquantuminformationscience.Itreferstothecor...

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Randomized channel-state duality Bin Yan1 2and Nikolai A. Sinitsyn1 1Theoretical Division Los Alamos National Laboratory Los Alamos New Mexico 87545 USA

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