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Discrimination between pure states and mixed states

Chi Zhang,1, Guoming Wang,1, and Mingsheng Ying1,

1State Key Laboratory of Intelligent Technology and Systems,

Department of Computer Science and Technology Tsinghua University, Beijing, China, 100084(Dated: February 1, 2008)

In this paper, we discuss the problem of determining whether a quantum system is in a pure state,or in a mixed state. We apply two strategies to settle this problem: the unambiguous discriminationand the maximum confidence discrimination. We also proved that the optimal versions of bothstrategies are equivalent. The efficiency of the discrimination is also analyzed. This scheme alsoprovides a method to estimate purity of quantum states, and Schmidt numbers of composed systems.

PACS numbers:

I. INTRODUCTION

In many applications of quantum information, one ofthe important elements which affect the result of quan-tum process, is the purity of the quantum states pro-duced or utilized. Hence, an interesting and importantproblem in quantum information is to estimate the purityof a quantum system [1, 2, 3, 4, 5]. This problem is alsostrongly related to the estimation of the entanglement ofmultiparty systems [6, 7, 8].

However, all above references considered this problemonly in the simplest case of qubits, where both the defini-tion and computation about purity are clear. Estimatingthe purity of a general quantum system is still open. Inthis paper, we first consider an extreme situation: givensome copies of a quantum state, the task of us is to de-termine whether the state is pure or mixed. The processis called discrimination between pure states and mixedstates. Then, by counting different results obtained inthe above discriminations, we offer an effective method

to estimate the purity of quantum states. The idea ofdiscrimination between pure states and mixed states, wasfirst mentioned in Ref.[8]. However, they did not studythe problem formally and systematically, which is ouraim in this paper. There are two different strategies todesign the discrimination: the unambiguous discrimina-tion [9], and the maximum confidence discrimination [10].In the strategy of unambiguous discrimination, one cantell whether the quantum system is in a pure state or amixed state without error, but a non-zero probability ofinconclusive answer is allowed. In this paper, in order tosimplify the presentation, we use the term unambigu-ous in a more general sense: it allows the success prob-

ability to be zero in some situation. On the other hand,in the maximum confidence discrimination, an inconclu-sive answer is not allowed, and after each discrimination,we must give a statement whether the quantum state ispure, or mixed. The discrimination is so named, becauseif an answer is given, the probability of obtaining a cor-

Electronic address: zangcy00@mails.tsinghua.edu.cnElectronic address: wgm00@mails.tsinghua.edu.cnElectronic address: yingmsh@tsinghua.edu.cn

rect conclusion is maximized.

It is convenient to introduce some notations here. Inthe Hilbert space Hn, we use Hnsym to denote its sym-metric subspace [11]. The orthogonal complement ofHnsym is called the asymmetric subspace of H

n, and

denoted as Hnasym. We use (Hnsym) and (H

nasym) to

represent the projectors of these two subspaces respec-tively. In this paper, we prove that, given n copies ofa quantum state in Hilbert space H, the optimal un-ambiguous discrimination and the maximum confidencediscrimination can be carried out by the same measure-ment {0 = (Hnsym), 1 = (H

nasym)}. The difference

between these two discriminations comes only from thedifferent explanations of the outcomes. In the unambigu-ous discrimination, the outcome 0 is an inconclusive an-swer, and the outcome 1 indicates that the system is in amixed state. The drawback of the unambiguous discrim-ination is that, if the quantum system is in a pure state,people always fail to give a confirm answer. However, inthe maximum confidence discrimination, the outcome 0

indicates that the quantum system is considered to be ina pure state, and the outcome 1 indicates a mixed state.

There are two natural assumptions in this paper.First, the purity of quantum states is invariant underany unitary operation. Suppose the purity of a quan-tum state is represented by (), it must satisfy that() = (U U), for any unitary operator U. We alsoassume that, when is a pure state, () = 1, other-wise 0 () < 1. For instance, the usually used purityof quantum states, () = Tr2(), clearly satisfies theseconditions. Second, the priori probability distributions ofquantum states are also assumed invariant under unitaryoperations. Let us denote the priori probability density

function as (), then () = (U U), for any unitaryoperator U.

Our present article is organized as follows. In sec-tion.II, we provide the optimal unambiguous discrimi-nation between pure states and mixed states. And, insection.IV, we provide the maximum confidence discrim-ination between pure states and mixed states. We alsogeneralize the unambiguous discrimination between purestates and mixed states to a semi-unambiguous esti-mation for ranks of quantum states in section.III, whichcan also be seen as an estimation for the Schmidt num-

http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2http://arxiv.org/abs/quant-ph/0611122v2mailto:zangcy00@mails.tsinghua.edu.cnmailto:wgm00@mails.tsinghua.edu.cnmailto:yingmsh@tsinghua.edu.cnmailto:yingmsh@tsinghua.edu.cnmailto:wgm00@mails.tsinghua.edu.cnmailto:zangcy00@mails.tsinghua.edu.cnhttp://arxiv.org/abs/quant-ph/0611122v2

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ber of bipartite quantum systems. Finally, in section.V,we provide a strategy to estimate the purity of quantumstates.

II. OPTIMAL UNAMBIGUOUS

DISCRIMINATION

In this section, we consider the unambiguous discrim-ination between pure states and mixed states. Supposewe are given n copies of a quantum state, which is inthe Hilbert space H. The unambiguous discrimination isdescribed by a POVM measurement on the Hilbert spaceHn. The measurement is comprised by three positiveoperators, p, m, and ?, satisfying that

Tr(pn) = 0, (1)

for any mixed state ,

|nm|n = 0, (2)

for any pure state |, and? = I m p. (3)

Therefore, if the outcome is p, the system is assured tobe in a pure state; if the outcome is m, the system is ina mixed state; and outcome ? denotes an inconclusiveanswer.

The efficiency of the discrimination is

p =

0()

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where Ini is the ni -dimensional identity matrix. Conse-

quently, from Eq.(12)

p(?|) =

n!

mi=1

ni !

=

Pm

i=1

ni=n

n!

mi=1 ni!

mi=1

nii

mi=1 ni!

n!

=

Pmi=1

ni=n

mi=1

nii .

(15)

III. SEMI-UNAMBIGUOUS ESTIMATION OF

SCHMIDT NUMBER

As we know, the entanglement of a bipartite quan-tum system is closely related to the purity of one of itssubsystems. Whether a subsystem is in a pure state isequivalent to whether the total quantum system is in

a product state. Hence, the measurement given in theabove section also provides an unambiguous estimationfor entanglement of bipartite quantum systems. More-over, in this section, we will provide a natural generaliza-tion, which can be called semi-unambiguous estimationof the Schmidt number of bipartite systems.

The Schmidt number of a bipartite system equals tothe rank of the quantum state in each of its subsys-tems. Hence, estimating the Schmidt number is equiv-alent to estimating the rank of quantum states. First,let us reconsider the discrimination between pure statesand mixed states. In the discrimination, the m resultmeans that the rank of the state is no less than 2, whilethe inconclusive answer can also be considered as a triv-ial conclusion that the rank of the state is no less than1. Although the discrimination does not offer the ex-act value of the rank of the quantum state, it offers alower bound for the rank. Moreover, the lower bound isassured to be correct. In this mean, we can call the dis-crimination between pure states and mixed states also asemi-unambiguous estimation for the rank of quantumstates. A more general semi-unambiguous estimationof the rank of quantum states can be defined as a POVMmeasurement on Hn with operators {1, 2, , m},where m is the dimension of H. The measurement sat-isfies that for any quantum state whose rank is k,Tr(i

n) = 0, for any i > k. Thus, whenever the out-

come k is observed, we can make sure that the rank ofis no less than k.

Before providing the semi-unambiguous estimation ofthe rank of quantum states. We first introduce some fun-damental knowledge about group representation theoryneeded here. For details, please see Ref. [13].

A Young diagram [] = [1, , k], where

i = nand 1 2 k > 0, is a graphical representation ofa partition of a natural number n. It consists of n cells,arranged in left-justified rows, where the number of cellsin the ith row is i.

A Young tableau is obtained by placing the numbers1, , n in the n cells of a Young diagram. If the num-bers form an increasing sequence along each row andeach column, the Young tableau is called standard Youngtableau. For a given Young diagram [], the number ofstandard Young tableau can be calculated with the hooklength formula, and denoted by f[]. In this paper, we use

T[]r to denoted the rth standard Young tableau, where

r = 1, , f[].The Hilbert space Hn, where the dimension of H is

m, can be decomposed into a set of invariant subspacesunder operation Un, for any unitary operation U on H.Each of the subspaces corresponds to a standard Young

tableau T[]r , where the number of rows in [] is no more

than m. So, we can denote the subspaces as H[]r , and

denote its projector as (H[]r ). Then, we have

Hn =[],r

H[]r . (16)

For instance, the symmetric subspace Hnsym is just one

of these subspaces, H

1 .For a quantum state in H, whose rank is k, the sup-

port space ofn is in the sum of subspaces H[]r , where

the number of rows in Young diagram [] is no greaterthan k. Therefore, a semi-unambiguous estimation ofthe rank of quantum states can be designed as a POVMmeasurement {1, , m}, such that

i =

h([])=i

r

(H[]r ), (17)

where h([]) is the number of rows in []. As said above,if the rank of is k, Tr(i

n) = 0, for any i > k.Thus, once an i result is observed, we can assert thatthe rank of is no less than i. For n copies of a bi-partite quantum system, through measuring any of itssubsystems with the measurement given in Eq.(17), wecan semi-unambiguously estimate the Schmidt number ofthe whole system.

IV. MAXIMUM CONFIDENCE

DISCRIMINATION

In this section, we consider a different strategyfor determining whether the quantum system is ina pure state, which is called maximum confidence

discrimination[10].The discrimination is still a POVM measurement

{p, m}. But when the outcome is m, the quantumsystem is believed in a mixed state; otherwise, the out-come is p, and the quantum state is considered to bepure. A maximum confidence strategy is to maximizethe reliability of the conclusion, i.e., let the following twoprobabilities be maximized,

p(pure|p) =

()=1

()Tr(np)d()1

()Tr(np)d, (18)

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and

p(mixed|m) =

()

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V. ESTIMATING PURITY OF STATES

The maximum confidence discrimination between purestates and mixed states provides a natural intuition forthe purity of a quantum system, i.e., the greater theprobability of getting a 0 result, the closer it is to apure state. Hence, by repetitively performing the mea-surement, and counting the proportion of 0 results, we

can estimate the probability of judging the system be-ing pure, which, in some sense, reflects some informationabout the purity of the system. However, a more inter-esting conclusion is that, no matter how people definethe purity of quantum states, as long as it satisfies thecondition of unitary invariant, it can be well estimatedthrough a set of maximum confidence discriminations.

On the assumption of unitary invariant, the purity ofa quantum state , () = (diag(1, , m)), where1, , m are the eigenvalues of . Hence, () is afunction of its eigenvalues. Estimating the purity of aquantum state can be reduced to estimating the eigen-values of . The characteristic polynomial of is a poly-

nomial, whose roots are the eigenvalues, i.e.,

det(xI ) =mi=1

(x i) =mj=0

ajxmj . (29)

If we can successfully estimate every coefficient aj , j =0, , m, the eigenvalues can be estimated by solving theequation

mj=0 ajx

mj = 0.Recall the famous Vietes theorem, it is easy to know

a0 = e0(1, , m) = 1

a1 = e1(1, , m) = m

i=1

i

a2 = e2(1, , m) =

1i1

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