THE ANALYTICAL EXPRESSION OF THE CHERNOFF POLARIZATION OFTHE WERNER STATE

IULIA GHIU1,*, AURELIAN ISAR2,3

1University of Bucharest, Faculty of Physics, Centre for Advanced Quantum Physics,PO Box MG-11, RO-077125, Bucharest-Magurele, Romania

∗Corresponding author, E-mail: iulia.ghiu@g.unibuc.ro2Horia Hulubei National Institute for Physics and Nuclear Engineering,

RO-077125, Bucharest-Magurele, Romania3Academy of Romanian Scientists, 54 Splaiul Independentei, RO-050094, Bucharest, Romania

Received March 16, 2016

We review the quantum Chernoff bound and the quantum degree of polarizationbased on this bound. Then we find the analytical expression of the quantum degree ofpolarization based on the quantum Chernoff bound for the Werner state, as a functionof the parameter that defines this state.

Key words: Chernoff bound, quantum degree of polarization, Werner state.

PACS: 42.50.Dv, 42.25.Ja.

1. INTRODUCTION

Entanglement plays an important role in quantum information theory, in par-ticular for quantum information protocols and tasks like quantum teleportation [1]and its generalizations [2–5], quantum cryptography [6], superdense coding [7], andquantum computation [8].

During the last decades a lot of attention was paid to the study of quantumcorrelations, including quantum entanglement and quantum discord, of bipartite ormultipartite states. The time evolution of quantum correlations in systems consistingof two-bosonic modes interacting with a thermal environment was studied in therecent years in Refs. [9–15].

The set of states that remain invariant under local unitary transformations arecalled Werner states [16]. Popescu [17] proved that the Werner state of two qubits isuseful for quantum teleportation. The Werner states have attracted a lot of attentionof the quantum information community. Recently we have investigated the quantumdegree of polarization based on the quantum Chernoff bound [18]. The quantumChernoff bound provides the minimal error probability of discriminating betweentwo quantum states when many identical copies are available [19].

In this work we analyze in detail the quantum degree of polarization of theWerner state. The paper is organized as follows. In Sec. 2 we review the quantum

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2 The analytical expression of the Chernoff polarization of the Werner state 769

Chernoff bound as well as the definition of the quantum degree of polarization basedon this bound. In Sec. 3 we employ the quantum Chernoff bound as a measure ofpolarization of the Werner state. We find the analytical expression of the parameters that minimizes a function, which is required in the evaluation of the polarization.This gives the exact expression of the Chernoff degree of polarization of the Wernerstate. Our conclusions are outlined in Sec. 4.

2. QUANTUM CHERNOFF BOUND

Suppose that we get N identical copies of a quantum system, which are pre-pared in the same unknown state, which is either ρ or σ. Our task is to determine thenature of the given state with the minimal probability of error. When the two statesare equiprobable, the minimal error probability of discriminating them is [20], [21]

P(N)min (ρ, σ) =

1

2

(1− 1

2||ρ⊗N − σ⊗N ||1

), (1)

where ||A||1 := Tr√

A†A is the trace norm of a trace-class operator A. If the twostates are pure |Φ⟩ and |Ψ⟩, then the minimal error probability (1) has the simplerexpression [20]:

P(N)min (|Φ⟩⟨Φ|, |Ψ⟩⟨Ψ|) = 1

2

(1−

√1−|⟨Φ|Ψ⟩|2N

).

In the asymptotic limit, i.e. N → ∞, an upper bound P(N)QCB of the minimal

probability of error (1) was found to decrease exponentially with N [19], [21]:

P(N)QCB(ρ, σ)∼ exp[−NξQCB(ρ, σ)] .

The positive quantity

ξQCB(ρ, σ) :=− ln

[mins∈[0,1]

Tr(ρsσ1−s

)](2)

is called the quantum Chernoff bound [19], [22].The quantum analogues of the classical Renyi overlaps are denoted by Qs(ρ, σ)

[23]:Qs(ρ, σ) := Tr(ρsσ1−s). (3)

The quantum Chernoff bound represents a generalization of a classical problemformulated and solved by Chernoff in 1952 [24], namely one has to find the minimalerror distribution for discriminating two probability distributions in the asymptoticlimit.

The quantum Chernoff bound was recently used for defining the quantum de-gree of polarization of a two-mode state of the quantum radiation field [23], [25]:

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770 Iulia Ghiu, Aurelian Isar 3

PC(ρ) := 1−maxσ∈U

[mins∈[0,1]

Qs(ρ, σ)

], (4)

σ being the state that remains invariant under any polarization transformation.

3. THE ANALYTICAL FORMULA OF THE CHERNOFF QUANTUM DEGREE OFPOLARIZATION FOR THE WERNER STATE

In this Section we evaluate the Chernoff degree of polarization for the Wernerstate, a state that is defined as follows:

ρW = a |Ψ− ⟩⟨Ψ− |+(1−a)1

4I, (5)

where |Ψ− ⟩ is the singlet state and a is a parameter that satisfies a ∈ [0,1].The maximization over σ ∈ U of the function Qs leads to [18]:

Qs(a) =1

4[(1+3a)s+3(1−a)s] . (6)

In Fig. 1 we plot the function Qs in terms of both s and the parameter a that definesthe Werner state.

According to Eq. (4) the expression of the Chernoff degree of polarization is[18]:

PC(ρW ) = 1− mins∈[0,1]

Qs(a). (7)

We need to find the analytical expression of the parameter s that minimizesQs(a). Further one evaluates the first order derivative:

∂Qs(a)

∂s=

1

4

[(1+3a)s ln(1+3a)+3(1−a)s ln(1−a)

]The expression of s for which one obtains the minimum is

s=ln[− ln(1+3a)

3 ln(1−a)

]ln(1−a)− ln(1+3a)

. (8)

The dependence of s in terms of the parameter a is shown in Fig. 2.In conclusion, the exact expression of the Chernoff degree of polarization for

the Werner state is:

PC(ρW ) = 1− Qs(a). (9)

The plot of the Chernoff degree of polarization of the Werner state is shown in Fig.2 of Ref. [18].

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4 The analytical expression of the Chernoff polarization of the Werner state 771

Qs

0.0

0.5

1.0a

0.0

0.5

1.0

s

0.7

0.8

0.9

1.0

Fig. 1 – The function Qs in terms of s and the parameter a that defines the Werner state (see Eq. (6)).

0.0 0.2 0.4 0.6 0.8 1.00.30

0.35

0.40

0.45

0.50

0.55

0.60

a

s

Fig. 2 – The plot of the parameter s for which one obtains the minimum in Eq. (7). The analyticalexpression of s is given by Eq. (8).

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772 Iulia Ghiu, Aurelian Isar 5

4. CONCLUSIONS

In this paper we have found the exact expression of the quantum degree ofpolarization based on the Chernoff bound for the Werner state. The investigation ofthis topic started in Ref. [18], where a numerical study was performed. Here we havepresented in detail how one can obtain the expression of a parameter denoted by s,that minimizes the function Qs(a) given by Eq. (6). After getting its formula, onecan employ it for computing the exact expression of the polarization of the Wernerstate.

Acknowledgements. The work of Iulia Ghiu was supported by the Romanian National Author-ity for Scientific Research through Grant PN-II-ID-PCE-2011-3-1012 for the University of Bucharest.Aurelian Isar acknowledges the financial support received from the Romanian Ministry of Educationand Research, through the Projects CNCS-UEFISCDI PN-II-ID-PCE-2011-3-0083 and PN 16 42 0101/2016.

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