The link between Fisher information and geometric discord

Lowe, A.; Yurkevich, I. V.

doi:10.1063/10.0010204

By considering an arbitrary two-qubit state, it is shown that the Fisher information is intrinsically linked to the geometric discord which allows a measure for quantum correlations beyond entanglement. The complex amplitude of oscillations of the probability density function is upper bounded by the geometric discord which subsequently results in the Fisher information being bounded by the geometric discord. This gives an experimental observable which can be used to quantify quantum correlations beyond entanglement. This observable can be used to witness quantum correlations in an interferometry experiment, and provide another avenue for quantum technologies to continue to develop.

1. INTRODUCTION

The link between quantum mechanics and measurement has been a widely debated topic since the advent of quantum theory. Specifically, the paradox proposed by Einstein, Podolsky, and Rosen (EPR)¹ led to far reaching philosophical discussions about the interpretation of quantum theory. This debate was laid to rest by the experiments on Bell states² which confirmed the violation of the Bell inequalities³ which proved that quantum mechanics appears to have non-local features.

Furthermore the notion of quantum correlations was believed to exist only in the form of quantum entanglement (QE). QE originally was the inspiration for the EPR paradox and was widely believed to be equivalent to any form of quantum correlation. This notion was disproved when quantum discord was introduced at the start of the 21st century by two independent papers.^4,5 Quantum discord shows that a system can have zero entanglement, and yet quantum correlations still persist. This has led to wide range of theoretical research in the past twenty years by attempting to understand how to classify these correlations, and importantly how to witness, and then make use of them.

The techniques for classifying quantum correlations are beginning to widen into the realm of classical information theory. This is to be expected, since the definition for quantum discord relies on the Shannon entropy, and when discussing quantum measurements, the von-Neumann entropy. Subsequently, this has been extended to classical Fisher information (FI) and in more recent times, quantum Fisher information.⁶ The link between these classifications, and whether they are useful for witnessing quantum correlations beyond entanglement remains an active area of research.

However, for all of this intensive research, experimentally witnessing quantum correlations remains a challenge. In particular, much of the literature focuses on witnessing entanglement,^7–9,10 and the literature that includes quantum discord is typically focused on few distinguishable particles,^11,12 although there have been claims to witness quantum discord through Aharanov–Bohm oscillations.¹³ This paper looks to extend upon a similar route, by providing a different witness for quantum correlations for a general two-qubit scenario. The main benefit of this research is the experimental convenience and simplicity.

It is important to first introduce the different definitions of quantum correlations beyond entanglement for clarity, and further reveal current theoretical methods for quantum metrology. This will allow direct comparisons with the results of this paper and the standard approaches.

2. QUANTUM DISCORD

If a system in the state ρ (density matrix) is partitioned into two subsystems, A and B, each being described by the marginals (reduced density matrices) ρ_A = tr_B ρ and ρ_B = tr_Aρ, the total correlation between two subsystems is measured by the quantum mutual information,

I (ρ) = S (ρ_{A}) + S (ρ_{B}) - S (ρ),

(1)

where the von Neumann entropy is S(ρ) = −trρlog₂ ρ. The total correlation can also be written using the conditional entropy S(B | A) = S(ρ)−S(ρ_A),

I (ρ) = S (ρ_{B}) - S (B | A) .

(2)

The classical correlation between subsystems is defined from a measurement prospective and it is given by

J (ρ) = S (ρ_{B}) - \sum_{a} p_{a} S (ρ_{B | a}),

(3)

where the second term is the post-measurement conditional entropy which depends on the measurement choice. In the case of a positive operator valued measurement (POVM) on the subsystem A with a set of local (acting on subsystem A) Kraus operators K_a, the post-measurement state becomes

ρ \to ρ^{'} = \sum_{a} K_{a} \cdot ρ \cdot K_{a}^{†}, \sum_{a} K_{a}^{†} K_{a} = 1,

(4)

where the conditional on the measurement outcome a density matrix is

ρ_{B | a} = \frac{1}{p_{a}} t r_{A} K_{a}^{†} K_{a} ρ, p_{a} = tr K_{a}^{†} K_{a} ρ .

(5)

The classical information can be accessed without change of the system state. The difference between the total correlation and the maximum of classical correlation is called quantum discord:

D (A : B) = min_{{K_{a}}} [I (ρ) - J (ρ)] .

(6)

It measures the information accessible without changing the state. In the case of projective measurement when Kraus operators become rank-one projectors,

K_{a} \to Π_{a} \otimes 1_{B}, Π_{a} Π_{a^{'}} = δ_{a a^{'}} Π_{a}, \sum_{a} Π_{a} = 1_{A},

(7)

one can use that after projective measurement,

ρ \to ρ' = \sum_{a} Π_{a} \otimes 1_{B} \cdot ρ \cdot Π_{a} \otimes 1_{B} = \sum_{a} p_{a} Π_{a} \otimes ρ_{B | a},

(8)

and

ρ_{B^{'}} = ρ_{B}, S (ρ^{'}) - S (ρ_{A^{'}}) = \sum_{a} p_{a} S (ρ_{B | a}),

(9)

to derive

D (A : B) = min_{{Π_{a}}} [I (ρ) - I (ρ^{'})] .

(10)

When the quantum mutual information equals the maximal classical mutual information, the quantum discord is zero. This is the definition of a classical state with no quantum correlations. This is possible if and only if there exists a projective measurement that does not disturb the state, i.e., ρ′ = ρ.

Minimization of the quantum mutual information is a challenging task. One may use another measure for quantum discord where optimization over projective measurements is an easier task to perform. We can use as a measure the quantity similar to geometric discord.¹⁴

D (ρ) = \frac{1}{2} min_{{Π_{a}}} tr {[ρ - ρ^{'}]}^{2},

(11)

where

ρ^{'} = \sum_{a} Π_{a} \otimes 1 \cdot ρ \cdot Π_{a} \otimes 1

⁠.

This measure does not have all properties required in an axiomatic construction of a measure (see, for example, discussion of a similar problem with the geometric discord,¹⁵ but it certainly quantifies the discord defined by Eq. (10).

3. QUANTUM METROLOGY

3.1. Fisher information

One of the main tasks in quantum information theory is to measure quantum correlations experimentally. The main task in metrology is to optimize measurements to obtain the best estimate of an unknown parameter defining interaction between the probe and its environment. The tasks are in fact related to each other as it will be shown below.

Sampling a random variable x described by a distribution function p(x) generates, after M measurements, a list of random numbers {x₁, x₂,

\dots

⁠, x_M}. If the distribution function depends on an unknown parameter

θ

⁠, one can use the data x = (x₁, x₂,

\dots

⁠, x_M) to reconstruct the value of this parameter of the distribution function p(x|

θ

⁠). Introducing the trial function, an estimator 0_est(x), such that its mean value is equal to the true value (unbiased estimator),

θ = ⟨ θ_{est} (x) ⟩ \equiv \int d^{M} x θ_{est} (x) \prod_{i = 1}^{M} p (x_{i} | θ),

(12)

one can show that the precision (Cramer–Rao bound),¹⁶

δ θ^{2} \equiv ⟨ {[θ_{est} (x) - θ]}^{2} ⟩ \geq \frac{1}{M F (θ)},

(13)

is related to another information-theoretic quantity—the Fisher information:

F (θ) = \int d x \frac{{[\partial_{θ} p (x | θ)]}^{2}}{p (x | θ)} .

(14)

The Fisher information measures sensitivity of the distribution function to the change of the parameter $θ$ to be estimated.

Working with quantum states, the distribution function is written according to the Born rule:

p (x | θ) = tr Π_{x} ρ (θ),

(15)

where ρ is the state (density matrix) of a probe and

Π_{x} = | x ⟩ ⟨ x |

is the projector onto the state vector

| x ⟩

corresponding to the observed outcomes x.

The upper bound of FI is known as the quantum Fisher information (QFI) F_q,

F (θ) \leq F_{q} (θ) = tr ρ (θ) L^{2} (θ),

(16)

where

L (θ)

is the symmetric logarithmic derivative,

\partial_{θ} ρ (θ) = \frac{1}{2} [L (θ) ρ (θ) + ρ (θ) L (θ)] .

(17)

The QFI for a separable state is bounded by

F_{q}^{sep} \leq N

⁠, where N is the number of distinguishable particles, leading to the shot noise limit

δ θ \geq N^{- 1 / 2}

⁠. For the metrological purposes, the QFI must be made as large as possible, i.e., given

θ

-dependent evolution of the state

ρ (θ) = U (θ) ρ U^{†} (θ),

(18)

one has to maximize the QFI preparing the best state. The optimal state depends on the task, i.e., evolution operator

U (θ)

⁠. This brings another challenge to answer the question whether a unitary operation on one subsystem can be hidden from the observer. Considering a bipartition into a target system (A) and control system (B), the unitary operation performed by one party,

ρ (θ) = U_{A} (θ) \otimes 1 \cdot ρ \cdot U_{A}^{†} (θ) \otimes 1,

(19)

must be detected by the final global measurement performed on the entire system. The answer to this question was suggested earlier by defining a new measure called Interferometric Power (IP).¹⁷ This approach relies on the minimization of the QFI over the evolution operators U_A:

P (ρ) = \frac{1}{4} min_{U_{A}} F (ρ, U_{A}) .

(20)

What was found is that the IP is zero for zero-discord states, and non-zero for discordant states.

Our claim is that the upper bound of the FI is proportional to the discord.

4. RELATION BETWEEN FI AND GEOMETRIC DISCORD

In an attempt to witness discord, it is important to start with a bipartite system in order to be consistent with the original definition. Since the measurement in geometric discord was performed on the A subsystem, the evolution of the unitary operator should occur on the A subsystem for consistency. This unitary evolution is given by

P (θ) = U (θ) ρ U {(θ)}^{†}, U (θ) = e^{i θ n \cdot σ / 2} \otimes 1,

(21)

where n denotes the unit Bloch vector and σ stands for the vector of Pauli matrices. The evolution operator can be written as a spectral decomposition in terms of two orthogonal projectors,

U (θ) = e^{i θ / 2} Π_{+} + e^{- i θ / 2} Π_{_},

(22)

onto the eigenstates of nσ,

Π_{\pm} = | \pm n ⟩ ⟨ \pm n | \otimes 1, n σ | \pm n ⟩ = \pm | \pm n ⟩ .

(23)

The probability function has two contributions,

p (x | θ) = tr Π_{x} ρ (θ) = C_{x} + A_{x} e^{i θ} + {\bar{A}}_{x} e^{- i θ},

(24)

where the non-oscillating part depends on the classical-quantum state ρ′ that would emerge if one performed a projective measurement with elements Π_±,

C_{x} = tr Π_{x} ρ^{'}, ρ^{'} = \sum_{a = \pm} Π_{a} ρ Π_{a},

(25)

while the oscillating part has a complex amplitude

A_{x} = tr Π_{x} \tilde{ρ}, \tilde{ρ} = Π_{+} ρ Π_{_} .

(26)

Applying the Cauchy-Schwarz inequality

| A_{x} |^{2} = {| tr Π_{x} \tilde{ρ} |}^{2} \leq tr {\tilde{ρ}}^{†} \tilde{ρ} = tr Π_{+} ρ Π_{_} ρ

(27)

= \frac{1}{2} tr (Π_{+} ρ Π_{_} + Π_{-} ρ Π_{+}) ρ

(28)

= \frac{1}{2} tr (ρ - ρ^{'}) ρ = \frac{1}{2} tr {(ρ - ρ^{'})}^{2},

(29)

where we used that

tr (ρ - ρ^{'}) ρ^{'} = 0

⁠. This result means that the amplitude of oscillations is bounded by the discord:

| A_{x} |^{2} \leq D (ρ) .

(30)

The fact that the visibility of the oscillations of the probability function is bounded by discord means that FI defining the precision of the measurements in metrology, is also bounded:

F (ρ) = 4 \int \frac{d x}{p (x | θ)} {| A_{x} |}^{2} \cos^{2} (θ + ϕ_{x}) \leq c (θ) D (ρ),

(31)

ϕ_{x}

is the phase of the complex amplitude A_x, and

c (θ)

is a non-universal constant. This result shows that operations on a target system can always be hidden when only classical states are used for detection.

The proposed protocol has obvious benefits of experimental ease, and could be used to further confirm quantum correlations beyond entanglement. Furthermore, this also highlights the benefits of using FI in terms of quantum metrology. Whilst this method slightly differs to the previous uses of FI and QFI, it is distinct in that it can quantify quantum correlations beyond entanglement.

5. DISCUSSION

The Fisher Information in application to interferometry defines the precision of the detection of an unknown parameter. For a qubit this parameter is the energy distance between two eigenenergies of the qubit usually associated with the effective magnetic field acting on a quantum two-level system. The FI depends on both the magnetic field (to be detected) and the final projective measurement performed on both qubits: the target (unitary evolving) and the control ones both sharing some quantum state. The task of detection requires optimization: minimization over the direction of the magnetic field (the intruder is trying to hide his local operation on the target system) and maximization over the global measurement (the observer prefers to make the most precise measurement). These two competing optimizations define the success of the detection. The figure of merit is the Fisher Information which governs the precision (Cramer-Rao bound).

We have demonstrated that there is an upper bound for the FI which is proportional to the geometric discord of the state used for detection. Classical (zero-discord) states are unable to reliably detect intrusion and subsequently the proper chosen local unitary operation on the target qubit can be hidden (undetected). Using discordant states guarantees the observation of intrusion by some optimal global measurement on both, the target and the control, qubits. This figure of merit can be used to quantify the discord of the quantum state.

The limitations to this theory are due to this only being valid for a two-qubit system. For future work, the approach will be extended to include multi-particle systems and, in particular, indistinguishable particles as this is the case in solid-state experiments. Whilst entanglement has been shown for indistinguishable systems, our approach provides a natural platform for the extension of the research to a description of indistinguishable qubits and their quantum discord.

6. CONCLUSIONS

This paper provides a general link for any two-qubit state between the Fisher Information of interferometric metrological problems and the geometric discord of the states used for detection. We analyzed the situation when intrusion into target subsystem (applying local unitary evolution) is intended to be hidden from observation while the final global measurement on both subsystems is optimized to make the detection possible. This problem is formulated as a competing optimization of the FI over the unitary operator, U_A, (minimization) and the global projective measurement, Π, (maximization). It has been shown that classical FI, defining metrological precision, is bounded by the discord of the states used for detection purpose, and, therefore, can be used as a measure for quantum correlations beyond entanglement for a two-qubit system. This has immediate experimental application via interferometry, and thus should readily be realizable.

ACKNOWLEDGMENTS

This work was supported by the Leverhulme Trust Grant No. RPG-2019-317.

REFERENCES

1.

A.

Einstein

,

B.

Podolsky

, and

N.

Rosen

, “

Can quantum-mechanical description of physical reality be considered complete?

,”

Phys. Rev.

47

,

777

(

1935

).

https://doi.org/10.1103/PhysRev.47.777

Google Scholar

Crossref

2.

J. S.

Bell

, “

On the einstein podolsky rosen paradox

,”

Phys. Phys. Fiz.

1

,

195

(

1964

).

https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195

Google Scholar

Crossref

3.

A.

Aspect

,

P.

Grangier

, and

G.

Roger

, “

Experimental realization of einstein-podolsky-rosen-bohm gedanken-experiment: A New violation of bell’s inequalities

,”

Phys. Rev. Lett.

49

,

91

(

1982

).

https://doi.org/10.1103/PhysRevLett.49.91

Google Scholar

Crossref

4.

H.

Ollivier

and

W. H.

Zurek

, “

Quantum discord: A measure of the quantumness of correlations

,”

Phys. Rev. Lett.

88

,

017901

(

2001

).

https://doi.org/10.1103/PhysRevLett.88.017901

Google Scholar

Crossref

PubMed

5.

L.

Henderson

and

V.

Vedral

, “

Classical, quantum and total correlations,

”

J. Phys. A

34

,

6899

(

2001

).

https://doi.org/10.1088/0305-4470/34/35/315

Google Scholar

Crossref

6.

S.

Braunstein

,

L.

Samuel

, and

C. M.

Caves

, “

Statistical distance and the geometry of quantum states

,”

Phys. Rev. Lett.

72

,

3439

(

1994

).

https://doi.org/10.1103/PhysRevLett.72.3439

Google Scholar

Crossref

PubMed

7.

O.

Gühne

and

G.

Tóth

, “

Entanglement detection

,”

Phys. Rep.

474

,

1

(

2009

).

https://doi.org/10.1016/j.physrep.2009.02.004

Google Scholar

Crossref

8.

G.

Tóth

and

I.

Apellaniz

, “

Quantum metrology from a quantum information science perspective

,”

J. Phys. A

47

,

424006

(

2014

).

https://doi.org/10.1088/1751-8113/47/42/424006

Google Scholar

Crossref

9.

B.

Morris

,

B.

Yadin

,

M.

Fadel

,

T.

Zibold

,

P.

Treutlein

, and

G.

Adesso

, “

Entanglement between identical particles is a useful and consistent resource

,”

Phys. Rev. X

10

,

041012

(

2020

).

https://doi.org/10.1103/PhysRevX.10.041012

Google Scholar

Crossref

10.

F.

Benatti

,

R.

Floreanini

, and

U.

Marzolino

, “

Entanglement in fermion systems and quantum metrology

,”

Phys. Rev. A

89

,

032326

(

2014

).

https://doi.org/10.1103/PhysRevA.89.032326

Google Scholar

Crossref

11.

B.

Bylicka

and

D.

Chruścińki

, “

Witnessing quantum discord in 2 × N systems

,”

Phys. Rev. A

81

,

062102

(

2010

).

https://doi.org/10.1103/PhysRevA.81.062102

Google Scholar

Crossref

12.

D.

Braun

,

G.

Adesso

,

F.

Benatti

,

R.

Floreanini

,

U.

Marzolino

,

M. W.

Mitchell

, and

S.

Pirandola

, “

Quantum-enhanced measurements without entanglement

,”

Rev. Mod. Phys.

90

,

035006

(

2018

).

https://doi.org/10.1103/RevModPhys.90.035006

Google Scholar

Crossref

13.

M. A.

Hunt

,

I. V.

Lerner

,

I. V.

Yurkevich

, and

Y.

Gefen

, “

How to observe and quantify quantum-discord states via correlations

,”

Phys. Rev. A

100

,

022321

(

2019

).

https://doi.org/10.1103/PhysRevA.100.022321

Google Scholar

Crossref

14.

B.

Dakić

,

V.

Vedral

, and

C.

Brukner

,

Necessary and sufficient condition for nonzero quantum discord

,

Phys. Rev. Lett.

105

,

190502

(

2010

).

https://doi.org/10.1103/PhysRevLett.105.190502

Google Scholar

Crossref

PubMed

15.

L.

Chang

and

S.

Luo

, “

Remedying the local ancilla problem with geometric discord,

”

Phys. Rev. A

87

,

062303

(

2013

).

https://doi.org/10.1103/PhysRevA.87.062303

Google Scholar

Crossref

16.

C.

Rao

, “

Information and accuracy attainable in the estimation of statistical parameters

,”

Bull. Calcutta Math. Soc.

37

,

81

(

1945

).

Google Scholar

17.

D.

Girolami

,

A. M.

Souza

,

V.

Giovannetti

,

T.

Tufarelli

,

J. G.

Filgueiras

,

R. S.

Sarthour

,

D. O.

Soares Pinto

,

I. S.

Oliveira

, and

G.

Adesso

, “

Quantum discord determines the interferometric power of quantum states

,”

Phys. Rev. Lett.

112

,

210401

(

2014

).

https://doi.org/10.1103/PhysRevLett.112.210401

Google Scholar

Crossref

Published open access through an agreement with Aston University

The link between Fisher information and geometric discord

1. INTRODUCTION

2. QUANTUM DISCORD

3. QUANTUM METROLOGY

3.1. Fisher information

4. RELATION BETWEEN FI AND GEOMETRIC DISCORD

5. DISCUSSION

6. CONCLUSIONS

ACKNOWLEDGMENTS

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

The link between Fisher information and geometric discord

1. INTRODUCTION

2. QUANTUM DISCORD

3. QUANTUM METROLOGY

3.1. Fisher information

4. RELATION BETWEEN FI AND GEOMETRIC DISCORD

5. DISCUSSION

6. CONCLUSIONS

ACKNOWLEDGMENTS

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only