Abstract
The Holevo bound is a keystone in many applications of quantum information theory. We propose “ maximal Holevo quantity for weak measurements” as the generalization of the maximal Holevo quantity which is defined by the optimal projective measurements. The scenarios that weak measurements is necessary are that only the weak measurements can be performed because for example the system is macroscopic or that one intentionally tries to do so such that the disturbance on the measured system can be controlled for example in quantum key distribution protocols. We evaluate systematically the maximal Holevo quantity for weak measurements for Bell-diagonal states and find a series of results. Furthermore, we find that weak measurements can be realized by noise and project measurements.
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Introduction
Weak measurements was introduced by Aharonov, Albert and Vaidman (AAV)1 in 1988. The standard measurements can be realized as a sequence of weak measurements which result in small changes to the quantum state for all outcomes2. Weak measurements realized by some experiments are also very useful for high-precision measurements3,4,5,6,7.
The quantum correlations of quantum states include entanglement and other kinds of nonclassical correlations. It is well known that the quantum correlations are more general than the well-studied entanglement8,9. Quantum discord, a quantum correlation measure differing from entanglement, is introduced by Oliver and Zurek10 and independently by Henderson and Vedral11. It quantifies the difference between the mutual information and maximum classical mutual information, i.e., it is a measure of the difference between total correlation and the classical correlation. Significant developments have been achieved in studying properties and applications of quantum discord. In particular, there are some analytical expressions for quantum discord for two-qubit states, such as for the 
states12,13,14,15,16,17. Besides, researches on the dynamics of quantum discord in various noisy environments have revealed many attractive features18,19,20. It is demonstrated that discord is more robust than entanglement for both Markovian and non-Markovian dissipative processes. As with projection measurements, weak measurements are also applied to study the quantification of quantum correlation. For example, the super quantum correlation based on weak measurements has attracted much attention21,22,23,24,25.
In general, maximum classical mutual information is called classical correlation which represents the difference in von Neumann entropy before and after the measurements11. A similarly defined quantity is the Holevo bound which measures the capacity of quantum states for classical communication26,27. The Holevo bound is an exceedingly useful upper bound on the accessible information that plays an important role in many applications of quantum information theory28. It is a keystone in the proof of many results in quantum information theory29,30,31,32,33,34.
The maximal Holevo quantity for projective measurements (MHQPM) has been investigated33. Due to the fundamental role of weak measurements, it is interesting to know how MHQPM will be if weak measurements are taken into account. Recently, it is shown that weak measurements performed on one of the subsystems can lead to “super quantum discord” which is always larger than the normal quantum discord captured by projective measurements21. It is natural to ask whether weak measurements can also capture more classical correlations. In this article, we shall give the definition of “super classical correlation” by weak measurements as the generalization of classical correlation defined for standard projective measurements. As the generalization of MHQPM, we propose “ maximal Holevo quantity for weak measurements (MHQWM)”. Interestingly, by tuning continuously from strong measurements to weak measurements, the discrepancy between MHQWM and MHQPM becomes larger. Such phenomenon also exits between super classical correlation and classical correlation. In comparison with super quantum discord which is larger than the standard discord, MHQWM and super classical correlation becomes less when weak measurements are applied, while they are completely the same for projective measurements. In this sense, weak measurements do not capture more classical correlations. It depends on the specified measure of correlations. We calculate MHQPM for Bell-diagonal states and compare the results with classical correlation. We give super classical correlation and MHQWM for Bell-diagonal states and compare the relations among super quantum correlations, quantum correlations, classical correlation, super classical correlation and entanglement. The dynamic behavior of MHQWM under decoherence is also investigated.
Results
Maximal holevo quantity for projective measurements and weak measurements
The quantum discord for a bipartite quantum state ρAB with the projection measurements 
performed on the subsystem 
is the difference between the mutual information I(ρAB)35 and classical correlation 
11:
where
with the minimization going over all projection measurements 
, where 
is the von Neumann entropy of a quantum state ρ, ρA, ρB are the reduced density matrices of ρAB and
The Holevo quantity of the ensemble 
33 that is prepared for A by B via B’s local measurements is given by
It denotes the upper bound of A’s accessible information about B’s measurement result when B projects its system by the projection operaters 
. The Maximal Holevo quantity for projective measurements (MHQPM)33 of the state ρAB over all local projective measurements on B’s system, denoted by C1(ρAB), is defined as
The weak measurement operators are given by2
where x is the measurement strength parameter, 
and 
are two orthogonal projectors with 
. The weak measurement operators satisfy: (i) 
, (ii) 
and 
.
Recently, super quantum discord for bipartite quantum state ρAB with weak measurements on the subsystem 
has been proposed21. Similarly to the definition of quantum discord, we give another form of definition of super quantum discord. We define super classical correlation 
for bipartite quantum state ΡAB with the weak measurements 
performed on the subsystem B as follow. The super quantum discord denoted by Dw(ρAB) is the difference between the mutual information I(ΡAB) and super classical correlation 
, i.e.,
where
with the minimization going over all weak measurements,
where 
is weak measurement operators performed on the subsystem B.
Now, let us define the Holevo quantity of the ensemble 
for weak measurements on the subsystem B,
It denotes the upper bound of A’s accessible information about B’s measurement results when B projects the system with the weak measurements operaters 
. We define maximum value of the Holevo quantity over all local weak measurements on B’s system to be the maximal Holevo quantity for weak measurements (MHQWM). MHQWM denoted by 
, is given by
Next, we consider MHQPM and MHQWM for two-qubit Bell-diagonal states,
where I is the identity matrix, −1 ≤ ci ≤ 1. The marginal states of ρAB are 
. The MHQPM for Bell-diagonal states is given as
where 
. We find that MHQPM C1(ρAB) equals to the classical correlation JB(ρAB),
The MHQWM of two-qubit Bell-diagonal states is given by
The super classical correlation of two-qubit Bell-diagonal states is given by
MHQWM 
equals to super classical correlation 
, i.e.,
Then, we compare MHQWM (super classical correlation), MHQPM (classical correlation), super quantum discord, quantum discord and entanglement of formation. For simplicity, we choose Werner states, c1 = c2 = c3 = −z,
where 
. Set 
. The Werner states have the form
where −1 ≤ α ≤ 1, I is the identity operator in the 4-dimensional Hilbert space and 
is the operator that exchanges A and B. The entanglement of formation Ef for the Werner states is given as 
, by 
. The MHQPM for werner states is given by, see Eq. (48) in section Method,
The MHQWM for werner states is given by, see Eq. (57) in section Method,
Quantum discord for Werner states is given by12
Super quantum discord for Werner states is given by21
In Fig. 1 we plot MHQWM, MHQPM, super quantum discord, quantum discord and entanglement of formation for Werner states. We find that super quantum discord , quantum discord, MHQPM and MHQWM have the relation, 
. For the case of projection measurements, 
, we have 
. MHQWM approaches to zero for smaller values of 
. MHQWM approaches to MHQPM and super quantum discord approaches to quantum discord for larger values of x. MHQWM and MHQPM are larger than the entanglement of formation for small z and smaller than the entanglement of formation for big z. It shows that MHQWM and MHQPM can not always capture more correlation than the entanglement as super quantum discord and quantum discord do.
MHQWM (super classical correlation) (dashed green line), MHQPM (classical correlation) (solid blue line), quantum discord(solid cyan line), super quantum discord (dashed black line) and entanglement of formation(solid red line) for the Werner states as a function of z: x = 0.25 and x = 2.5.
As a natural generalization of the classical mutual information, the classical correlation represents the difference in von Neumann entropy before and after projection measurements, i.e.,
Similarly, the super classical correlation represents the difference in von Neumann entropy before and after weak measurements, i.e.,
As weak measurements disturb the subsystem of a composite system weakly, the information is less lost and destroyed by weak measurements on the subsystem alone. That is the physical interpretation that the super classical correlation is smaller than the classical correlation, 
. According to this fact, we can infer that weak measurements can capture more quantum correlation than projection measurements. In fact, the super quantum correlation 
is lager than the quantum correlation 
. There is a similarity to the Holevo quantity which measures the capacity of quantum states for classical communication.
Dynamics of MHQWM of Bell-diagonal states under local nondissipative channels.
We will consider the system-environment interaction28 through the evolution of a quantum state ρ under a trace-preserving quantum operation ε(ρ),
where 
is the set of Kraus operators associated to a decohering process of a single qubit, with 
. We will use the Kraus operators in Table 136 to describe a variety of channels considered in this work.
The decoherence processes BF, PF and BPF in Table 1 preserve the Bell-diagonal form of the density operator ρAB. For the case of GAD, the Bell-diagonal form is kept for arbitrary γ and p = 1/2. In this situation, we can write the quantum operation ε(ρ) as
where the values of the 
, 
, 
are given in Table 236.
When 
, 
, 
, respectively, we have that 
are the maximal values among 
, 
, 
in each line of Tabel 2 . As ε(ρ) are also Bell-diagonal states, from Eqs. (46), (48), (49), (57), (58) we find that classical correlation, MHQPM, super classical correlation and MHQWM for Bell-diagonal states through any channel of bit flip, phase flip, bit-phase flip remain unchanged. In particular, for Werner states, we find that classical correlation, MHQPM, super classical correlation and MHQWM for Werner states keep unchanged under all channels of bit flip, phase flip, bit-phase flip.
The MHQPM of the Werner states under generalized amplitude damping is given by
The MHQWM of the Werner states under generalized amplitude damping is given by
In Fig. 2, as an example, the dynamic behaviors of the MHQWM and MHQPM for the Werner states under the generalized amplitude damping channel are depicted for x = 0.5 and x = 1. Against the decoherence, when x increases, MHQWM become greater. MHQWM approaches to MHQPM for larger x under the generalized amplitude damping channel. MHQWM and MHQPM increase as z increases. Then as γ increases, MHQWM and MHQPM decrease.
The MHQWM (super classical correlation) {x = 0.5 (blue surface), x = 1(gray surface)} and the MHQPM (classical correlation)(orange surface) for the Werner states under generalized amplitude damping channel as a function of z and γ.
Weak measurements can be realized by noise and project measurements
Now we study the realization of weak measurements by means of depolarizing noise and project measurements. The depolarizing noise is an important type of quantum noise that transforms a single qubit state into a completely mixed state I/2 with probability p and leaves a qubit state untouched with probability 1 − p. The operators for single qubit depolarizing noise are given by37
where p = 1−e−τt. Then the Bell-diagonal states under the depolarizing noise acting on the first qubit of quantum state ρAB are given by37
As ε(ρAB) is also a Bell-diagonal state, after projective measurements on B, see Eq. (41) in section Method, the state ε(ρAB) becomes the following ensemble with 
and
Comparing Eq. (36) with the ensemble after weak measurements Eq. (52) in section Method, when 
, we obtain that weak measurements can be realized by means of depolarizing noise and projective measurements.
Discussion
We have evaluated analytically MHQPM for Bell-diagonal states and find that it equals to the classical correlation. We have given the definition of “super classical correlation” by weak measurements as the generalization of classical correlation defined by standard projective measurements. We have evaluated super classical correlation for Bell-diagonal states and find that it is smaller than the classical correlation and approaches the classical correlation by tuning the weak measurements continuously to the projective measurements. We have shown the physical implications that weak measurements can capture more quantum correlation than projective measurements.
As the generalization of the MHQPM defined by projective measurements, we have also proposed MHQWM by weak measurements. We have evaluated MHQWM for Bell-diagonal states and find that it is smaller than MHQPM in general. Moreover, it has been shown that MHQWM equals to super classical correlation.
As applications, the dynamic behavior of the MHQWM under decoherence has been investigated. For some special Bell-diagonal states, we found that MHQWM remain unchanged under all channels of bit flip, phase flip and bit-phase flip.
The dynamical behaviors of the MHQWM for Werner states under the generalized amplitude damping channel have been investigated. Under the generalized amplitude damping channel, MHQWM becomes greater when x increases and approaches to MHQPM for larger x. MHQWM increases as z increases. MHQWM decreases as γ increases. Above all, it has been shown that weak measurements can be realized by means of depolarizing noise and projective measurements.
The Holevo bound is a keystone in quantum information theory and plays important roles in many quantum information processing. While MHQPM provides us different perspectives about classical correlations. The behaviors of the MHQWM vary a lot with the strength of the weak measurements. Those measures can be applied to various protocols in quantum information processing and identify the importance of the classical correlations in those protocols.
Methods
Calculation of the MHQPM for Bell-diagonal states.
We compute the MHQPM C1(ρAB) of Bell-diagonal states. Let 
be the local measurements on the system B along the computational base 
. Any von Neumann measurement on the system B can be written as
for some unitary V ∈ U(2). Any unitary V can be written as
with 
, 
and 
After the measurements Bk, the state ρAB will be changed to the ensemble 
with
After some algebraic calculations12, we obtain 
and
where
Therefore,
Denote 
. Then
and
It can be directly verified that 
. Let
then we have 
Hence we get 
and θ ∈ [0, C]. It can be verified that 
is a monotonically decreasing function of θ in the interval of 
. The minimal value of 
can be attained at the point C,
By Eqs. (43) and (47), we obtain
As 
, the classical correlation JB(ρAB) is given by
Calculation of the MHQWM for Bell-diagonal states
Let 
be the local measurements for the part B along the computational base 
. Then any weak measurement operators on the system B can be written as
for some unitary V ∈ U(2) of the form Eq. (38).
After weak measurements the resulting ensemble is given by 
. We need to evaluate 
and p(+x). By using the relations12,
and 
, 
, 
for 
, 
, from Eqs. (12) and (13), we obtain 
and
where 
, 
and 
. Therefore, we see that
Denote 
. Then
and
Let 
then 
. Hence we get 
and θ ∈ [0, C]. It can be verified that 
is a monotonically decreasing function of θ in the interval of [0, C]. The minimal value of 
can be attained at point C,
By Eqs. (53) and (56), we obtain
As 
, the super classical correlation 
is given by
Additional Information
How to cite this article: Wang, Y.-K. et al. Maximal Holevo Quantity Based on Weak Measurements. Sci. Rep. 5, 10727; doi: 10.1038/srep10727 (2015).
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Acknowledgements
This work was supported by the Science and Technology Research Plan Project of the Department of Education of Jilin Province in the Twelfth Five-Year Plan, the National Natural Science Foundation of China under grant Nos. 11175248, 11275131, 11305105.
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Y.-K. W., S.-M. F., Z.-X. W, J.-P. C. and H. F. calculated and analyzed the results. Y.-K. W. and H. F. co-wrote the paper. All authors reviewed the manuscript and agreed with the submission.
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Wang, YK., Fei, SM., Wang, ZX. et al. Maximal Holevo Quantity Based on Weak Measurements. Sci Rep 5, 10727 (2015). https://doi.org/10.1038/srep10727
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DOI: https://doi.org/10.1038/srep10727
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