PHYSICAL REVIEW A 76, 014307 共2007兲

Invariant information and complementarity in high-dimensional states Wei Song and Zeng-Bing Chen Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 共Received 23 June 2007; published 26 July 2007兲 Using a generalization of the invariant information introduced by Brukner and Zeilinger 关Phys. Rev. Lett. 83, 3354 共1999兲兴 to high-dimensional systems, we introduce a complementarity relation between the local and nonlocal information for d ⫻ d systems under an isolated environment, where d is prime or a power of a prime number. We also analyze the dynamics of the local information in the decoherence process. DOI: 10.1103/PhysRevA.76.014307

PACS number共s兲: 03.67.⫺a, 03.65.Ud

When dealing with classical measurement Shannon information 关1兴 is a natural measure of our ignorance about the properties of a system. However, Shannon information is only applicable when measurement reveals a preexisting property. In sharp contrast to classical measurements, a quantum measurement does not reveal a preexisting property. For example, if we want to read out the information encoded in a qubit, we have to project the state of the qubit onto the measurement basis 兵兩0典, 兩1典其 which will give us a bit value of either 0 or 1. The qubit might be in the eigenstate of the measurement apparatus only in the special case; in general, the value obtained by the measurement has an element of irreducible randomness. Thus we cannot get the value of the bit or even hidden property of the system existing before the measurement is performed. To overcome this difficulty, Brukner and Zeilinger introduced operationally invariant information 关2兴 as the measure of local information. This measure of information is invariant under the transformation from one complete set of complementary variables to another. It is also conserved in time if there is no information exchange between the system and the environment. In this paper, using a generalization of Brukner and Zeilinger’s result to the higher-dimensional case, we derive a complementarity relation between the local and nonlocal information for d ⫻ d systems. The measure is obtained by summing over the measurement outcome of a set of mutually complementary observables 共MCOs兲 关3,4兴. Before describing the details of our derivations, let us briefly review the definition of MCOs. In a d-dimensional Hilbert space we call two observables K and M mutually complementary if all their respective, complete, orthonormal eigenvectors fulfill 1 兩具K,k兩M,m典兩2 = , d

∀ k,m = 1, . . . ,d.

共1兲

It is known that if d is prime or the power of a prime, the number of MCOs is d + 1 关3–8兴. Consider a measurement of an observable M = 兺dj=1a j⌸ j. Each outcome j is detected with probability p j = Tr ␳⌸ j. According to Ref. 关2兴, we define the local invariant information for a d-level system as d+1 d

I = N兺

兺

␣=1 j=1

冉

p␣ j −

1 d

冊

2

,

共2兲

where ␣ = 1 , . . . , d + 1 and j = 1 , . . . , d label complementary observables and their eigenvectors, respectively, and N is the 1050-2947/2007/76共1兲/014307共4兲

normalization factors. The derivation of our results use the fact that 关3,4兴 1 Tr兵⌸␣ j⌸␤k其 = ␦␣␤␦ jk + 共1 − ␦␣␤兲. p After some algebra, we get I=

冉

冊

1 d log2 d Tr ␳2 − , d−1 d

共3兲

共4兲

where we choose log2 d as the normalization factor to ensure that a d-level pure state carries log2 d bits of information. If kk 共Tr ␳2 − 21k 兲 which corresponds to the local ind = 2k, I = 22k−1 variant information for composite k-qubit systems. The special choice for k = 1 , 2 recovers the results obtained in Ref. 关2兴. Next, we want to use this measure to establish a complementary relation between the local and nonlocal information for d ⫻ d systems. Complementarity is an important concept in quantum theory. Besides the most well-known complementarity principle introduced by Bohr 关9兴, many other kinds of complementarity relations have also been discussed 关10–13兴. In particular, for two-state systems, elegant relations between two complementary observables have been derived 关14–16兴. Additionally, Jakob and Bergou 关17兴 have recently derived a complementarity relation for an arbitrary pure state of two qubits. Subsequently, their result was generalized to multiqubit systems 关18,19兴. Recently, Cai et al. 关20兴 also established an elegant complementarity relation between local and nonlocal information for qubit systems. Below we show that a complementarity relation also holds for d ⫻ d systems. Now we analyze a two-qutrit system to give an example. Suppose the system initial state is a pure state 兩␺典12 and ␳i 共i = 1 , 2兲 is the reduced density matrix of each individual qutrit. The total information contained in the 3 ⫻ 3 system is in two forms. One is the local form, which is the information content of each individual qutrit. The other is the nonlocal form, which is entanglement between the two qutrits. If the 3 ⫻ 3 system is isolated—i.e., it starts with an initial pure and then is subject to unitary transformations only—then the sum of the local and nonlocal information should be conserved. Here, we adopt the operationally invariant information derived above to measure local information. In the case of the 3 ⫻ 3 pure state 兩␺典12, the local information contained in

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©2007 The American Physical Society

PHYSICAL REVIEW A 76, 014307 共2007兲

BRIEF REPORTS 2 3 共 2 1兲 qutrits 1 and 2 is I1 + I2 = 兺i=1 2 log2 3 Tr␳i − 3 . If being measured by 2-tangle 关21兴, the pairwise entanglement of the pure 2 = 2共1 − Tr␳2i 兲, where C12 is the generalstate 兩␺典12 is ␶12 = C12 ized notion of concurrence for 3 ⫻ 3 pure states 关22,23兴. Thus we have the complementarity relation

冉

冊

冊

d log2 d ␶12 艋 2 log2 d. d−1

共6兲

1−F 共I − 兩⌽+典具⌽+兩兲 + F兩⌽+典具⌽+兩, d2 − 1

␶共␳12兲 =

冦

0 , d1

冉 冊 冉 冊冉 1 , 3

1 3 F− 3

2

冊

1 艋F艋1 . 3

冧

艋

冉

冊

d log2 d ␶12 d−1

冉

冊

2d2 1 log2 d Tr␳2 − 2 . 2 d −1 d

共9兲

0.4

0.6

0.8

1

FIG. 1. 共Color online兲. Plots of I共␳1兲 + I共␳2兲 + 共 d−1 log2 d兲␶12 共red line兲 and I共␳12兲 共blue line兲 for d = 3. d

Note that inequality 共9兲 does not hold for arbitrary d ⫻ d states. It is tempting to define mutual invariant information like the von Neumann entropy. However, this is not reasonable because there does not exist a subadditivity relation for invariant information—i.e., I共␳12兲 艋 I共␳1兲 + I共␳2兲. It only satisfies an additivity relation like I共␳ 丢 ␴兲 = I共␳兲 + I共␴兲. In general, we can derive a lower and upper bound of I共␳12兲 − I共␳1兲 − I共␳2兲. First, we can image a d ⫻ d mixed state ␳12 that is a part of a pure state 兩␺典12R, where R denotes a reference system. Usually the reduced density matrix ␳R also lies in a 共d ⫻ d兲-dimensional Hilbert space. For the composite d2 ⫻ d2 pure state 兩␺典12R under the cut 12: R, we have I共␳12兲 + I共␳R兲 +

冉

冊

2d2 log2 d ␶12:R = 4 log2 d, d2 − 1

共10兲

where we have used the complementarity relation for pure states. On the other hand, we have I共␳1兲 + I共␳2兲 +

冉

冊

d log2 d ␶12 艋 2 log2 d. d−1

共11兲

Combining Eqs. 共10兲 and 共11兲, we get I共␳12兲 − I共␳1兲 − I共␳2兲 艌 2 log2 d − I共␳R兲 −

共8兲

Straightforward calculation shows that the local information for qutrit 1 and qutrit 2 are both 0. Then the value of I共␳1兲 + I共␳2兲 + 共 23 log2 3兲␶12 can be calculated using Eq. 共8兲. On the other hand, we consider the local information of the composite two-qutrit system. Using formula 共4兲, we obtain 2 2 − 91 兲 = 81F −18F+1 log2 3. For a vivid compariI共␳12兲 = 49 共Tr␳12 32 son, we plot I共␳1兲 + I共␳2兲 + 共 23 log2 3兲␶12 and I共␳12兲 in Fig. 1. From Fig. 1 we find that there always exists the relation I共␳1兲 + I共␳2兲 + 共 23 log2 3兲␶12 ⬍ I共␳12兲. Based on this fact, we conjecture that for a general d ⫻ d isotropic mixed state the following inequality is always true: I共␳1兲 + I共␳2兲 +

0.2

共7兲

兴 this and F 苸 关0 , 1兴. For F 苸 关 where state is known to be separable. The tangle for these isotropic states has been obtained in Ref. 关24兴. For d = 3 the tangle is d 兩ii典 兩⌽+典 = 冑1d 兺i=1

0 F艋

0.5 F

Even though this inequality holds for arbitrary d ⫻ d states, for some special cases we might get a more stringent bound. Consider a class of isotropic mixed states for d ⫻ d systems. These are states invariant under ␳ → 兰dU共U 丢 U*兲␳共U† 丢 U*†兲 and can be expressed as

␳iso共F兲 =

2 1.5

共5兲

Equation 共5兲 suggests that the nonlocal information has a close relation to entanglement, which is a reasonable fact. As every term of I1, I2, and ␶12 is convex, the total expression is. Therefore, I1 + I2 + 共 23 log2 3兲␶12 ⬍ 2 log2 3 for a mixed 3 ⫻ 3 state ␳12. This is easy to understand from a physical picture. If the system is not isolated, the 共2 log2 3兲-bit information is not only contained in the system, but also in its correlations with the outside environment. A similar result also holds for d ⫻ d systems if we notice the following fact. For an 2 ⬅ 2共1 − Tr ␳21兲 arbitrary d ⫻ d pure state, we have ␶12 = C12 2 = 2共1 − Tr ␳2兲. Thus, for a d ⫻ d mixed state ␳12, the following inequality holds:

冉

2.5

1

3 I1 + I2 + log2 3 ␶12 = 2 log2 3. 2

I1 + I2 +

3

+

2d2 共log2 d兲␶12:R d2 − 1

d 共log2 d兲␶12 , d−1

共12兲

It is easy to see that the lower bound depends not only on the entanglement of ␳12, but also on the entanglement of ␳12 with the reference system. It should be a noted that this bound can be a negative value. For example, consider a sepa5 4 兩01典具01兩 兩00典具00兩 + 12 rable two-qubit mixed state ␳12 = 12 2 1 + 12 兩10典具10兩 + 12 兩11典具11兩; one can verify that I共␳12兲 − I共␳1兲 5 . The upper bound of I共␳12兲 − I共␳1兲 − I共␳2兲 is eas− I共␳2兲 = − 54 ily obtained. We should only consider a d ⫻ d maximally entangled pure state 兩␺典12 = 冑1d 兺dj=1兩ii典. The local information I共␳1兲 and I共␳2兲 for this special pure state are both 0, while I共␳12兲 reaches a maximum value 2 log2 d. Therefore we have I共␳12兲 − I共␳1兲 − I共␳2兲 艋 2 log2 d. The above analysis suggests that we cannot treat invariant information in the same way as the von Neumann entropy; sometimes, we might resort to the von Neumann entropy to gain a more comprehensive insight into the quantum information. Invariant information provides a complement of the von Neumann entropy in a description of quantum information, but cannot substitute the role of von

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PHYSICAL REVIEW A 76, 014307 共2007兲

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0.4

0

1 0

0

0.2

0.4

0.6

0.8

1

1 0.8

2 1

1.5 1 0

0.75 0.25

FIG. 2. 共Color online兲. The plots of the local information for the state 兩␺典12 = a兩00典 + 冑1 − a2兩11典 in the three decoherence processes. The right pannels are plotted in the density form of the left ones.

0.6 0.4

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0

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Neumann entropy. Perhaps a deeper investigation of invariant information is desired. So far, we only consider systems in an isolated environment. One can easily visualize that the local information will decrease in open systems. In order to gain more insight into the above question, we analyze the behavior of simple twoqubit entangled states under decoherence. We choose the depolarization, dephasing, and dissipation channels as our toy model for decoherence. Consider the two-qubit state 兩␺典12 = a兩00典 + 冑1 − a2兩11典 where a is a real number with 0 艋 a 艋 1. We define p as the degree of decoherence of an individual qubit, which lies between 0 and 1. Here the value 0 means no decoherence and 1

means complete decoherence. The depolarization process with a decoherence degree p is described by I 兩i典具j兩 → 共1 − p兲兩i典具j兩 + p␦ij . 2

共13兲

The dephasing process is represented by 兩i典具j兩 → 共1 − p兲兩i典具j兩 + p␦ij兩i典具j兩.

共14兲

The dissipation is an energy-loosing process and thus changes the state to a specific state. We choose 兩0典 as the ground state. Then the dissipation process is described by

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兩i典具i兩 → 共1 − p兲兩i典具i兩 + p兩0典具0兩,

PHYSICAL REVIEW A 76, 014307 共2007兲

BRIEF REPORTS

兩i典具j兩 → 共1 − p兲1/2兩i典具j兩,

共15兲

After the action of the depolarization channel, the local information of the two-qubit system is given by 2 I = 共4Tr␳2 − 1兲 3 =

再冋

2 4 a2共1 − a2兲共2p4 − 8p3 + 10p2 − 4p兲 3

册 冎

1 + p4 − p3 + 2p2 − 2p + 1 − 1 . 4

共16兲

We plot Eq. 共16兲 in Fig. 2共a兲 and the local information under the dephasing and dissipation processes are plotted in Figs. 2共b兲 and 2共c兲, respectively. It is shown in Fig. 2 that the minimum value of the local information of the two-qubit systems is 0 for the depolarization channel while it is always a positive value for the other two channels. The main reason is due to the fact that the depolarization channel will transform the initial state into a totally mixed state after infinite time, while the local information for maximally mixed state is 0. Before concluding we would like to stress that our complementarity relation is based on invariant information. This perspective is different from the methods used in Refs. 关25–28兴, where the authors utilized the von Neumann entropy to investigate the complementarity between local and

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nonlocal information. The distinct advantage by using invariant information is based on the following facts. The first is that invariant information always implies additivity, which is a desired property to establish complementarity relations. The second is that invariant information can be directly related to the tangle of d ⫻ d pure states. The third nice feature of the invariant information is that its definition is operational. It is obtained by synthesizing the errors of a specially chosen set of measurements performed on the system. Thus we might get a different insight into the complementarity relation between the local and nonlocal information. Summarizing, we have generalized the invariant information introduced by Brukner and Zeilinger. For d ⫻ d mixed states 共where d is prime or the power of a prime兲, using the invariant information, we establish a complementarity relation between the local and nonlocal information under the isolated environment. Furthermore, we show that the nonlocal information has a direct relation with the entanglement of the system. Some differences between the invariant information and von Neumann entropy are also discussed. We also investigate the dynamics of the local information in the open systems through a simple example. W.S. thanks Jian-Ming Cai for valuable discussions. This work was supported by the NNSF of China, the CAS, and the National Fundamental Research Program 共under Grant No. 2006CB921900兲.

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