aa r X i v : . [ qu a n t - ph ] J a n Information and fidelity in pro jectivemeasurements
Hiroaki Terashima
Department of Physics, Faculty of Education, Gunma University,Maebashi, Gunma 371-8510, Japan
Abstract
In this study, we explicitly calculate information and fidelity of an r -rank projective measurement on a completely unknown state in a d -dimensional Hilbert space. We also show a tradeoff between infor-mation and fidelity at the level of a single outcome and discuss theefficiency of measurement with respect to fidelity. PACS : 03.65.Ta, 03.67.-a
Keywords : quantum measurement, quantum information, projective measurement
In quantum theory, a measurement that provides information about a phys-ical system inevitably changes the state of the system depending on theoutcome of the measurement. This is an interesting property of quantummeasurement not only in the foundations of quantum mechanics but alsoin quantum information processing and communication [1] such as quantumcryptography [2–5]. Therefore, there have been many discussions regard-ing the tradeoffs between information gain and state change using variousformulations [6–15]. For example, Banaszek [7] has shown an inequality be-tween mean estimation fidelity and mean operation fidelity that quantifiesinformation gain and state change, respectively.In connection with such tradeoffs, the author [16, 17] has recently dis-cussed tradeoffs together with physical reversibility [18, 19] of measurementin the context of reversibility in quantum measurement [20–34]. In par-ticular, the author [17] has shown tradeoffs among information gain, statechange, and physical reversibility in the case of single-qubit measurements.An important feature of these tradeoffs is that they occur at the level ofa single outcome without averaging all possible outcomes [6, 7, 9, 13]. This1eature originates from the fact that the physical reversibility of measure-ments suggests quantifying the information gain and the state change foreach single outcome, because in physically reversible measurements, a staterecovery with information erasure (see the Erratum of [22]) occurs becauseof the post-selection of outcomes. However, the explicit calculations in theprevious studies [16,17] were only performed with two-level systems or qubits.In this study, we calculate information gain and state change in a projec-tive measurement of rank r on a d -level system assumed to be in a completelyunknown state. We evaluate the amount of information gain by a decreasein Shannon entropy [10, 33] and the degree of state change by fidelity [35] toexpress them as functions of r and d . These results lead to a tradeoff betweeninformation gain and state change at a single outcome level. We also considerthe efficiency of the measurement with respect to fidelity. Of course, projec-tive measurements are not physically reversible [18]. However, they wouldcorrespond to special points as the most informative but the least reversiblemeasurements in the tradeoffs among information gain, state change, andphysical reversibility in general measurements on a d -level system.The rest of this paper is organized as follows: Section 2 explains theprocedure to quantify information gain and state change and calculates themin the case of an r -rank projective measurement on a d -level system. Section 3discusses a tradeoff between information gain and state change and considersefficiency of the measurement with respect to the state change. Section 4summarizes our results. We evaluate the amount of information provided by a quantum measurementas follows. Suppose that the pre-measurement state of a system is known tobe one of the predefined pure states {| ψ ( a ) i} , a = 1 , . . . , N , with equal prob-ability p ( a ) = 1 /N [16, 17, 33], although the index a of the pre-measurementstate is unknown to us. Thus, the lack of information about the state of thesystem can be evaluated by Shannon entropy as H = − X a p ( a ) log p ( a ) = log N (1)before measurement, where we have used the Shannon entropy rather thanthe von Neumann entropy of the mixed state ˆ ρ = P a p ( a ) | ψ ( a ) ih ψ ( a ) | be-cause what we are uncertain about is the classical variable a rather than the2redefined quantum state | ψ ( a ) i . If the pre-measurement state is completelyunknown, as is usually the case in quantum measurement, then the set of thepredefined states, {| ψ ( a ) i} , consists of all possible pure states of the systemwith N → ∞ . Each state can be expanded by an orthonormal basis {| k i} as | ψ ( a ) i = X k c k ( a ) | k i (2)with k = 1 , , . . . , d , where d is the dimension of the Hilbert space associatedwith the system. The coefficients { c k ( a ) } obey the normalization condition X k | c k ( a ) | = 1 . (3)We next perform a quantum measurement on the system to obtain theinformation about its state. A quantum measurement is generally describedby a set of measurement operators { ˆ M m } [1, 36] that satisfies X m ˆ M † m ˆ M m = ˆ I, (4)where ˆ I is the identity operator. If the system to be measured is in a state | ψ i , the measurement yields an outcome m with probability p m = h ψ | ˆ M † m ˆ M m | ψ i (5)and then causes a state reduction of the measured system into | ψ m i = 1 √ p m ˆ M m | ψ i . (6)Here we consider performing a projective measurement because it is themost informative. In particular, we perform a measurement where the pro-cess yielding a particular outcome m is described by a projection operator ofrank r ( r = 1 , , . . . , d ); that is, the measurement operator corresponding tothe outcome m is written without loss of generality asˆ M m = κ m ˆ P ( r ) = κ m r X k =1 | k ih k | (7)3y relabeling the orthonormal basis, where κ m is a complex number. Theother measurement operators are irrelevant as long as condition (4) is satis-fied, since our interest is only at the level of a single outcome. The measure-ment then yields the outcome m with probability p ( m | a ) = | κ m | r X k =1 | c k ( a ) | ≡ | κ m | q m ( a ) (8)when the pre-measurement state is | ψ ( a ) i from Eqs. (2) and (5). Since theprobability for | ψ ( a ) i is p ( a ) = 1 /N , the total probability for the outcome m becomes p ( m ) = X a p ( m | a ) p ( a ) = 1 N X a | κ m | q m ( a ) = | κ m | q m , (9)where the overline denotes the average over a , f ≡ N X a f ( a ) . (10)On the contrary, Bayes’ rule states that given the outcome m , the probabilityfor the pre-measurement state | ψ ( a ) i is given by p ( a | m ) = p ( m | a ) p ( a ) p ( m ) = q m ( a ) N q m . (11)Thus, the lack of information about the pre-measurement state can be eval-uated by Shannon entropy as H ( m ) = − X a p ( a | m ) log p ( a | m ) (12)after the measurement yields the outcome m . Therefore, we define informa-tion gain by the measurement with the single outcome m as the decrease inShannon entropy [10, 33] I ( m ) ≡ H − H ( m ) = q m log q m − q m log q m q m , (13)which is free from the divergent term log N → ∞ in Eq. (1) owing to theassumption that the probability distribution p ( a ) is uniform.4n order to explicitly calculate the information gain (13), we introduceparametrization of the coefficients { c k ( a ) } . Let α k ( a ) and β k ( a ) be the realand imaginary parts of c k ( a ), respectively: c k ( a ) = α k ( a ) + iβ k ( a ) . (14)The normalization condition (3) then becomes X k (cid:2) α k ( a ) + β k ( a ) (cid:3) = 1 . (15)Note that this is the condition for a point to be on the unit sphere in 2 d dimensions. This means that { α k ( a ) } and { β k ( a ) } can be expressed by hy-perspherical coordinates ( θ , θ , . . . , θ d − , φ ) as [33] α ( a ) = sin θ d − sin θ d − · · · sin θ sin θ sin θ cos φ,β ( a ) = sin θ d − sin θ d − · · · sin θ sin θ sin θ sin φ,α ( a ) = sin θ d − sin θ d − · · · sin θ sin θ cos θ ,β ( a ) = sin θ d − sin θ d − · · · sin θ cos θ , (16)... α d ( a ) = sin θ d − cos θ d − ,β d ( a ) = cos θ d − , where 0 ≤ φ < π and 0 ≤ θ p ≤ π with p = 1 , , . . . , d −
2. The index a now represents the angles ( θ , θ , . . . , θ d − , φ ) and thus the summation over a is replaced with an integral over the angles as1 N X a −→ ( d − π d Z π dφ d − Y p =1 Z π dθ p sin p θ p . (17)From Eqs. (8) and (10), we get q m ( a ) = d − Y p =2 r − sin θ p ( r < d )1 ( r = d ) (18)and q m = rd (19)5sing the integral formula Z π dθ sin n θ = √ π Γ (cid:0) n +12 (cid:1) Γ (cid:0) n +22 (cid:1) , (20)where n > − n ). Similarly, using Z π dθ sin n θ log sin θ = √ π Γ (cid:0) n +12 (cid:1) Γ (cid:0) n +22 (cid:1) " ( − n +1 + n X k =1 ( − n + k +1 k ln 2 (21)for n > − x = ln x/ ln 2, we obtain q m log q m = − rd ln 2 h η ( d ) − η ( r ) i , (22)where η ( n ) ≡ n X k =1 k . (23)Therefore, the total probability (9) and the information gain (13) are calcu-lated to be p ( m ) = | κ m | rd (24)and I ( m ) = log dr − h η ( d ) − η ( r ) i , (25)respectively. Figure 1 shows the information gain I ( m ) as a function of rank r for d = 2 , , , ,
10. As shown in Fig. 1, the information gain monotonicallydecreases as r increases and becomes 0 at r = d . Note that when r = d , themeasurement corresponds to an uninformative identity operation, since themeasurement operator (7) reduces to the identity operator ˆ I except for theconstant κ m . In contrast, when r is fixed, the information gain monotonicallyincreases as d increases. Thus, taking the limit of Eq. (25) as d goes to infinityat r = 1, we find the upper bound on information gain as I ( m ) → − γ ) ≃ . , (26)where γ is Euler’s constant. 6 I( m ) r d=2d=4d=6d=8d=10 Figure 1: Information gain I ( m ) when the projective measurement yields theoutcome m as a function of rank r for d = 2 , , , , | ψ ( a ) i and the measurementoutcome is m , the post-measurement state is | ψ ( m, a ) i = 1 p p ( m | a ) κ m ˆ P ( r ) | ψ ( a ) i (27)according to Eqs. (6) and (7). To quantify this state change, we use fidelity [1,35] between the pre-measurement and post-measurement states, namely F ( m, a ) = (cid:12)(cid:12) h ψ ( a ) | ψ ( m, a ) i (cid:12)(cid:12) = p q m ( a ) . (28)This fidelity decreases as the measurement increasingly changes the state ofthe system. Averaging it over a with the probability (11), we evaluate thedegree of state change as F ( m ) = X a p ( a | m ) (cid:2) F ( m, a ) (cid:3) = q m q m (29)after the measurement yields the outcome m , where for simplicity, we haveaveraged the squared fidelity rather than the fidelity. The fidelity (29) canbe explicitly calculated using the parameterization (16) as F ( m ) = r + 1 d + 1 , (30)7 F ( m ) r d=2d=4d=6d=8d=10 Figure 2: Fidelity F ( m ) when the projective measurement yields the outcome m as a function of rank r for d = 2 , , , , q m = r ( r + 1) d ( d + 1) . (31)Figure 2 shows the fidelity F ( m ) as a function of rank r for d = 2 , , , , r andbecomes 1 at r = d . Moreover, when r is fixed, fidelity monotonically de-creases as d increases and becomes 0 in the limit d → ∞ .In terms of the density operator of the system, the measurement changesthe maximally mixed state in d dimensions, ˆ ρ = P a p ( a ) | ψ ( a ) ih ψ ( a ) | = ˆ I/d ,into that in r dimensions, decreasing the von Neumann entropy of the systemby log d − log r = log ( d/r ). However, the information gain (25) is less thanlog ( d/r ) because of our formulation of information resource [1], i.e. a set ofpredefined states with Shannon entropy rather than a density operator withvon Neumann entropy. Within this formulation, the second term in Eq. (25)comes from the indistinguishability of non-orthogonal quantum states. Tosee this, consider the orthonormal basis {| k i} with k = 1 , , . . . , d in Eq. (2)as the set of predefined states, instead of all possible pure states {| ψ ( a ) i} . Inthis distinguishable case, the information gain is equal to just the decreasein the von Neumann entropy log ( d/r ). Therefore, the reduced informationgain (25) is due to the indistinguishability of predefined states. In otherwords, quantum measurement with no a priori information about the stateof the system is not optimal as quantum communication between the system8 F ( m ) I(m) d=2d=4d=6d=8d=10
Figure 3: Fidelity F ( m ) as a function of information gain I ( m ) for d =2 , , , , From the explicit formulae for the information gain (25) and fidelity (30), wefind a tradeoff between information and fidelity in projective measurements.Figure 3 shows the fidelity F ( m ) as a function of the information gain I ( m )for d = 2 , , , ,
10. As the measurement provides more information aboutthe state of a system, the process of measurement changes the state to agreater extent, as shown in Fig. 3. It should be emphasized that this tradeoffis at a single outcome level in the sense that there is no average over outcome.In addition, another relationship between information gain and statechange can be shown by defining the efficiency of measurement as the ra-tio of the information gain to the fidelity loss [16, 17], E F ( m ) ≡ I ( m )1 − F ( m ) . (32)Figure 4 shows the efficiency of measurement, E F ( m ), as a function of rank r for d = 2 , , , ,
10, although it is ill-defined at r = d because of I ( m ) =1 − F ( m ) = 0. The efficiency is a monotonically decreasing function for each9 E F ( m ) r d=2d=4d=6d=8d=10 Figure 4: Efficiency of measurement E F ( m ) as a function of rank r for d =2 , , , , d and has a maximal value 3[1 − / (2 ln 2)] at r = 1 in d = 2. This meansthat among the various projective measurements, a projective measurementon a two-level system or qubit is the most efficient with respect to fidelity.Nevertheless, it is the least efficient among single-qubit measurements, asdiscussed in Ref. [17]. We calculated the information gain and fidelity of a projective measurementon a system where the pre-measurement state was assumed to be in a com-pletely unknown state. They are expressed as functions of the dimensions d of the Hilbert space associated with the system and rank r of the projectionoperator associated with the measurement, as in Eqs. (25) and (30). Theseresults show a tradeoff between information and fidelity at the level of a sin-gle outcome without averaging all outcomes, as shown in Fig. 3. We alsodiscussed the efficiency of the measurement by using the ratio of informationgain to fidelity loss. In terms of this efficiency, a projective measurement ona two-level system or qubit is the most efficient among the various projectivemeasurements.Although here we have considered only projective measurements, thereare many measurements that are not projective, e.g., photodetection pro-10esses in photon counting [16]. Such measurements can be less informativebut more reversible than projective measurements. However, in general mea-surements on a d -level system, it would be difficult to find tradeoffs amonginformation gain, fidelity, and physical reversibility because they are all func-tions of d − Acknowledgments
This study was supported by a Grant-in-Aid for Scientific Research (GrantNo. 20740230) from Japan Society for the Promotion of Science.
References [1] M. A. Nielsen and I. L. Chuang,
Quantum Computation and QuantumInformation (Cambridge University Press, Cambridge, 2000).[2] C. H. Bennett and G. Brassard, in