Symmetry-based estimation of lower bound on secure key rate of noisy private states
SSymmetry-based estimation of lower bound on secure key rate of noisy private states
Jan Tuziemski and Pawe(cid:32)l Horodecki
Faculty of Applied Physics and Mathematics, Technical University of Gda´nsk, PL-80-952 Gda´nsk, Poland andNational Quantum Information Centre of Gda´nsk, PL-81-824 Sopot, Poland (Dated: November 11, 2018)Quantum private states are the states that represent some amount of perfect secure key. A simplesymmetry of any generalised private quantum state (ie. the states that represent perfect key butnot fully random) is provided and extended on Devetak-Winter so called ccq (classical-classical-quantum) and cqq (classical-quantum-quantum) lower bound on secure key. This symmetry isused to develop a practical method of estimating the Alice measurement that is optimal form theperspective of single shot Devetak-Witner lower bound on secure key. The method is particularlygood when the noise does not break the symmetry of the state with respect to the lower boundformula. It suggest a general paradigm for quick estimation of quantum communication rates underthe symmetry of a given resource like state and/or channel.
I. INTRODUCTION
Entanglement is considered as a resource in quantum communication and computing. It has many intriguingproperties that make in some cases quantum physics predictions drastically different from classical ones (see [1]).Any BB84 type protocol [2] is formally equivalent to some version of entangled based protocol of the type E91 [3].On the level of uncorrelated sources Bennett et al. pointed out this fact on specific scheme [4] which was laternaturally extended to a quantum privacy amplification QPA [5] based entanglement distillation protocol [6]. Thegeneral intrinsic connection between BB84 secret key generation and possibility of maximal entanglement distillationfrom correlated sources of noisy entanglement was provided explicitly by elegant error correction type analysis [7]which in particular illuminated this aspect hidden in previous proofs. However the fundamental intuition behindthe BB84 secret key generation - entanglement distillation equivalence is already present in the case of uncorrelatedsource for which QPA protocol works. The latter is a protocol that distills maximally entangled states out of a mixedstates in a well defined way. In QPA it is eavesdropper that is representing the noise and the distillation procedureis aimed at remove the correlations with eavesdropper in the process that produces a pure output state - maximallyentangled state that is a source of perfect key. It seemed that this QPA is necessary to get privacy. However thereexists nondistillable entanglement called bound entanglement [8] for which by the very definition QPA in its originalform can not work. It turns out that there is another possibility of distilling secret key by distilling private states [9]that are generalisations of maximally entangled states - they provide secret key under the measurement in a fixed basison some part of the system. The measurement basis may be unique and that is what makes the private states moregeneral form maximally entangled ones for which there are infinite many pairs of local bases that provide secure key.The distillation of private states allow to provide secret key form bound entanglement (see [9]) showing in particulara possibility of drastic separation between amount of pure entanglement that can be distilled from a quantum state(called distillable entanglement and denoted by D) and amount of secret key that can be distilled from the same state(called disiilalble secret key and denoted by K). Recently the separation
D < K was experimentally demonstratedtogether with the illustration how inefficient may be the original entanglement distillation based scheme, if comparedwith p-bit based protocols [10].In this article we present the new symmetry of the states with perfect secure key called generalized private states,which states that in the most popular Devetak-Winter protocol secret key rate is invariant in both scenarios of CCQand CQQ type if the measurement bases, chosen in a wrong way, are rotated around the axis corresponding to thesecure basis by any angle. Then we use this symmetry to propose a new method of searching the optimal basis, whichallows to obtain optimal amount of distillable key with respect to the Devetak-Winter secret key rate, in the casewhen a state possessing this symmetry was rotated by an unknown angle. We investigate the influence of various qubitchannels on the result of the method and show that the proposed method is an optimal one as long as a channel isbistochastic. Finally we derive the error estimation of numerical implementation of the procedure and show examplesof the results.
II. A USEFUL SYMMETRY OF THE GENERALISED PRIVATE STATES
According to [11] any state containing perfectly secure key corresponds to a pure state shared by three partiesAlice, Bob and Eve. Unlike the eavesdropper subsystem E with a Hilbert space H E the subsystems of Alice and Bob a r X i v : . [ qu a n t - ph ] J un are composite and correspond to the tensor products of Hilbert spaces H A ⊗ H A (cid:48) and H B ⊗ H B (cid:48) respectively. Wecall the subsystem corresponding to the pair A,B the key part since this is the part that is used for key generationby local von Neumann measurements while the pair A’B’ is called the shield part since it is in a sense responsible forprotecting the key. The above structure allows to write explicitly the pure state of the three parties which representsa perfectly secure random a perfectly secure random statistics (cid:126)p = [ p , ..., p d − ]. It is a pure state | Ψ AA (cid:48) BB (cid:48) (cid:105) of thefollowing form which we shall call generalised private states : | Ψ AA (cid:48) BB (cid:48) E (cid:105) = (cid:88) ij c i c j | ii (cid:105) AB ⊗ [ U ( i ) A (cid:48) B (cid:48) ⊗ I E ] | Ψ A (cid:48) B (cid:48) E (cid:105) (1)for some fixed | Ψ A (cid:48) B (cid:48) E (cid:105) , some unitaries U ( i ) A (cid:48) B (cid:48) and probabilities { p i = | c i | } . The basis | ii (cid:105) AB is called the securebasis since after performing the local von Neumann measurements in that basis Alice and Bob share the correlatedprobability distribution { p ijAB = δ ij p i } d − i =0 which is completely uncorrelated form the system E. Here we do not assumethem to be necessarily p i = d as it is in the case of the private states . In fact the density matrix corresponding to thestate vector (1) is of the form: ρ priv = (cid:88) ij c i c j [ | ii (cid:105) (cid:104) jj | ] AB ⊗ [ | Ψ i (cid:105) (cid:104) Ψ j | ] A (cid:48) B (cid:48) E , (2)where we put | Ψ i (cid:105) = [ U A (cid:48) B (cid:48) ⊗ I E ] | Ψ (cid:105) A (cid:48) B (cid:48) E dropping the superscripts A (cid:48) B (cid:48) E . If Alice and Bob measure the subsystems AB in some basis {| e i (cid:105) A } , {| f i (cid:105) B } and trace the shield subsystems A (cid:48) B (cid:48) they get the so called the form of the CCQstate with respect to the (product) basis B AB = {| e i f i (cid:105) AB ≡ | e i (cid:105) A ⊗ | f i (cid:105) B } : ρ CCQ, B AB ABE ( {| e i , f j (cid:105)} ) = d − (cid:88) i =0 q i | e i (cid:105) (cid:104) e i | A ⊗ | f i (cid:105) (cid:104) f i | B ⊗ ρ iE with some probability distribution q i . It happens that if they choose the basis {| e i f j (cid:105) AB } to be just equal to thesecure one B AB = {| ij (cid:105) AB } then the above state reduces to the product form ρ CCQ, B AB ABE = ( d − (cid:88) i =0 p i [ | ii (cid:105) (cid:104) ii | ] AB ) ⊗ ρ E with q i = p i . Because of the explicitly product form - no correlations of the E system with the key part AB arepresent here. Before proving some new property let us remind that the so called CQQ state with respect to the basis B A = {| e i (cid:105) A which results form Alice local von Neumann measurement and tracing out both A’B’: ρ CQQ, B A ABE = d − (cid:88) i =0 q i | e i (cid:105) (cid:104) e i | A ⊗ ρ iBE (3)Note that measuring the private state in the local basis B A being just the reduction of the product secure basis B AB we get still the CCQ state in the form (3) rather than the general CQQ state (note that any CCQ state is aCQQ one but not vice versa) which is a consequence of the private character of the state. A. The Devetak-Winter protocol rates
We have a natural definition of the key rates in one-way protocols obtained by measuring the state first in somelocal basis B A or a product one B AB which will produce the CQQ or CCQ state respectively and then calculating thedifference of the Holevo functions of the states: K B A DW ( ρ ABE ) = I A : B ( ρ CQAB ) − I A : E ( ρ CQAE ) (4)with ρ CQAB , ρ CQAE being a suitable reductions of the state ρ CQQ, B A ABE resulting form the original state ρ ABE after the localAlice measurement associated with the basis B A . In full analogy we have K B AB DW ( ρ ABE ) = I A : B ( ρ CCAB ) − I A : E ( ρ CQAE ) (5)with the suitable reductions of the state ρ CCQ, B AB ABE resulting form the original state ρ ABE after the product of thetwo local Alice and Bob measurements corresponding to the bases B AB . The role of the function f is played just bythe mutual information function I . B. General symmetry rule and its simple application
In what follows we shall use the notation ˆ U ( X ) = U XU † and ˆ M ( { P k } )( X ) = (cid:80) k P k XP k with a projectors P k = P ( e k ) = | e k (cid:105)(cid:104) e k | for any orthonormal basis { e k } . We have a simple Observation .-
Consider a function f defined on any CQ state on the composite system XYσ
CQXY = (cid:88) k P k ⊗ σ k (6)Assume that the function f is invariant under some subgroup R X of unitary operations R X ∈ R X on the system X ie. ∀ R X ∈R X f ( ˆ R X ⊗ ˆ I Y ( σ CQXY ) = f ( σ CQXY ) (7)Given any state ρ XY which is invariant in an analogous way ∀ R X ∈R X ˆ R X ⊗ ˆ I Y ( ρ XY ) = ρ XY (8)we have the following identity f ([ ˆ M X ( { ˆ R X ( P k ) } ) ⊗ ˆ I Y ]( ρ XY )) = f ([ ˆ M X ( { P k } ) ⊗ ˆ I Y ]( ρ XY )) (9)for all R X ∈ R X and all { P k } constructed from any orthonormal bases { e k } . Proof. -
It is obvious to see that ,,internal” ˆ R † X is absorbed by the QQ state ρ XY while the external conjugatedone ˆ R X is absorbed by the invariance of the function f .We have immediate conclusion: Conclusion.-
The functions (4) and (5), if calculated on a given generalised private state (2), are invariant underthe rotations ˆ R A ⊗ ˆ I BE and ˆ R A ⊗ ˆ R B ⊗ ˆ I E respectively where R A , R B are any unitary operations which are diagonalin the local bases | i (cid:105) A , | j (cid:105) B forming a secure basis of the states (2). Proof .-
The role of the pair of the subsystems { X, Y } is played by { A, BE } or { AB, E } respectively and the roleof the subgroup are all the unitary operations diagonal in the bases described in the conclusion. C. Consequences
In the case, when the key part dimension d = 2, this feature can be interpreted graphically. Let us consider CQQcase. Then Alice can choose two angels ( θ, ϕ ) to determine her measurement basis using the eigenvectors of the σ ˆ n operator | e ( θ, ϕ ) (cid:105) = (cid:20) cos θ e i ϕ sin θ (cid:21) , | e ( θ, ϕ ) (cid:105) = (cid:20) sin θ − e i ϕ cos θ (cid:21) . (10)We define the function K D ( θ , ϕ ) = K DWD (cid:16) ρ C ( θ,ϕ ) QQpriv (cid:17) . Here superscript C ( θ, ϕ ) denotes that to calculate CQQ statebase vectors determined by angels ( θ, ϕ ) were used. In spherical coordinate system, in which | e (0 , (cid:105) , | e (0 , (cid:105) coincide with the base vectors used in (2) the function K D ( θ , ϕ ) is ϕ independent, i.e. becomes function only of θ angle. III. PROCEDURE
As follows from the previous section, points on the sphere[16] possessing the same vale of K D establish a circle,whose center is located at intersection of Z axis and the sphere. Moreover, each circle has a center in the same point(all circles are concentric). At this point vale of K D is maximal. Using facts presented in Subsection II C one is ableto find such angles θ Max , ϕ Max , for which the measurement in basis given by (10) will lead to the maximal value of K D , without a priori knowledge of this basis or rotations by which the state was changed. Suppose that the originalideal state was rotated by unknown transformation U A ⊗ I BA (cid:48) B (cid:48) which eventually changed its optimal measurementbasis on Alice side from {| ˆ z (cid:105) , | ˆ z (cid:105)} to {| ˆ n (cid:105) , | ˆ n (cid:105)} where we define ˆ n = O ˆ z as: | ˆ n (cid:105) (cid:104) ˆ n | = U | ˆ z (cid:105) (cid:104) ˆ z | U † = 12 ( I + ( O ˆ z ) (cid:126)σ ) | ˆ n (cid:105) (cid:104) ˆ n | = U | ˆ z (cid:105) (cid:104) ˆ z | U † = 12 ( I − ( O ˆ z ) (cid:126)σ ) . (11)The procedure is as follows.1. First one chooses arbitrary values of angles θ , ϕ and establishes two base vectors | e ( θ , ϕ ) (cid:105) and | e ( θ , ϕ ) (cid:105) .This basis is used to perform measurement and to obtain value of K D ( θ , ϕ ) equal K D .2. Then one changes the value of θ to θ and creates a set of base vectors {| e ( θ , ϕ i ) (cid:105) , | e ( θ , ϕ i ) (cid:105)} Ni =1 . Vectorsfrom this set differ in the value of ϕ i angle by arbitrary constant factor πN so that 0 ≤ ϕ i < π . One can ascribeeach vector from the set to a corresponding point on the sphere. These points lay on a circle, whose centre islocated at the point ascribed to vector | e ( θ , ϕ ) (cid:105) .3. Subsequently, using the vectors from the set, the measurements are performed and for each pair of vectors | e ( θ , ϕ i ) (cid:105) , | e ( θ , ϕ i ) (cid:105) the value of K Di is calculated. A set of values K D,N = { K D , . . . , K DN } is created.The aim of these measurements is to find two points laying on a chosen circle, characterized by values of ϕ i angle, for which value of K Di is equal to earlier calculated value K D . It is not difficult to see that it is alwayspossible to achieve this purpose when we assume continuity of ϕ (or arbitrary small resolution in ϕ i ). AccordingSubsection II C, because the sphere is covered with circles with the same value of K D , any other circle laying onthe sphere can have 0, 1, 2 or infinity intersection points. Thus it is always possible to find such values of θ , θ ,which ensure that points with the same value of K D are found. Just as for a plane, three points on the sphereare enough to unambiguously determine the circle. The radius and the centre of the circle are found solving thesystem of equations: d ( θ Max , ϕ
Max , θ , ϕ ) = R (12) d ( θ Max , ϕ
Max , θ , ϕ ) = R (13) d ( θ Max , ϕ
Max , θ , ϕ ) = R, (14)where d is spherical distance defined as [13]: d = arccos( P · Q ) , (15)here P , Q are two points on the sphere characterized by angles θ i , ϕ i and θ j , ϕ j , respectively. In therms ofCartesian coordinates ( x = sin θ cos ϕ , y = sin θ sin ϕ and z = cos θ ) expression (15) is of a form: d ( θ i , ϕ i , θ j , ϕ j ) == arccos(sin θ i sin θ j cos( ϕ i − ϕ j ) + cos θ i cos θ j ) . (16)According to Subsection II C centre of the circle determined in this way is associated with the basis (characterized byangles θ Max , ϕ Max ), in which K D ( θ, ϕ ) has maximal value. Fig. 1 presents main ideas of the proposed procedure.The proposed procedure can be slightly modified. Finding two points with value of K Di exactly equal K D can cause a problem and such solution is not a practical one. To overcome this difficulty, instead of finding two pointswith the same value of K D , one finds points K D , K D from the set K D,N , for which the values of K D , K D are theclosest to the K D i.e. for which ∆ K D = K D − K D and ∆ K D = K D − K D are minimal. Subsequently theinterpolating function I { K D,N } ( θ , ϕ ) from the set K D,N is created. To construct the interpolation function Hermitepolynomials of a required order are used. Thus one can write for i = { , } : K D == K Di + ∆ K Di == K D ( θ , ϕ i ) + ∂I { K D,N } ( θ , ϕ ) ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ i ∆ ϕ i + ∂ I { K D,N } ( θ , ϕ ) ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ i ∆ ϕ i == I { K D,N } ( θ , ϕ i ) + ∂I { K D,N } ( θ , ϕ ) ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ i ∆ ϕ i + ∂ I { K D,N } ( θ , ϕ ) ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ i ∆ ϕ i , (17)where we use the fact that I { K D,N } ( θ , ϕ i ) = K Di (i.e. the interpolation function reproduces the values of K Di fromthe set K D,N in the probe points). One solves equation (17) for ∆ ϕ i , i = { , } . In general, equation (17) can havetwo different solutions. However, in such a case one chooses smaller ∆ ϕ and ∆ ϕ (because equation (17) is Taylorexpansion of function I { K D,N } ( θ , ϕ ) near ϕ = ϕ i ). By solving modified systems of equations: d ( θ Max , ϕ
Max , θ , ϕ ) = R (18) d ( θ Max , ϕ
Max , θ , ϕ + ∆ ϕ ) = R (19) d ( θ Max , ϕ
Max , θ , ϕ + ∆ ϕ ) = R, (20)one obtains values of θ Max , ϕ
Max .The proposed approach enables to find the basis optimizing the value of K D by performing only local measurements. IV. ERROR ESTIMATION
Due to approximation (finite sum) and possible numerical errors, it is never possible to solve (17) exactly. As aresult angles θ (cid:48) M , ϕ (cid:48) M will not lead to the maximal value of distillable key. In this section the estimation of this erroris provided. Let us denote (see Fig. 2): ˜ ϕ = ϕ + ∆ ϕ = ϕ (cid:48) + ∆ ϕ (cid:48) ˜ ϕ = ϕ + ∆ ϕ = ϕ (cid:48) + ∆ ϕ (cid:48) θ M + ∆ θ M = θ (cid:48) M ϕ M + ∆ ϕ M = ϕ (cid:48) M . (21)Without loss of generality, we can arrange ˜ ϕ , ˜ ϕ so that ˜ ϕ > ˜ ϕ . Spherical distance between points characterizedby angles ( θ i , ϕ i ), ( θ j , ϕ j ) is given by (15). We assume that ( θ i , ϕ i ) = (0 , θ (cid:48) M ] = R (22)arccos [sin θ sin θ (cid:48) M cos ( ˜ ϕ − ϕ (cid:48) M ) + cos θ cos θ (cid:48) M ] = R (23)arccos [sin θ sin θ (cid:48) M cos ( ˜ ϕ − ϕ (cid:48) M ) + cos θ cos θ (cid:48) M ] = R. (24)Combining 23 and 24 we get: cos ( ˜ ϕ − ϕ (cid:48) M ) = cos ( ˜ ϕ − ϕ (cid:48) M ) . (25)Because ˜ ϕ (cid:54) = ˜ ϕ and ϕ (cid:48) M ∈ (0 , π ] there are two possibilities: ˜ ϕ − ϕ (cid:48) M = − ( ˜ ϕ − ϕ (cid:48) M ) or ˜ ϕ − ϕ (cid:48) M = − ( ˜ ϕ − ϕ (cid:48) M − π ) . We set ϕ (cid:48) M := ϕ (cid:48) M mod πϕ (cid:48) M = ˜ ϕ + ˜ ϕ ϕ M + ∆ ϕ M = ϕ + ϕ ϕ + ∆ ϕ , (26)so ∆ ϕ M = ∆ ϕ +∆ ϕ . However, we know only ϕ (cid:48) i , ∆ ϕ (cid:48) i but we can estimate (see Fig. 2) as | ∆ ϕ i | < ∆ ϕ − | ∆ ϕ (cid:48) i | . As aresult ∆ ϕ M < ϕ − | ∆ ϕ (cid:48) | − | ∆ ϕ (cid:48) | . (27)Combining equations (22) and (23) cot θ (cid:48) M = cot θ ϕ − ϕ (cid:48) M ) . (28)As a consequence of the equalitycos ( ˜ ϕ − ϕ (cid:48) M ) = cos (cid:18) ϕ + ∆ ϕ − ϕ + ∆ ϕ + ϕ + ∆ ϕ (cid:19) =cos (cid:18) ϕ + ∆ ϕ − ϕ − ∆ ϕ (cid:19) , (29)we obtain the following relation θ M = arccot (cid:20) cot θ (cid:18) ϕ − ϕ (cid:19)(cid:21) (30)and θ M + ∆ θ (cid:48) M = arccot (cid:20) cot θ ϕ − ϕ (cid:48) M ) (cid:21) . (31)In order to obtain the upper bound on ∆ θ (cid:48) M we have to find ¯ θ M - an lower bound on θ M . Then the following relationholds: ¯ θ M + ∆ θ (cid:48) M < θ M + ∆ θ (cid:48) M = arccot (cid:20) cot θ ϕ − ϕ (cid:48) M ) (cid:21) , (32)so ∆ θ (cid:48) M < arccot (cid:2) cot θ cos ( ˜ ϕ − ϕ (cid:48) M ) (cid:3) − ¯ θ M . We have to estimate the difference ϕ − ϕ using known quantities˜ ϕ , ˜ ϕ . There are two different possibilities: in the first one ˜ ϕ − ˜ ϕ < π whereas in the second ˜ ϕ − ˜ ϕ > π . Let usconsider the first one. Because arccot x ∈ ( − π , π ] is a decreasing function for x ∈ ( −∞ , ∪ (0 , ∞ ), in order to find¯ θ M we have to increase cos (cid:0) ϕ − ϕ (cid:1) . For our purposes we shall assume the worst case, namely ˜ ϕ < ϕ and ˜ ϕ > ϕ .Then ϕ − ϕ > ˜ ϕ − ˜ ϕ . From previous considerations the following relation holds: ˜ ϕ i + ∆ ϕ > ϕ i > ˜ ϕ i − ∆ ϕ i . As aresult ˜ ϕ − ˜ ϕ + 2∆ ϕ > ϕ − ϕ > ˜ ϕ − ˜ ϕ . Using this inequality we get θ M > ¯ θ M = arccot (cid:20) cot θ (cid:18) ˜ ϕ − ˜ ϕ + 2∆ ϕ (cid:19)(cid:21) . (33)If ˜ ϕ − ˜ ϕ > π the similar line of reasoning leads to θ M > ¯ θ M = arccot (cid:20) cot θ (cid:18) ˜ ϕ − ˜ ϕ − ϕ (cid:19)(cid:21) . (34)Finally one gets ∆ θ (cid:48) M < arccot (cid:20) cot θ ϕ − ϕ (cid:48) M ) (cid:21) − ¯ θ Mi , (35)where ¯ θ Mi , i = { , } is given by (33) or (34). Using perturbed points ( θ i , ˜ ϕ i ) one obtains the point ( θ M + ∆ θ M , ϕ M +∆ ϕ M )which differs from the real point ( θ M , ϕ M ) by (∆ θ M , ∆ ϕ M ), where ∆ θ M , ∆ ϕ M are given by (31) and (27). Inthe new coordinate system associated with the point ( θ M + ∆ θ M , ϕ M + ∆ ϕ M ) the error is given by:∆ θ = arccos [ sin θ (cid:48) M sin ( θ (cid:48) M − ∆ θ M ) cos ∆ φ M ++ cos θ (cid:48) M cos ( θ (cid:48) M − ∆ θ M )] . (36) V. CONDITIONS FOR INVARAINCE OF KEY RATE IN CASE OF LOCAL ACTION OF PAULICHANNELS
According to [11], using appropriate unitary operation U = 1 A ⊗ (cid:80) i | i (cid:105) (cid:104) i | ⊗ U ( i ) A (cid:48) B (cid:48) (called twisting) it is possible towrite a particular private state as: ρ priv = | Ψ + (cid:105) (cid:104) Ψ + | AB ⊗ σ A (cid:48) B (cid:48) , (37)where | Ψ + (cid:105) is one of the four Bell states | Ψ ± (cid:105) , | φ ± (cid:105)| Ψ ± (cid:105) = | (cid:105) ± | (cid:105)√ , | φ ± (cid:105) = | (cid:105) ± | (cid:105)√ . (38)After sending (37) down the channel Λ A ⊗ I where Λ A ( · ) = (cid:80) i p i K i ( · ) K † i with K i = { I, σ x , σ y , σ z } (note that twistingcommutes with the action of the channel) one obtains a state˜ ρ priv = ( p | Ψ + (cid:105) (cid:104) Ψ + | AB + p | Ψ − (cid:105) (cid:104) Ψ − | AB p | φ + (cid:105) (cid:104) φ + | AB + p | φ − (cid:105) (cid:104) φ − | AB ) ⊗ σ A (cid:48) B (cid:48) . (39)The purification of this state is given by (cid:12)(cid:12)(cid:12) ˜Ψ (cid:69) priv = (cid:20) p √ | (cid:105) AB + | (cid:105) AB ) | (cid:105) ¯ E p √ | (cid:105) AB − | (cid:105) AB ) | (cid:105) ¯ E p √ | (cid:105) AB + | (cid:105) AB ) | (cid:105) ¯ E p √ | (cid:105) AB − | (cid:105) AB ) | (cid:105) ¯ E (cid:21) ⊗ | ψ σ (cid:105) A (cid:48) B (cid:48) E , (40)where ¯ E, E denote Eves’ subsystem. It follows from (4) that we can trace over subsystems A’B’E (due to additivityof Von Neumann entropy for tensor product states K B A DW is independent of subsystems A’B’E). In order to find thevalue of the key rate due to Devetak - Winter protocol (4) we calculate the cqq state using base vectors defined by(10). The nonzero elements of the reduced AB matrix are given by12 a = cos θ p + p ) + sin θ p + p )12 a = sin θ p + p ) + cos θ p + p )12 a = sin θ p + p ) + cos θ p + p )12 a = cos θ p + p ) + sin θ p + p )12 a = a ∗ = e iϕ sin θ θ p − p ) + e − iϕ sin θ θ p − p )12 a = a ∗ = e iϕ sin θ θ p − p ) + e − iϕ sin θ θ p − p ) . (41)As a result entropies of Alice and Bob are equal S A = S B = 1. The reduced AB matrix is block diagonal so itseigenvalues are λ , = 14 (cid:18) (cid:113) cos θ ( p + p − p − p ) + sin θ | e iϕ ( p − p ) + e − iϕ ( p − p ) | (cid:19) λ , = 14 (cid:18) − (cid:113) cos θ ( p + p − p − p ) + sin θ | e iϕ ( p − p ) + e − iϕ ( p − p ) | (cid:19) . (42)As a consequence, I A : B will be independent of ϕ angle if and only if p − p or p = p . In order to minimize jointentropy S A : B one has to set θ = 0 or θ = π depending on { p , p , p } or { p , p , p } . Setting θ = 0 will be optimalif ( p + p − p ) > p = p or ( p + p − p ) > p = p , otherwise θ = π . In order to show that I A : ¯ E is independent of ϕ and let us consider a state resulting from the measurement performed on ¯ E subsystem. Thisoperation does not increase the value of I A : ¯ E (which we denote as I MA : ¯ E ) so we have I A : ¯ E ≥ I MA : ¯ E and due to (4) K B A DW (˜ ρ priv ) = I A : B − I A : ¯ E ≥ I A : B − I MA : ¯ E . (43)After measurement the reduced A ¯ E matrix has following eigenvalues λ , = p , λ , = p , λ , = p , λ , = p whereasthe eigenvalues of ¯ E matrix are given by λ = p , λ = p , λ = p , λ = p so we obtain that I A : ¯ E is independent of θ . As a result if p = p or p = p the distillable key will preserve its invariance and the proposed procedure will bevalid. VI. NUMERICAL RESULTS
In this section we provide some examples of the results obtained by implementing the above procedure numerically.
Rotated ρ SW AP state. -
Consider a private state introduced in [9] and realized experimentally [12] ρ SW AP = 14 | Ψ − (cid:105) (cid:104) Ψ − | AB ⊗ | Ψ − (cid:105) (cid:104) Ψ − | A (cid:48) B (cid:48) +14 | Ψ + (cid:105) (cid:104) Ψ + | AB ⊗ I A (cid:48) B (cid:48) − | Ψ + (cid:105) (cid:104) Ψ + | AB ⊗ | φ − (cid:105) (cid:104) φ − | A (cid:48) B (cid:48) . (44)This state was rotated and then the optimizing procedure was applied. The example of the results is shown in Fig. 3. Depolaraizig channel. -
The procedure was checked using the rotated state ˜ ρ SW AP = Λ A ⊗ I BA (cid:48) B (cid:48) , where Λ( ρ ) = p I + (1 − p ) ρ . The example of the results is shown in Fig. 4. Phase flip channel. -
Another test was performed using the rotated ˜ ρ SW AP = Λ A ⊗ I BA (cid:48) B (cid:48) , where Λ( ρ ) = pρ + (1 − p ) σ z ρσ z . The example of the results is shown in Fig. 5. Rotated mixture of ρ SW AP and ρ MSW AP states. -
Another example of the private states is a state ρ MSW AP = 12 | φ − (cid:105) (cid:104) φ − | ⊗ (cid:18) | (cid:105) (cid:104) | + | Ψ + (cid:105) (cid:104) Ψ + | (cid:19) ++ 12 | φ + (cid:105) (cid:104) φ + | ⊗ (cid:18) | (cid:105) (cid:104) | | Ψ − (cid:105) (cid:104) Ψ − | (cid:19) . (45)This state plays a role in bound entangled secure key [14]. We checked the procedure using rotated mixture of twoprivate states ρ priv = pρ SW AP +(1 − p ) ρ MSW AP and the rotated ˜ ρ SW AP = Λ A ⊗ I BA (cid:48) B (cid:48) , where Λ( ρ ) = pρ +(1 − p ) σ z ρσ z .The example of the results is shown in Fig. 6. Qubit channel with trigonometrical parametrization. -
Consider a channel given by the Kraus operators [15] K = (cid:104) cos u v (cid:105) I + (cid:104) sin u v (cid:105) σ z K = (cid:104) cos u v (cid:105) σ x − i (cid:104) sin u v (cid:105) σ y (46)which transforms the Bloch vector (cid:126)r = [ r x , r y , r z ] T of the state into (cid:126)r (cid:48) = [cos ur x , cos vr y , cos u cos vr z + sin u sin v ] T .It follows from Section V that in general this channel does not preserve the invariance of distillable key. In this casethe procedure fails. The example of the results is shown in Fig. 7. VII. CONCLUSIONS
In the present paper the new symmetry of the states with perfect secure key called generalized private stateshave been provided which says that the most popular Devetak-Winter protocol secret key rate is invariant in bothscenarios of CCQ and CQQ type if the measurement bases, chosen in a wrong way, are rotated (in a sense of angularmomentum) around the axis corresponding to the secure basis by any angle. The symmetry has a particularly goodinterpretation when seen on a sphere since then the wrong is any basis corresponding to ˆ n with an angle θ to the ˆ z axis (corresponding to the secure basis) while the symmetry rotation is just the rotation by the φ angle around theˆ z axis. We have also proven that for the qubit key part the optimality of the ˆ z axis as the secure basis is preservedafter the action of any bistochastic channel (i.e. the one represented by random Pauli rotations).The symmetry of the ideal p-bit lead us to the heuristic scheme of estimation of the optimal axis (with respect tothe Devetak-Winter secret key rate K DW ) which is valid for any state that has this type of symmetry of the key rateunder the rotation around the optimal basis. Namely given the density matrix, may be even in a numerical form,instead of searching over all sphere Alice may perform the analysis of the key over a ring around some chosen axis ˆ z (cid:48) on the sphere and guess the optimal measurement axis only on the data based on this ring.The method generally has a ,,dualistic” character with respect to the channel action. If the Alice subsystem as thedirection of the optimal axis unperturbed, than the results are good if the ˆ z (cid:48) is chosen to be far form the (unknown)optimal one ˆ z while if there is a perturbation of the optimal direction ˆ z (in a sense of the shrinking of that directionon a Bloch sphere) then the closer is the chosen axis to the original one the result is better. Basing on the polynomialapproximation of the key function on the chosen ring there is also the possibility of the derivation of the error bar FIG. 1: Local Alice’s sphere with two different coordinate systems. Z axis corresponds to θ Max , ϕ
Max angles whereas Z’ to θ , ϕ angles. Black circle shows the path along which angle ϕ changes. At the intersection point between Z axis and thesphere K D ( θ Max , ϕ
Max ) = K DMax . The intersection point between Z’ axis and the sphere is denoted by ( θ , ϕ ). At this point K D ( θ , ϕ ) = K D . Points with the same value of K D (laying on a circle, whose center is located at the point K D ) are denotedby ( θ , ϕ ) and ( θ , ϕ ). At the first point K D ( θ , ϕ ) = K D , whereas at the second point K D ( θ , ϕ ) = K D .FIG. 2: Starting point for the procedure ( θ , ϕ ), in which K ( θ , ϕ ) = K D . Point ( θ , ϕ (cid:48) ), K D = K ( θ , ϕ (cid:48) ) is closest tothe point ( θ , ϕ ) in the given set because∆ K D = | K D − K D | is minimal (see text for details). Using the interpolationfunction one obtains that K ( θ , ϕ (cid:48) + ∆ ϕ (cid:48) ) = K D , however, due to numerical error, in fact ϕ (cid:48) + ∆ ϕ (cid:48) (cid:54) = ϕ , where the equality K D ( θ , ϕ (cid:48) + ∆ ϕ (cid:48) ) = K ( θ , ϕ ) really holds. The case of the second point ( θ , ϕ (cid:48) + ∆ ϕ (cid:48) ) is similar. The error of the procedureis given by (36). of the procedure. The analysis of examples shows that the error bar in general bounds the actual value of the errormade in the procedure.We believe that the present method may be especially useful when the large sample of data are provided and quickestimation of optimal Alice measurement is needed. Acknowledgments . We thank Ewa and J¸edrzej Tuziemscy for help in preparation of Figiures. Calculations werecarried out at the Academic Computer Center in Gda´nsk. This work was supported by 7th Framework ProgrammeFuture and Emerging Technologies project Q-ESSENCE. [1] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod. Phys. , 865 (2009)[2] C. H. Bennett and G. Brassard, Proceedings of the IEEE International Conference on Computers, Systems and SignalProcessing IEEE Computer Society, New York, 1984, pp. 175179.[3] A. K. Ekert, Phys. Rev. Lett. , 661 (1991). [4] C. H. Bennett, G. Brassard, and N. D. Mermin, 1992, Phys. Rev. Lett. , 557 (1992).[5] D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Phys. Rev. Lett. , 2818 (1996).[6] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. , 722(1996).[7] P. W. Shor, J. Preskill, Phys. Rev. Lett. , 441 (2000)[8] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. , 5239 (1998).[9] K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim, Phys. Rev. Lett. , 160502 (2005).[10] K. Dobek, M. Karpinski, R. Demkowicz-Dobrzanski, K. Banaszek, P. Horodecki, Phys. Rev. Lett. , 030501 (2011).[11] K. Horodecki, M. Horodecki, P. Horodecki, J. Oppenheim IEEE Trans. Inf. Theory , 1898 (2009).[12] K. Dobek, M. Karpinski, R. Demkowicz-Dobrzanski, K. Banaszek, P. Horodecki Phys. Rev. Lett. , 030501 (2011).[13] T. Rowland Spherical Distance.
MathWorld–A Wolfram Web Resource, created by Eric W. Weisstein.http://mathworld.wolfram.com/SphericalDistance.html[14] K. Horodecki, . Pankowski, M. Horodecki, P. Horodecki IEEE Trans. Inf. Theory , 2621 (2008).[15] M. B. Ruskai, S. Szarek, E. Werner Lin. Alg. Appl. , 159 (2002).[16] Points on the sphere correspond to angles used in (10). The term sphere should not be confused with the Bloch sphere ofthe state. FIG. 3: The estimated error of the procedure (using formula (36)) - left plot, and the error of the procedure (absolute value ofthe difference between rotation angle and the angle calculated by the procedure ∆ θ = | θ − θ M (cid:48) | ) -right plot vs. the numberof points r=10n (n=1 denotes that 10 points were used) for the rotated ρ SWAP state. Rotation angle θ = π . Different colorscorrespond to different choices of θ angle in (20): blue - θ =0.0025 π , red - θ =0.005 π , orange - θ =0.0075 π , red - θ =0.01 π . In agreement with (36), for all cases the estimated error of the procedure constitute an upper bound on the error. The errorof the procedure is orders of magnitude smaller than its estimated value.FIG. 4: The estimated error of the procedure (using formula (36)) - left plot and, the error of the procedure (absolute value ofthe difference between rotation angle and the angle calculated by the procedure ∆ θ = | θ − θ M (cid:48) | ) -right plot vs. the numberof points r=10n (n=1 denotes that 10 points were used) for the state ˜ ρ SWAP = Λ A ⊗ I BA (cid:48) B (cid:48) , where Λ( ρ ) = p I + (1 − p ) ρ .Rotation angle θ = π , p = . Different colors correspond to different choices of θ angle in (20): blue - θ =0.0025 π , red - θ =0.005 π , orange - θ =0.0075 π , red - θ =0.01 π . In agreement with (36), for all cases the estimated error of the procedureconstitute an upper bound on the error. The error of the procedure is orders of magnitude smaller than its estimated value.FIG. 5: The estimated error of the procedure (using formula (36)) - left plot and the error of the procedure (absolute value ofthe difference between rotation angle and the angle calculated by the procedure ∆ θ = | θ − θ M (cid:48) | ) -right plot vs. the numberof points r=10n (n=1 denotes that 10 points were used) for the state ˜ ρ SWAP = Λ A ⊗ I BA (cid:48) B (cid:48) , where Λ( ρ ) = pρ + (1 − p ) σ z ρσ z .Rotation angle θ = π , p= . Different colors correspond to different choices of θ angle in (20): blue - θ =0.0025 π , red - θ =0.005 π , orange - θ =0.0075 π , red - θ =0.01 π . In agreement with (36), for all cases the estimated error of the procedureconstitute an upper bound on the error. The error of the procedure is orders of magnitude smaller than its estimated value. FIG. 6: The estimated error of the procedure (using formula (36)) - left plot and the error of the procedure (absolute value ofthe difference between rotation angle and the angle calculated by the procedure ∆ θ = | θ − θ M (cid:48) | ) -right plot vs. the number ofpoints r=10n (n=1 denotes that 10 points were used) for the state ρ priv = pρ SWAP + (1 − p ) ρ MSWAP . Rotation angle θ = π ,p= . Different colors correspond to different choices of θ angle in (20): blue - θ =0.0025 π , red - θ =0.005 π , orange - θ =0.0075 π , red - θ =0.01 π . In agreement with (36), for all cases the estimated error of the procedure constitute an upperbound on the error. The error of the procedure is orders of magnitude smaller than its estimated value.FIG. 7: The estimated error of the procedure (using formula (36)) and the error of the procedure (absolute value of thedifference between rotation angle and the angle calculated by the procedure ∆ θ = | θ − θ M (cid:48) | ) vs. the number of points r=10n(n=1 denotes that 10 points were used) for the state ρ priv = (Λ A ⊗ BA (cid:48) B (cid:48) ) ρ SWAP , Λ A ( ρ ) = K † ρK + K † ρK where K i aregiven by (46). Rotation angle θ = π , u=0.1 π , v=0.05 π . Different colors correspond to different choices of θ angle in (20):blue - θ =0.0025 π , red - θ =0.005 π , orange - θ =0.0075 π , red - θ =0.01 ππ