The Relationship Between Discrete and Continuous Entropy in EPR-Steering Inequalities. Abstract

Transcription

1 The Reationship Between Discrete and Continuous Entropy in EPR-Steering Inequaities James Schneeoch 1 1 Department of Physics and Astronomy, University of Rochester, Rochester, NY arxiv: v1 [quant-ph] 9 Dec 2013 (Dated: December 11, 2013) Abstract This document expands upon the reationship between discrete and continuous entropy given in (Phys. Rev. Lett ), Vioating Continuous Variabe Einstein-Podosky-Rosen Steering with Discrete Measurements. We provide a detaied derivation for the inequaity reating the continuous conditiona entropy to its discrete approximation, and show how this connection works between discrete and continuous entropic quantities in genera. In addition, we use this connection to show how to derive the continuous variabe Einstein-Podosky-Rosen steering inequaity with discrete measurements as seen in (Phys. Rev. Lett ), and make an additiona comment which strengthens this resut. 1

2 EXTENDED PROOF OF ENTROPY CONNECTION INEQUALITIES To derive our continuous variabe Einstein-Podosky-Rosen (EPR)-steering inequaity (20)[1], we used a fundamenta connection between continuous and discrete entropies (9) to show that any two continuous random variabes x and y that can be discretized into equay spaced windows of size x and y satisfy the foowing inequaity; h(y x) H(Y X)+og( y), (1) where the base of the ogarithms here and throughout this paper are equa to the base in which one chooses to measure the entropy. The Fundamenta Connection between Continuous and Discrete Entropies Consider an experiment to measure random variabe x which can take the vaue of any rea number with probabiity density ρ(x). The experiment is ony capabe of measuring x to discrete windows X of size x. The probabiity of measuring x to be within window X is P(X ) = dx ρ(x), (2) where the region of integration is the range of vaues of x between x 1 x and 2 x x, and x is the vaue of x at the center of the window X. The Shannon entropy of this discrete probabiity distribution is given by H(X) = P(X )og(p(x )), (3) and the Shannon entropy of the continuous probabiity density function ρ(x) is expressed as h(x) = dx ρ(x)og(ρ(x)). (4) We now define the probabiity density function ρ (x) as the probabiity distribution of x conditioned on having been measured within window X. The continuous entropy h (x) is defined as the entropy of ρ (x) where ρ (x) = ρ(x) P(X ) (5) 2

3 for a vaues of x in the window X, and is zero otherwise. By breaking up the continuous entropy h(x) into a sum over a windows, h(x) = dx ρ(x)og(ρ(x)), (6) and expressing h(x) in terms of P(X ) and ρ (x), h(x) = ( dx ρ (x)p(x ) ) og ( ρ (x)p(x ) ) = ( ) P(X ) dx ρ (x) og(ρ (x))+og(p(x ), (7) and then in terms of h (x), h(x) = P(X ) dx ρ (x)og(ρ (x)) P(X )og(p(x )), (8) x we obtain the fundamenta connection between discrete and continuous entropies; h(x) = P(X )h (x)+h(x). (9) Continuing the Extended Proof This connection (9) exists for joint entropies as we as for margina entropies. Using this, we now define h m (x,y) as the entropy of the joint distribution ρ m (x,y) in which x is conditioned on being measured within window X and y is conditioned on being measured within window Y m. The conditiona entropies h(y x) and H(Y X) are defined as differences between joint and margina entropies [2], h(y x) h(x,y) h(x), H(Y X) H(X,Y) H(X). (10a) (10b) By using the fundamenta connection (9) for both margina and joint entropies, aong with the definition of conditiona entropy (10a), we can show that h(y x) =,m P(X,Y m )h m (x,y) P(X )h (x)+h(y X) = P(X ) m P(Y m X )h m (x,y) P(X )h (x)+h(y X). (11) 3

4 Conditioning on additiona events on average reduces the entropy. This is a consequence both of Jensen s inequaity and the fact that the entropy is a concave function [2]. As such, we can say that where ρ(x) = P(X )ρ (x), (12) we have the inequaity h(x) P(X )h (x). (13) Simiary, where [3] ρ (x) = m P(Y m X )ρ m (x), (14) we have the inequaity h (x) m P(Y m X )h m (x). (15) When this reation (15) is substituted as a minimum vaue of h (x) in the expression for h(y x), (11), it can be shown that h(y x) P(X,Y m )h m (y x)+h(y X). (16),m The uniform distribution maximizes the entropy, so that when a windows y m are of equa size, we have h m (y x) og( y), which competes our proof of (1). This approach aows us to summarize the connection between continuous and discrete entropic quantities incuding the mutua information, h(x : y) defined as h(x)+h(y) h(x, y). In short: h(x) H(X)+og( x), h(x,y) H(X,Y)+og( x y), h(x y) H(X Y)+og( x), h(x : y) H(X : Y). This aso transates to three or more variabes easiy, giving us h(x,y,z) H(X,Y,Z)+og( x y z), h(x,y z) H(X,Y Z)+og( x y), h(x y,z) H(X Y,Z)+og( x), h(x : y,z) H(X : Y,Z), 4

5 and in particuar, we have ( n h( x y) H( X Y)+og x i ), (17) where x is a vector of the random variabes (x 1,x 2,...,x n ). It remains to be shown whether such a simpe reationship exists between the conditiona mutua informations h(x : y z) and H(X : Y Z). Since the conditiona mutua information h(x : y z) can be either greater or ess than the ordinary mutua information, h(x : y), [2] the reationship between h(x : y z) and H(X : Y Z) may not be straightforward. i=1 INCORPORATING THE ENTROPY CONNECTION INTO OUR STEERING IN- EQUALITY Where x Ai and x Bi are another ordinary pair of random variabes, we can substitute the arger discrete approximations (1) for each continuous conditiona entropy in the steering inequaity derived by Waborn et. a [4], h(x Bi x Ai )+h(k Bi k Ai ) og(πe), (18) to derive our entropic EPR steering inequaity suitabe for experimenta investigations of continuous variabe entangement. For a particuar spatia degree of freedom i {1,..., n}, (i.e. a particuar dimension in space) we ve shown that ( ) πe H(X Bi X Ai )+H(K Bi K Ai ) og. (19) x Bi k Bi When different spatia degrees of freedom are statisticay independent of one another, the entropies add, giving us the n-dimensiona discrete steering inequaity H( X B X A )+H( K B K A ) n i=1 ( og πe x Bi k Bi ). (20) Comment on Steering Inequaity Derivation Thoughthe steering inequaity just arrived at (20) is quite vaid, it is in fact stronger than the prior derivation suggests. The prior derivation requires the assumption that different spatia degrees of freedom be independent of one another, so that the steering inequaity 5

6 in mutipe dimensions (20) is just the sum of the steering inequaities in each dimension (19). We can in fact do away with this assumption, by noting that without any additiona assumptions, Waborn et. a s steering inequaity [4] in n spatia degrees of freedom is h( x B x A )+h( k B k A ) nog(πe), (21) because Bia ynicki-birua and Mycieski s entropic uncertainty reation [5] in n spatia degrees of freedom is h( x)+h( k) nog(πe), (22) which requires no additiona assumptions to prove. Knowingthis, wecanusethesteering inequaity (21)andtheconnection(17)toprove our steering inequaity (20) without any extra assumptions. This strengthens the significance of our experimenta vioation of (20), so that, oophoes aside, it is a soid demonstration of EPR-steering without specia caveats. [1] J. Schneeoch, P. B. Dixon, G. A. Howand, C. J. Broadbent, and J. C. Howe, Phys. Rev. Lett. 110, (2013). [2] T. M. Cover and J. A. Thomas, Eements of Information Theory, 2nd ed. (Wiey and Sons, New York, 2006). [3] ρ m (x) is just what you get when you integrate over a vaues of y the probabiity density function ρ m (x,y). [4] S. P. Waborn, A. Saes, R. M. Gomes, F. Toscano, and P. H. Souto Ribeiro, Phys. Rev. Lett. 106, (2011). [5] I. Bia ynicki-birua and J. Mycieski, Communications in Mathematica Physics 44, 129 (1975), /BF