A. Uhlmann, \The `transition probability' on the state space of a -algebra," Rep. Math. Phys. 9, 273 (1976).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....distinguishability measure for quantum states, the delity, is given by the formidable looking expression F ( 1 ; 2 ) 2.34) 27 where 1 and 2 are a pair of density operators. The delity (2. 34) was introduced by Jozsa [63] but the original idea came from the work of Uhlmann [138] who generalized the notion of the transition probability h j i for pure states to general states over C algebras. For this reason we will refer to the delity F as the Jozsa Uhlmann delity. The main appeal of the Jozsa Uhlmann delity lies in the result known as the Uhlmann theorem [138] We ....

....[138] who generalized the notion of the transition probability h j i for pure states to general states over C algebras. For this reason we will refer to the delity F as the Jozsa Uhlmann delity. The main appeal of the Jozsa Uhlmann delity lies in the result known as the Uhlmann theorem [138]. We state this theorem in the form given by Jozsa [63] Theorem 2.3.2 (Uhlmann) Let 1 and 2 be density operators on a Hilbert space H . Then F ( 1 ; 2 ) max 1 ; 2 jh 1 j 2 ij ; 2.35) where the maximum is taken over all puri cations 1 and 2 of 1 and 2 respectively in an ....

A. Uhlmann, \The `transition probability' on the state space of a -algebra," Rep. Math. Phys. 9, 273 (1976).

.... their delity is de ned as F ( 1 ; 2 ) sup jh 1 j 2 ij where the supremum is taken over all puri cations j i i of i in the same Hilbert space [19] Jozsa [19] gave a simple proof, for the nite dimensional case, of the following remarkable equivalence rst established by Uhlmann [25]. Theorem II.4 (Jozsa) For any two density matrices 1 ; 2 on the same nite dimensional space H, F ( 1 ; 2 ) Tr ( 1 2 1 ) i 2 = k 1 2 k Using this equivalence, Fuchs and van de Graaf [20] relate delity to the trace distance. Theorem II.5 (Fuchs, van de ....

A. Uhlmann, \The `transition probability' in the state space of a -algebra.," Reports on Mathematical Physics, vol. 9, pp. 273-279, 1976.

.... is defined as F (#1 , #2) sup ##1 #2 # 2 , where the supremum is taken over all purifications # i # of # i in the same Hilbert space [12] Jozsa [12] gave a simple proof, for the finite dimensional case, of the following remarkable equivalence first established by Uhlmann [27]. Theorem 2.3 (Jozsa) For any two density matrices #1 , #2 on the same finite dimensional space H, F (#1 , #2) # Tr ( # #1 #2 # #1) 1 2 # 2 = # # #1 # #2 # 2 t . Using this equivalence, Fuchs and van de Graaf [11] relate fidelity to the trace distance. Theorem 2.4 (Fuchs, van ....

A. Uhlmann. The `transition probability' in the state

.... H, their delity is de ned as F ( 1 ; 2) sup jh 1 j 2 ij 2 ; where the supremum is taken over all puri cations j i i of i in the same Hilbert space [12] Jozsa [12] gave a simple proof, for the nite dimensional case, of the following remarkable equivalence rst established by Uhlmann [27]. Theorem 2.3 (Jozsa) For any two density matrices 1 ; 2 on the same nite dimensional space H, F ( 1 ; 2) h Tr ( p 1 2 p 1) 1 2 i 2 = k p 1 p 2 k 2 t : Using this equivalence, Fuchs and van de Graaf [11] relate delity to the trace distance. Theorem 2.4 (Fuchs, van ....

A. Uhlmann. The `transition probability' in the state space of a -algebra. Reports on Mathematical Physics, 9:273-279, 1976.

....trace in a different environment basis (related to the first by that same unitary) 2. 5 Fidelity A good measure F (ae 1 ; ae 2 ) of overlap between quantum states one satisfying a set of intuitively appealing axioms such as F (ae; ae) 1; F (ae; j ih j) h jaej i [12] is termed by Uhlmann [13] transition probability , for which I prefer Jozsa s term fidelity . We may define it by: F (ae 1 ; ae 2 ) j max jOE 2 i jhOE 1 jOE 2 ij 2 ; 2.31) where jOE 1 i is an arbitrary purification of ae 1 and jOE 2 i ranges over all possible purifications of ae 2 . This quantity was introduced ....

....ae 2 . This quantity was introduced (in the more general w algebra context) 40 by Bures [14] who showed that it was equal to (tr( q ae 1=2 1 ae 2 ae 1=2 1 ) 2 (2. 32) Jozsa [15] presents version of the proof which does not use the language of algebra representation theory; Uhlmann [13] also contains a proof. For pure states, the fidelity is just jh 1 j 2 ij 2 ; for one pure and one mixed state, it is h 1 jae 2 j 1 i. We will define the fidelity so as to take values not just on density operators, but on all positive operators, by extending the notion of purification to ....

[Article contains additional citation context not shown here]

A. Uhlmann, "The "transition probability" in the state space of a -algebra," Reports on Mathematical Physics, vol. 9, pp. 273--279, 1976.

....because by expanding a Hilbert space one can only add distinguishability. However, the accessible information for two pure states can be calculated exactly; it is given by the appropriate analog to Eq. 1. 10) Using a result for the maximal possible overlap between two puri cations due to Uhlmann [7], one arrives at the tightest possible upper bound of this form. Statistical Overlap The second most important way of de ning a notion of statistical distinguishability (see Section 3.3) concerns the following scenario. One imagines a nite number of copies, N , of a quantum system secretly ....

....conceptually as itself) The optimal statistical overlap is given by F ( 0 ; 1 ) tr q 1=2 1 0 1=2 1 = tr q 1=2 0 1 1=2 0 ; 1. 15) 5 a quantity known as the delity for quantum states, which has appeared in other mathematical physics contexts [7, 9]. Here we start the trend in notation that the same function is used to denote both the classical distinguishability and its quantum version; notice that the former has the probability distributions as its argument and the latter has the density operators themselves. This measure of ....

[Article contains additional citation context not shown here]

A. Uhlmann, \The `transition probability' in the state space of a  -algebra," Reports on Mathematical Physics, vol. 9, pp. 273-279, 1976.

....) k = 1; N (3) be the reduced state in the k th signal position after coding and decoding; i.e. ae k is the decoded version of the k th transmitted state ae i k . Let F (ae 1 ; ae 2 ) i trace(ae 1=2 1 ae 2 ae 1=2 1 ) 1=2 j 2 (4) denote the Bures Uhlmann fidelity function [15, 16, 17]. The coding decoding scheme has fidelity 1 Gamma ffl if it satisfies the following fidelity requirement: There is an N 0 such that for all N N 0 , X oe N Prob(oe N ) N Y k=1 F (ae i k ; ae k ) 1 Gamma ffl (LOCAL FID) 5) Note that high fidelity according to (LOCAL FID) allows ....

....mixture of j 1 i and j 2 i. Thus to get the greatest benefit from Schumacher compression, the purifications should be chosen so that their ensemble has least von Neumann entropy; i.e. the two purifications should be as parallel as possible. According to Bures and Uhlmann s basic 11 theorem [15, 16, 17], the minimum possible angle min between purifications of ae 1 and ae 2 is given by cos 2 min = F (ae 1 ; ae 2 ) Moreover, a 50 50 mixture of states at angle min has entropy S min = H 1 cos min 2 ; 1 Gamma cos min 2 ; which gives the Schumacher limit of ....

A. Uhlmann, "The "transition probability" in the state space of a -algebra," Reports on Mathematical Physics, vol. 9, pp. 273--279, 1976.

....true that repeating the measurement is guaranteed to give the same result when the operation is F 1=2 b : I conjecture there is no operation which can provide this guarantee in the case of nonorthogonal F b . I will use the fidelity F (ae; oe) tr q ae 1=2 oeae 1=2 ) 2 [7] [8], 9] in specifying a measure of disturbance for quantum states. For pure states ae = j ih j, this is just h joej i. It is unity when ae = oe, and zero when their supports are orthogonal. It is therefore a reasonable measure of how similar two quantum states are. We may define 1 Gamma F (ae; ....

A. Uhlmann, "The "transition probability" in the state space of a -algebra," Reports on Mathematical Physics, vol. 9, pp. 273--279, 1976.

....by using (13) to define the transition probability. This enlarges the set of observables O n of (7) and one gets O n 1 . 5 As a first indication that things go together with the definition (13) we compare it with the transition probability in the state space of unital algebras, 19] [36]. The rather different definitions in these references turned out to coincide for unital C algebras, 33] 11] For two states, 1 ; 2 of a unital C algebra A one knows [3] p( 1 ; 2 ) inf 1 (A) 2 (A Gamma1 ) A 0 A; A Gamma1 2 A (14) A similar relation is true with ....

....fix point set being the embedded state space of the embedded Jordan algebra. The next aim is to compare and to compute some transition probabilities. Let and be two states of Omega : Omega Gamma H) Their transition probability can expressed by their density operators, D and D , 10] [36], p( 1=2 = tr q (D 1=2 D D 1=2 ) 30) A nice way to describe this is by the help of an operator version of the non commutative geometrical mean [32] S#R : R 1=2 (R Gamma1=2 S R Gamma1=2 ) 1=2 R 1=2 ; R 0; S 0 (31) Then, assuming and faithful, consider the ....

[Article contains additional citation context not shown here]

A. Uhlmann, The "Transition Probability" in the State Space of a -Algebra, Rep. Math. Phys. 9 (1976) 273

No context found.

A. Uhlmann. The `transition probability' in the state space of a #-algebra. Reports on Mathematical Physics, 9:273--279, 1976.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC