R. Jozsa, "Fidelity for mixed quantum states.," Journal of Modern Optics, vol. 41(12), pp. 2315--2323, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....basis of 1 . Note that #### is an admissible state corresponding to the maximally chaotic density operator (dim 1 ) 1 1H 1 . Then the channel fidelity [15] is defined by 1 , T 2 ) 1 # 2 # , 18) where the quantity on the r.h.s. of (18) is the mixed state fidelity [16]. We do not need all of the properties of the channel fidelity (18) but see Ref. 15] except the following: 2 (F(T 1 , T 2 ) # ## 1 , 19) which is a simple corollary of the results of Fuchs and van de Graaf [17] We also note that the channel fidelity has the natural property ....

.... 2 ) # ## 1 , 19) which is a simple corollary of the results of Fuchs and van de Graaf [17] We also note that the channel fidelity has the natural property that 1 , T 2 ) 1 if and only if T 1 T 2 (this is a straightforward consequence of the properties of the mixed state fidelity [16]) We can rewrite w 1 and w 2 from (15) in terms of #, # 1 , and # 2 : # i = # 1 2 )w i (1 # 1 2 ) i = 1, 2. 20) Then we can use the well known inequalities #AB# # #A##B# #A # A# = where is the usual operator norm [9] to get . 21) Combining this ....

R. Jozsa, "Fidelity for mixed quantum states," J. Mod. Opt. 41, 2315--2323 (1994).

....Sec. 2.3.3. 2.3.2 Jozsa Uhlmann delity Another useful 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 ....

....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 extended Hilbert space K . Proof: Without loss of ....

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R. Jozsa, \Fidelity for mixed quantum states," J. Mod. Opt. 41, 2315 (1994).

....encoding theorem in case of the uniform distribution on X, without using Lindblad Uhlmann monotonicity, with the constant 2 ln 2 replaced with the somewhat weaker constant 4. Another primitive we use is derived from the work of Lo and Chau [17] and Mayers [18] and combines results of Jozsa [19], and Fuchs and van de Graaf [20] Consider two bi partite pure states such that one party sharing the states cannot locally distinguish between the two states with signi cant probability. Then the other party can locally transform any of the states to a state that is close to the other. Theorem ....

....pure states such that one party sharing the states cannot locally distinguish between the two states with signi cant probability. Then the other party can locally transform any of the states to a state that is close to the other. Theorem I. 6 (Local transition theorem) based on [17] 18] [19], 20] Let 1 ; 2 be two mixed states with support in a Hilbert space H, K any Hilbert space of dimension at least dim(H) and j i i any puri cations of i in K. Then, there is a local unitary transformation U on K that maps j 2 i to j 2 i = I U j 2 i such that 2 j k t 2 k ....

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R. Jozsa, \Fidelity for mixed quantum states.," Journal of Modern Optics, vol. 41(12), pp. 2315-2323, 1994.

....string) is on average a good approximation of any encoded state. Thus, in certain situations, we may dispense with the encoding altogether, and use the single state # instead. We also use another primitive derived from the work of Lo and Chau [14] and Mayers [15] which combines results of Jozsa [10], and Fuchs and van de Graaf [8] Consider two bi partite pure states such that one party sharing the states cannot locally distinguish between the two states with significant probability. Then the other party can locally transform any of the states to a state that is close to the other. Theorem ....

....two bi partite pure states such that one party sharing the states cannot locally distinguish between the two states with significant probability. Then the other party can locally transform any of the states to a state that is close to the other. Theorem 1. 4 (Local transition theorem) based on [14, 15, 10, 8]) Let # 1 , # 2 be two mixed states with support in a Hilbert space H, K any Hilbert space of dimension at least dim(H) and # i # any purifications of # i in H# K. Then, there is a local unitary transformation U on K that maps # 2 # to # # 2 # = I# U # 2 # such that # ....

R. Jozsa. Fidelity for mixed quantum states. Journal of Modern Optics, 41(12):2315--2323, 1994.

....get the average encoding theorem as a special case. This more general theorem seems to be of independent interest. A classical version of the theorem can be found in, e.g. 9] We also use another primitive derived from the work of Lo and Chau [17] and Mayers [18] which combines results of Jozsa [12], and Fuchs and van de Graaf [11] Consider two bi partite pure states such that one party sharing the states cannot locally distinguish between the two states with significant probability. Then the other party can locally transform any of the states to a state that is close to the other. Theorem ....

....two bi partite pure states such that one party sharing the states cannot locally distinguish between the two states with significant probability. Then the other party can locally transform any of the states to a state that is close to the other. Theorem 1. 6 (Local transition theorem) based on [17, 18, 12, 11]) Let #1 , #2 be two mixed states with support in a Hilbert space H, K any Hilbert space of dimension at least dim(H) and # i # any purifications of # i in H# K. Then, there is a local unitary transformation U on K that maps #2# to # # 2 # = I# U #2# such that # # #1 ....

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R. Jozsa. Fidelity for mixed quantum states. Journal of Modern Optics, 41(12):2315--2323, 1994.

....In this case, the distance between the two density matrices is measured by the delity F ( 1 ; 2 ) The delity is de ned as F ( 1 ; 2) max j 1 i;j 2 i jh 1 j 2 ij 2 , over all choices of j 1 i and j 2 i that give density matrices 1 and 2 when a part of system is traced out. Lemma 3. [13] Let 1 , 2 be two mixed states with support in a Hilbert space H, K any Hilbert space of dimension at least dim(H) and j i i any puri cations of i in H K. Then, there is a local unitary transformation U on K that maps j 2 i to j 0 2 i = I U j 2 i such that jh 1 j 0 2 ij 2 = F ....

R. Jozsa. Fidelity for mixed quantum states. Journal of Modern Optics, 41:2315-2323, 1994.

....get the average encoding theorem as a special case. This more general theorem seems to be of independent interest. A classical version of the theorem can be found in, e.g. 9] We also use another primitive derived from the work of Lo and Chau [17] and Mayers [18] which combines results of Jozsa [12], and Fuchs and van de Graaf [11] Consider two bi partite pure states such that one party sharing the states cannot locally distinguish between the two states with signi cant probability. Then the other party can locally transform any of the states to a state that is close to the other. Theorem ....

....two bi partite pure states such that one party sharing the states cannot locally distinguish between the two states with signi cant probability. Then the other party can locally transform any of the states to a state that is close to the other. Theorem 1. 6 (Local transition theorem) based on [17, 18, 12, 11]) Let 1 ; 2 be two mixed states with support in a Hilbert space H, K any Hilbert space of dimension at least dim(H) and j i i any puri cations of i in H K. Then, there is a local unitary transformation U on K that maps j 2i to j 0 2 i = I U j 2 i such that j 1ih 1 j 0 2 0 ....

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R. Jozsa. Fidelity for mixed quantum states. Journal of Modern Optics, 41(12):2315-2323, 1994.

....to do the enviroment partial 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 . ....

R. Jozsa, "Fidelity for mixed quantum states," Journal of Modern Optics, vol. 41(12), pp. 2314--2323, 1994.

....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 ....

....has appeared before as the distance function d 2 B ( 0 ; 1 ) 2 2F ( 0 ; 1 ) 3.29) of Bures [80, 81] the generalized transition probability for mixed states prob( 0 1 ) F ( 0 ; 1 ) 2 (3. 30) of Uhlmann [7] and in the same form as Uhlmann s Jozsa s criterion [9] for delity of signals in a quantum communication channel. Note that Jozsa s delity [9] is actually the square of the quantity called delity here. The delity was found to be of use within that context because it is symmetric in 0 and 1, because it is invariant under unitary operations, ....

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R. Jozsa, \Fidelity for mixed quantum states," Journal of Modern Optics, vol. 41(12), pp. 2315-2323, 1994.

.... 2 : This may be applied directly to obtain the lemma. A very simple but useful lemma implies that if two operations have high entanglement delity on the same density operator, the nal density operators have high delity with each other. The notion of delity used here is treated in [33] [34], and [35] It may be de ned by F ( 1 ; 2 ) max j 1 i; 2 i jh 1 j 2 ij 2 ; 34) where j i i are puri cations of i . In terms of this delity, the lemma is: Lemma 4: If A,B are trace preserving and F e ( A) 1 1 and F e ( B) 1 2 then F (A( B( 1 1 2 . Proof: ....

R. Jozsa, \Fidelity for mixed quantum states," Journal of Modern Optics, vol. 41(12), pp. 2314-2323, 1994.

....bipartite state j 2 i by applying only transformations on her part. The delity between density matrices 1 and 2 is F ( 1 ; 2 ) max j 1 i;j 2 i jh 1 j 2 ij 2 , over all choices of j 1 i and j 2 i that give density matrices 1 and 2 when a part of system is traced out. Lemma 2 [8] Let 1 , 2 be two mixed states with support in a Hilbert space H, K any Hilbert space of dimension at least dim(H) and j i i any puri cations of i in H K. Then, there is a local unitary transformation U on K that maps j 2 i to j 0 2 i = I U j 2 i such that jh 1 j 0 2 ij 2 = ....

R. Jozsa. Fidelity for mixed quantum states. Journal of Modern Optics, 41:2315-2323, 1994.

....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(ae) ....

.... output density operator A b (ae ff ) conditional on the measurement result b, averaged over both the input ensemble and the measurement result: D 2 : 1 Gamma Z d(ff) X b F (ae ff A b (ae ff ) 8) Since F is not linear, these do not define the same quantity; by the concavity of fidelity [9], D 2 D 1 . One might also consider the disturbance measures obtained by replacing the first argument of the fidelity function, ae ff in the above formulae, by the ensemble average density operator R d(ff)ae ff : These measures seem much less natural (and, again by concavity, each is less than ....

R. Jozsa, "Fidelity for mixed quantum states," Journal of Modern Optics, vol. 41(12), pp. 2314--2323, 1994.

....variations consists in having the photons travel in the reverse direction compared with the original BCJL protocol. This is natural for many cryptographic applications. Nevertheless, all these variations fail as well for different reasons related to subtle points in quantum information theory [5, 6] that only began to be understood at the time the BCJL paper was written. A proof that none of these variations work will be the subject of a forthcoming paper: the current paper focuses on the correct attack against the original BCJL protocol. Lo and Chau ask the question: Is quantum bit ....

....[10] that (1 Gamma BW ) K=2. Therefore, we have BW (1 Gamma 4 Theta 2 Gammaffn ) Furthermore, in [4, 12] it is shown that the minimum of BW over all possible measurement is the fidelity F between ae 0 and ae 1 . So, we have F (1 Gamma 4 Theta 2 Gammaffn ) A theorem due to Uhlmann [6, 11] says that the fidelity between two mixed states ae 0 and ae 1 on a system H is given by F = max jhOE 0 jOE 1 ij 2 where the maximum is taken over the purifications OE 0 and OE 1 of ae 0 and ae 1 respectively. Let OE 0 and OE 1 be two purifications such that jhOE 0 jOE 1 ij 2 = F (1 Gamma ....

R. Jozsa, "Fidelity for mixed quantum states", Journal of Modern Optics, vol. 41(12), pp. 2315 -- 2323, 1994.

....2 D(H) Then 2 2 p F ( k k tr 2 p 1 F ( 6. If j i; j i 2 H K satisfy trK j ih j = trK j ih j, then there exists U 2 U(K) such that (I U)j i = j i. Proofs of the facts comprising Theorem 1 can be found as follows: 1. see page 430 of [20] 2. and 3. see [1] or [23] 4. see [22], 5. see [15] and 6. see [21] 2.2 Quantum circuits The computational model upon which quantum interactive proof systems are based is the (acyclic) quantum circuit model. See [1; 8; 23; 31] for background information regarding quantum circuits. A family fQxg of quantum circuits is said to be ....

R. Jozsa. Fidelity for mixed quantum states. Journal of Modern Optics, 41(12):2315-2323, 1994.

....two density matrices ae 0 and ae 1 is defined by B(ae 0 ; ae 1 ) def = min E2M B(p 0 (E) p 1 (E) 3.13) 64 where the POVM E ranges over the set of all possible measurements M. Surprisingly, it turns out that B is equivalent to another measure, studied by Wootters [WO81] and Jozsa [JO94], among others. Suppose j 0 i and j 1 i are pure states. When we think of these two state vectors geometrically, a natural notion of distinguishability is the angle between j 0 i and j 1 i, or any simple function of this angle. In particular, using the formula jh 0 j 1 ij = j 0 jj 1 j cos ....

....jh 0 j 1 ij = j 0 jj 1 j cos ff and recalling that state vectors are usually normalized, we can define overlap(j 0 i; j 1 i) jh 0 j 1 ij as a measure of distinguishability. The question is: what to do for mixed states The answer was given by Uhlmann [UH76] and simplified by Jozsa [JO94]. If ae 0 is the density matrix of a mixed state in the Hilbert space H 1 , then we can always extend the Hilbert space such that ae 0 becomes a pure state in the combined Hilbert space H 1 Omega H 2 . More precisely, we can always find an extension H 2 of H 1 and a pure state j 0 i 2 H ....

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JOZSA, R., "Fidelity for mixed quantum states", Journal of Modern Optics 41, 12 (1994), pp. 2315--2323.

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

R. Jozsa, \Fidelity for mixed quantum states," Journal of Modern Optics, vol. 41(12), pp. 2314-2323, 1994.

....) 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 ....

R. Jozsa, "Fidelity for mixed quantum states," Journal of Modern Optics, vol. 41(12), pp. 2314--2323, 1994.

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R. Jozsa, "Fidelity for mixed quantum states.," Journal of Modern Optics, vol. 41(12), pp. 2315--2323, 1994.

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R. Jozsa, "Fidelity for mixed quantum states.," Journal of Modern Optics, vol. 41(12), pp. 2315--2323, 1994.

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Jozsa, R., "Fidelity for mixed quantum states", Journal of Modern Optics, vol. 41(12), pp. 2315 -- 2323, 1994.

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