E.H. Lieb, Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture, Advances in Mathematics, 11, 267-288 (1973).  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... ; T ; 1 2 log(det ; n 2 log( Proposition 3.1. Let 0 ; 0 and let 1. The potential function ; is a convex function with respect to ( DUAL DIFFERENTIAL GEOMETRY 121 For proving Proposition 3. 1, we employ the following Lemma due to Lieb [11]. Lemma 3.2 (Lieb) The function from S n M(n;R) to the non negative reals R [ f0g is de ned by S n M(n;R) 3 (X; Y ) 7 T r[X p Y T X q Y ] 2 R [ f0g (18) is jointly convex in (X; Y ) whenever 0 p; 0 q and p q 1. Proof of Proposition 3.1. If we put X = 2 ....

E.H. Lieb, Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture, Advances in Mathematics, 11, 267-288 (1973).

....M. Horodecki [11] has independently derived the lower bound, using the nonincrease of the Holevo quantity under completely positive maps. This nonincrease is an easy consequence of the monotonicity of relative entropy under such maps [24, 25] and therefore of Lieb s fundamental concavity theorem [26]) A good treatment of all of these is to be found in [27] We were also aware of the possibility of such a derivation of the lower bound. Indeed, we originally informally formulated the argument as conditional upon giving the classical capacity, which was not then known to be true. Realizing ....

E. Lieb, \Convex trace functions and the Wigner-Yanase-Dyson conjecture," Advances in Mathematics, vol. 11, pp. 267-288, 1973.

....an ensemble for ae; P i p i ae i = ae; then (E) j S(ae) Gamma X i p i S(ae i ) X i p i S(ae i ; ae) 9.6) X i p i S(E(ae i ) E(ae) j (E(E) 9. 7) The key to Uhlmann s result, as for the data processing inequality for the coherent information, is Lieb s celebrated concavity result [65], 66] which implies double convexity of the relative information. 9.3.2 Quantum Fano inequality for ensembles The average entropy P i p i S(E(j i ih i j) will play the role that S e played in the Fano inequality for entanglement fidelity, since it is the noise quantity that is subtracted ....

E. Lieb, "Convex trace functions and the Wigner-Yanase-Dyson conjecture," Advances in Mathematics, vol. 11, pp. 267--288, 1973.

....[11] has independently derived the lower bound of Theorem 17, using the nonincrease of the Holevo quantity under completely positive maps. This nonincrease is an easy consequence of the monotonicity of relative entropy under such maps [24, 25] and therefore of Lieb s fundamental concavity theorem [26]) A good treatment of all of these is to be found in [27] A special case of Theorem 17 is the lower bound of S(ae) qubits per source signal on the rate of compression of ensembles of pure states. This lower bound was established for preparation blind encodings and unitary decodings in [1] ....

E. Lieb, "Convex trace functions and the Wigner-Yanase-Dyson conjecture," Advances in Mathematics, vol. 11, pp. 267--288, 1973.

....P 2 bi are proportional to each other. This implies that knowing which of them occured gives us no additional information. In fact, in the general case Proposition 6 follows quickly from the following theorem of T. Ando [13] which is easily seen to be equivalent to Lieb s concavity theorem ( [14]; see also discussions in [15] especially p. 273, and [16] Theorem 7 (Ando) For 0 t 1, the map: A; B) A t Omega B 1 Gammat (23) is jointly concave on pairs of positive operators A; B. Proof of Proposition 6: Consider the map from operators to the reals given by: F(A) h jA 1=2 j i ....

E. Lieb, "Convex trace functions and the Wigner-Yanase-Dyson conjecture," Advances in Mathematics, vol. 11, pp. 267--288, 1973.

.... may be interesting to note that the proof of the converse did not use the Holevo bound [6] which usually is derived from Uhlmann s monotonocity of the quantum I divergence [17] which in turn is a consequence of a deep concavity result, the Wigner Yanase Dyson conjecture, first proved by Lieb [12]) Thus in turn we get a new proof of the Holevo bound. Finally some general remarks: note that our proof of the coding theorem follows the original idea of Feinstein [5] the proof of the converse is after Wolfowitz [21] In fact we exploited a close analogy between the classical and the quantum ....

E. H. Lieb, "Convex Trace Functions and the Wigner--Yanase--Dyson Conjecture", Advan. Math. 11 (1973), 267--288

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