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Multiplicativity of the 2-norm: The case of the depolarized Werner-Holevo maps

Mathematical Physics & Probability

Speaker: Spiros Michalakis, UC Davis
Location: 2112 MSB
Start time: Tue, Jun 5 2007, 2:10PM

One of the big open problems in Quantum Information Theory is the so-called "additivity conjecture", which states that the combined capacity of two independent quantum channels is the sum of the individual capacities of the channels. In other words, entanglement between the input states cannot create an advantage when transmitting information through two independent quantum channels. An equivalent way to formulate this problem is called the "multiplicativity conjecture": One can always find a sequence of real numbers p_n > 1 and p_n --> 1, such that the Schatten p_n-norm (a discrete L_p norm) of the tensor product of two completely positive maps is multiplicative. It was recently shown that for p > 4.79 one can construct counterexamples to multiplicativity. Nevertheless, there is strong consensus that if the multiplicativity conjecture holds then it should hold for all 1 <= p <= 2. Moreover, there is good reason to expect that if the 2-norm is multiplicative, then all p-norms with 1 < p < 2 satisfy multiplicativity. It is natural then to study the multiplicativity of the 2-norm for generalizations of the maps that gave us counterexamples for p > 4.79. We show that for these maps, the depolarized Werner-Holevo maps, multiplicativity of the 2-norm holds.