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The adaptive classical capacity of a quantum channel, or information capacities of three symmetric pure states in three dimensions
 IBM. J. Res. & Dev
, 2004
"... We investigate the capacity of three symmetric quantum states in three real dimensions to carry classical information. Several such capacities have already been defined, depending on what operations are allowed in the sending and receiving protocols. These include the C1,1 capacity, which is the cap ..."
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Cited by 8 (1 self)
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We investigate the capacity of three symmetric quantum states in three real dimensions to carry classical information. Several such capacities have already been defined, depending on what operations are allowed in the sending and receiving protocols. These include the C1,1 capacity, which is the capacity achievable if separate measurements must be used for each of the received states, and the C1, ∞ capacity, which is the capacity achievable if joint measurements are allowed on the tensor product of all the received states. We discover a new classical information capacity of quantum channels, the adaptive capacity C1,A, which lies strictly between the C1,1 and the C1, ∞ capacities. The adaptive capacity allows what is known as the LOCC (local operations and classical communication) model of quantum operations for decoding the channel outputs. This model requires each of the signals to be measured by a separate apparatus, but allows the quantum states of these signals to be measured in stages, with the first stage partially reducing their quantum states, and where measurements in subsequent stages may depend on the results of a classical computation taking as input the outcomes of the first round of measurements. We also show that even in three dimensions, with the information carried by an ensemble containing three pure states, achieving the C1,1 capacity may require a POVM with six outcomes. 2 1
Qubit Channels Which Require Four Inputs to Achieve Capacity: Implications for Additivity Conjectures
, 2008
"... An example is given of a qubit quantum channel which requires four inputs to maximize the Holevo capacity. The example is one of a family of channels which are related to 3state channels. The capacity of the product channel is studied and numerical evidence presented which strongly suggests additiv ..."
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An example is given of a qubit quantum channel which requires four inputs to maximize the Holevo capacity. The example is one of a family of channels which are related to 3state channels. The capacity of the product channel is studied and numerical evidence presented which strongly suggests additivity. The numerical evidence also supports a conjecture about the concavity of output entropy as a function of entanglement parameters. However, an example is presented which shows that for some channels this conjecture does not hold for all input states. A numerical algorithm for finding the capacity and optimal inputs is presented and its relation to a relative entropy optimization discussed.
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"... The adaptive classical capacity of a quantum channel, or Information capacities of three symmetric pure states in three dimensions P. W. Shor We investigate the capacity of three symmetric quantum states in three real dimensions to carry classical information. Several such capacities have already be ..."
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The adaptive classical capacity of a quantum channel, or Information capacities of three symmetric pure states in three dimensions P. W. Shor We investigate the capacity of three symmetric quantum states in three real dimensions to carry classical information. Several such capacities have already been defined, depending on what operations are allowed in the protocols that the sender uses to encode classical information into these quantum states, and that the receiver uses to decode it. These include the C 1,1 capacity, which is the capacity achievable if separate measurements must be used for each of the received states, and the C 1, � capacity, which is the capacity achievable if joint measurements are allowed on the tensor product of all of the received states. We discover a new classical information capacity of quantum channels, the adaptive capacity C 1,A, which lies strictly between the C 1,1 and the C 1, � capacities. The adaptive capacity allows the use of what is known as the LOCC (local operations and classical communication) model of quantum operations for decoding the channel outputs. This model requires each of the signals to be measured by a separate apparatus, but allows the quantum states of these signals to be measured in stages, with the first stage partially reducing their quantum states; measurements in subsequent stages may depend on the results of a classical computation taking as input the outcomes of the first round of measurements. We also show that even in three dimensions, with the information carried by an ensemble containing three pure states, achieving the C 1,1 capacity may require a positive operator valued measure (POVM) with six outcomes. 1.
Conic Geometric Programming
, 2013
"... We introduce and study conic geometric programs (CGPs), which are convex optimization problems that unify geometric programs (GPs) and conic optimization problems such as semidefinite programs (SDPs). A CGP consists of a linear objective function that is to be minimized subject to affine constraint ..."
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We introduce and study conic geometric programs (CGPs), which are convex optimization problems that unify geometric programs (GPs) and conic optimization problems such as semidefinite programs (SDPs). A CGP consists of a linear objective function that is to be minimized subject to affine constraints, convex conic constraints, and upper bound constraints on sums of exponential and affine functions. The conic constraints are the central feature of conic programs such as SDPs, while upper bounds on combined exponential/affine functions are generalizations of the types of constraints found in GPs. The dual of a CGP involves the maximization of the negative relative entropy between two nonnegative vectors jointly, subject to affine and conic constraints on the two vectors. Although CGPs contain GPs and SDPs as special instances, computing global optima of CGPs is not much harder than solving GPs and SDPs. More broadly, the CGP framework facilitates a range of new applications that fall outside the scope of SDPs and GPs. Specifically, we demonstrate the utility of CGPs in providing solutions to problems such as permanent maximization, hittingtime estimation in dynamical systems, the computation of the capacity of channels transmitting quantum information, and robust optimization formulations of GPs.
Conic Geometric Programming
, 2013
"... We introduce and study conic geometric programs (CGPs), which are convex optimization problems that unify geometric programs (GPs) and conic optimization problems such as linear programs (LPs) and semidefinite programs (SDPs). A CGP consists of a linear objective function that is to be minimized sub ..."
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We introduce and study conic geometric programs (CGPs), which are convex optimization problems that unify geometric programs (GPs) and conic optimization problems such as linear programs (LPs) and semidefinite programs (SDPs). A CGP consists of a linear objective function that is to be minimized subject to affine constraints, convex conic constraints, and upper bound constraints on sums of exponential and affine functions. The conic constraints are the central feature of conic programs such as LPs and SDPs, while upper bounds on combined exponential/linear functions are generalizations of the types of constraints found in GPs. The dual of a CGP involves the maximization of the negative relative entropy between two nonnegative vectors jointly, subject to affine and conic constraints on the two vectors. Although CGPs contain GPs and SDPs as special instances, computing global optima of CGPs is not much harder than solving GPs and SDPs. More broadly, the CGP framework facilitates a range of new applications that fall outside the scope of SDPs and GPs. Specifically, we demonstrate the utility of CGPs in providing solutions to problems such as permanent maximization, hittingtime estimation in dynamical systems, the computation of the capacity of channels transmitting quantum information, and robust optimization formulations of GPs.