The general stable quantum memory unit is a hybrid consisting of a classical digit with a quantum digit (qudit) assigned to each classical state. The shape of the memory is the vector of sizes of these qudits, which may differ. We determine when N copies of a quantum memory /spl Ascr/ embed in N(1+/spl ogr/(1)) copies of another quantum memory /spl Bscr/. This relationship captures the notion that/spl Bscr/ is as at least as useful as /spl Ascr/ for all purposes in the bulk limit. We show that the embeddings exist if and only if for all p/spl ges/1, the p-norm of the shape of /spl Ascr/ does not exceed the p-norm of the shape of /spl Bscr/. The log of the p-norm of the shape of /spl Ascr/ can be interpreted as the maximum of S(/spl rho/)+H(/spl rho/)/p (quantum entropy plus discounted classical entropy) taken over all mixed states /spl rho/ on /spl Ascr/. We also establish a noiseless coding theorem that justifies these entropies. The noiseless coding theorem and the bulk embedding theorem together say that either /spl Ascr/ blindly bulk-encodes into /spl Bscr/ with perfect fidelity, or A admits a state that does not visibly bulk-encode into/spl Bscr/with high fidelity. In conclusion, the utility of a hybrid quantum memory is determined by its simultaneous capacity for classical and quantum entropy, which is not a finite list of numbers, but rather a convex region in the classical-quantum entropy plane.
The general stable quantum memory unit is a hybrid consisting of a classical digit with a quantum digit (qudit) assigned to each classical state. The shape of the memory is the vector of sizes of these qudits, which may differ. We determine when N copies of a quantum memory /spl Ascr/ embed in N(1+/spl ogr/(1)) copies of another quantum memory /spl Bscr/. This relationship captures the notion that/spl Bscr/ is as at least as useful as /spl Ascr/ for all purposes in the bulk limit. We show that the embeddings exist if and only if for all p/spl ges/1, the p-norm of the shape of /spl Ascr/ does not exceed the p-norm of the shape of /spl Bscr/. The log of the p-norm of the shape of /spl Ascr/ can be interpreted as the maximum of S(/spl rho/)+H(/spl rho/)/p (quantum entropy plus discounted classical entropy) taken over all mixed states /spl rho/ on /spl Ascr/. We also establish a noiseless coding theorem that justifies these entropies. The noiseless coding theorem and the bulk embedding theorem together say that either /spl Ascr/ blindly bulk-encodes into /spl Bscr/ with perfect fidelity, or A admits a state that does not visibly bulk-encode into/spl Bscr/with high fidelity. In conclusion, the utility of a hybrid quantum memory is determined by its simultaneous capacity for classical and quantum entropy, which is not a finite list of numbers, but rather a convex region in the classical-quantum entropy plane.
The general stable quantum memory unit is a hybrid consisting of a classical digit with a quantum digit (qudit) assigned to each classical state. The shape of the memory is the vector of sizes of these qudits, which may differ. We determine when N copies of a quantum memory A embed in N(1+o(1)) copies of another quantum memory B. This relationship captures the notion that B is as at least as useful as A for all purposes in the bulk limit. We show that the embeddings exist if and only if for all p >= 1, the p-norm of the shape of A does not exceed the p-norm of the shape of B. The log of the p-norm of the shape of A can be interpreted as the maximum of S(ρ) + H(ρ)/p (quantum entropy plus discounted classical entropy) taken over all mixed states ρon A. We also establish a noiseless coding theorem that justifies these entropies. The noiseless coding theorem and the bulk embedding theorem together say that either A blindly bulk-encodes into B with perfect fidelity, or A admits a state that does not visibly bulk-encode into B with high fidelity. In conclusion, the utility of a hybrid quantum memory is determined by its simultaneous capacity for classical and quantum entropy, which is not a finite list of numbers, but rather a convex region in the classical-quantum entropy plane.
In this thesis we analyse the type of states and ensembles which achieve the capacity for certain quantum channels carrying classical information. We first concentrate on the product-state capacity of a particular quantum channel, that is, the capacity which is achieved by encoding the output states from a source into codewords comprised of states taken from ensembles of non-entangled states and sending them over copies of the quantum channel. Using the "single-letter" formula proved independently by Holevo and by Schumacher and Westmoreland we obtain the product-state capacity of the qubit quantum amplitude-damping channel, which is determined by a transcendental equation in a single real variable and can be solved numerically. We demonstrate that the product-state capacity of this channel can be achieved using a minimal ensemble of non-orthogonal pure states. Next we consider the classical capacity of two quantum channels with memory, namely a periodic channel with quantum depolarising channel branches and a convex combination of quantum channels. We prove that the classical capacity for each of the classical memory channels mentioned above is, in fact, equal to the respective product-state capacities. For those channels this means that the classical capacity is achieved without the use of entangled input-states. Next we introduce the channel coding theorem for memoryless quantum channels, providing a known proof by Winter for the strong converse of the theorem. We then consider the strong converse to the channel coding theorem for a periodic quantum channel.
The readout of a classical memory can be modelled as a problem of quantum channel discrimination, where a decoder retrieves information by distinguishing the different quantum channels encoded in each cell of the memory (Pirandola 2011 Phys. Rev. Lett. 106 090504). In the case of optical memories, such as CDs and DVDs, this discrimination involves lossy bosonic channels and can be remarkably boosted by the use of nonclassical light (quantum reading). Here we generalize these concepts by extending the model of memory from single-cell to multi-cell encoding. In general, information is stored in a block of cells by using a channel-codeword, i.e. a sequence of channels chosen according to a classical code. Correspondingly, the readout of data is realized by a process of 'parallel' channel discrimination, where the entire block of cells is probed simultaneously and decoded via an optimal collective measurement. In the limit of a large block we define the quantum reading capacity of the memory, quantifying the maximum number of readable bits per cell. This notion of capacity is nontrivial when we suitably constrain the physical resources of the decoder. For optical memories (encoding bosonic channels), such a constraint is energetic and corresponds to fixing the mean total number of photons per cell. In this case, we are able to prove a separation between the quantum reading capacity and the maximum information rate achievable by classical transmitters, i.e. arbitrary classical mixtures of coherent states. In fact, we can easily construct nonclassical transmitters that are able to outperform any classical transmitter, thus showing that the advantages of quantum reading persist in the optimal multi-cell scenario.
Motivated by deterministic identification via classical channels, where the encoder is not allowed to use randomization, we revisit the problem of identification via quantum channels but now with the additional restriction that the message encoding must use pure quantum states, rather than general mixed states. Together with the previously considered distinction between simultaneous and general decoders, this suggests a two-dimensional spectrum of different identification capacities, whose behaviour could a priori be very different. We demonstrate two new results as our main findings: first, we show that all four combinations (pure/mixed encoder, simultaneous/general decoder) have a double-exponentially growing code size, and that indeed the corresponding identification capacities are lower bounded by the classical transmission capacity for a general quantum channel, which is given by the Holevo-Schumacher-Westmoreland Theorem. Secondly, we show that the simultaneous identification capacity of a quantum channel equals the simultaneous identification capacity with pure state encodings, thus leaving three linearly ordered identification capacities. By considering some simple examples, we finally show that these three are all different: general identification capacity can be larger than pure-state-encoded identification capacity, which in turn can be larger than pure-state-encoded simultaneous identification capacity.
Applications of quantum technology often require fidelities to quantify performance. These provide a fundamental yardstick for the comparison of two quantum states. While this is straightforward in the case of pure states, it is much more subtle for the more general case of mixed quantum states often found in practice. A large number of different proposals exist. In this review, we summarize the required properties of a quantum fidelity measure, and compare them, to determine which properties each of the different measures has. We show that there are large classes of measures that satisfy all the required properties of a fidelity measure, just as there are many norms of Hilbert space operators, and many measures of entropy. We compare these fidelities, with detailed proofs of their properties. We also summarize briefly the applications of these measures in teleportation, quantum memories and quantum computers, quantum communications, and quantum phase-space simulations.
Motivated by deterministic identification via channels, where the encoder cannot use randomisation, we revisit the problem of identification via quantum channels with the additional restriction that the message encoding must use pure quantum states, rather than general mixed states. Together with the previously considered distinction between simultaneous and general decoders, this suggests a two-dimensional spectrum of different identification capacities, whose behaviour could a priori be very different. We demonstrate two new main results: first, we show that all of the four combinations (pure/mixed encoder, simultaneous/general decoder) have a double-exponentially growing code size, and that indeed the corresponding identification capacities are lower bounded by the classical transmission capacity for a general quantum channel, which is given by the Holevo-Schumacher- Westmoreland theorem. Secondly, we show that the simultaneous identification capacity of a quantum channel equals the simultaneous identification capacity with pure state encodings, thus leaving three linearly ordered identification capacities. By considering some simple examples we finally show that these three are all different: general identification capacity can be larger than pure-state-encoded identification capacity which in turn can be larger than pure-state-encoded simultaneous identification capacity,