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Zyczkowski Subnormalized states and tracenonincreasing maps
 J. Math. Phys
, 2007
"... We investigate the set of completely positive, trace–non–increasing linear maps acting on the setMN of mixed quantum states of size N. Extremal point of this set of maps are characterized and its volume with respect to the Hilbert–Schmidt (Euclidean) measure is computed explicitly for an arbitrary N ..."
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We investigate the set of completely positive, trace–non–increasing linear maps acting on the setMN of mixed quantum states of size N. Extremal point of this set of maps are characterized and its volume with respect to the Hilbert–Schmidt (Euclidean) measure is computed explicitly for an arbitrary N. The spectra of partially reduced rescaled dynamical matrices associated with trace–non–increasing completely positive maps belong to the N– cube inscribed in the set of subnormalized states of size N. As a byproduct we derive the measure inMN induced by partial trace of mixed quantum states distributed uniformly with respect to HS–measure inM N 2.
Rieffel “Quantum operations that cannot be implemented using a small mixed environment
 J. Math. Phys
, 2002
"... To implement any quantum operation (a.k.a. “superoperator ” or “CP map”) on a ddimensional quantum system, it is enough to apply a suitable overall unitary transformation to the system and a d 2dimensional environment which is initialized in a fixed pure state. It has been suggested that a ddimen ..."
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To implement any quantum operation (a.k.a. “superoperator ” or “CP map”) on a ddimensional quantum system, it is enough to apply a suitable overall unitary transformation to the system and a d 2dimensional environment which is initialized in a fixed pure state. It has been suggested that a ddimensional environment might be enough if we could initialize the environment in a mixed state of our choosing. In this note we show with elementary means that certain explicit quantum operations cannot be realized in this way. Our counterexamples map some pure states to pure states, giving strong and easily manageable conditions on the overall unitary transformation. Everything works in the more general setting of quantum operations from ddimensional to d ′dimensional spaces, so we place our counterexamples within this more general framework. 1 Quantum operations Quantum operations (see e.g. [3]) are also known as “superoperators”, “superscattering operators ” or “completely positive maps ” (“CP maps”). They can be viewed as a generalization of unitary transformations and are the most general transformations that can be applied to a quantum system in an unknown (possibly mixed) state. More precisely, quantum operations are the most general transformations that can be implemented deterministically, thus excluding operations which only succeed with a certain probability, like those depending on a measurement outcome. Under a quantum operation pure states are frequently mapped to mixed states. All quantum operations on a ddimensional system can be implemented as the partial trace of a unitary operator acting on the system together with an auxiliary system (the “environment”). The question is how small an environment suffices to implement all possible quantum operations on a ddimensional
How often is a random quantum state kentangled?
, 2010
"... The set of tracepreserving, positive maps acting on density matrices of size d forms a convex body. We investigate its nested subsets consisting of kpositive maps, where k = 2,...,d. Working with the measure induced by the Hilbert–Schmidt distance we derive asymptotically tight bounds for the volu ..."
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The set of tracepreserving, positive maps acting on density matrices of size d forms a convex body. We investigate its nested subsets consisting of kpositive maps, where k = 2,...,d. Working with the measure induced by the Hilbert–Schmidt distance we derive asymptotically tight bounds for the volumes of these sets. Our results strongly suggest that the inner set of (k +1)positive maps forms a small fraction of the outer set of kpositive maps. These results are related to analogous bounds for the relative volume of the sets of kentangled states describing a bipartite d × d system. PACS numbers: 03.65.Aa, 03.67.Mn, 02.40.Ft 1.
Limits and restrictions of private quantum channel
, 2005
"... We study private quantum channels on a single qubit, which encrypt given set of plaintext states P. Specifically, we determine all achievable states ρ (0) (average output of encryption) and for each particular set P we determine the entropy of the key necessary and sufficient to encrypt this set. It ..."
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We study private quantum channels on a single qubit, which encrypt given set of plaintext states P. Specifically, we determine all achievable states ρ (0) (average output of encryption) and for each particular set P we determine the entropy of the key necessary and sufficient to encrypt this set. It turns out that single bit of key is sufficient when the set P is two dimensional. However, the necessary and sufficient entropy of the key in case of three dimensional P varies continuously between 1 and 2 bits depending on the state ρ (0). Finally, we derive private quantum channels achieving these bounds. We show that the impossibility of universal NOT operation on qubit can be derived from the fact that one bit of key is not sufficient to encrypt qubit.
WebWatcher: a tour guide for the world wide web
 In Proceedings of International Joint Conference on Artificial Intelligence (IJCAI
, 1997
"... It is well known that n bits of entropy are necessary and sufficient to perfectly encrypt n bits (onetime pad). Even if we allow the encryption to be approximate, the amount of entropy needed doesn’t asymptotically change. However, this is not the case when we are encrypting quantum bits. For the p ..."
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It is well known that n bits of entropy are necessary and sufficient to perfectly encrypt n bits (onetime pad). Even if we allow the encryption to be approximate, the amount of entropy needed doesn’t asymptotically change. However, this is not the case when we are encrypting quantum bits. For the perfect encryption of n quantum bits, 2n bits of entropy are necessary and sufficient (quantum onetime pad), but for approximate encryption one asymptotically needs only n bits of entropy. In this paper, we provide the optimal tradeoff between the approximation measure ǫ and the amount of classical entropy used in the encryption of single quantum bits. Then, we consider nqubit encryption schemes which are a composition of independent singlequbit ones and provide the optimal schemes both in the 2 and the ∞norm. Moreover, we provide a counterexample to show that the encryption scheme of AmbainisSmith [3] based on smallbias sets does not work in the ∞norm.
Visualizing two qubits
 J. Math. Phys
"... We show that notions of entanglement, witnesses, and certain Bell inequalities can be visualized in three dimensions. This allows us to give “proofs by inspection ” of the result that for two qubits, Peres test is iff, and to “solve by inspection ” the optimization problem of the CHSH inequality vio ..."
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We show that notions of entanglement, witnesses, and certain Bell inequalities can be visualized in three dimensions. This allows us to give “proofs by inspection ” of the result that for two qubits, Peres test is iff, and to “solve by inspection ” the optimization problem of the CHSH inequality violation. Finally, we give numerical evidence that, remarkably, allowing Alice and Bob to use more measurements, three rather than two, does not help them to distinguish any new entangled SLOCC equivalence class beyond the CHSH class. 1
Concurrence and foliations induced by some 1qubit channels
, 2003
"... We start with a short introduction to the roof concept. An elementary discussion of phasedamping channels shows the role of antilinear operators in representing their concurrences. A general expression for some concurrences is derived. We apply it to 1qubit channels of length two, getting the ind ..."
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We start with a short introduction to the roof concept. An elementary discussion of phasedamping channels shows the role of antilinear operators in representing their concurrences. A general expression for some concurrences is derived. We apply it to 1qubit channels of length two, getting the induced foliations of the state space, the optimal decompositions, and the entropy of a state with respect to these channels. For amplitudedamping channels one obtains an expression for the Holevo capacity allowing for easy numerical calculations. 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.
Entropy of quantum channel in the theory of quantum information
 Ph.D. thesis Kraków
, 2011
"... Quantum channels, also called quantum operations, are linear, trace preserving and completely positive transformations in the space of quantum states. Such operations describe discrete time evolution of an open quantum system interacting with an environment. The thesis contains an analysis of prope ..."
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Quantum channels, also called quantum operations, are linear, trace preserving and completely positive transformations in the space of quantum states. Such operations describe discrete time evolution of an open quantum system interacting with an environment. The thesis contains an analysis of properties of quantum channels and different entropies used to quantify the decoherence introduced into the system by a given operation. Part I of the thesis provides a general introduction to the subject. In Part II, the action of a quantum channel is treated as a process of preparation of a quantum ensemble. The Holevo information associated with this ensemble is shown to be bounded by the entropy exchanged during the preparation process between the initial state and the environment. A relation between the Holevo information and the entropy of an auxiliary matrix consisting of square root fidelities between the elements of the ensemble is proved in some special cases. Weaker bounds on the Holevo information are also established.
Lower Bounds on the Quantum Capacity and Error Exponent of General Memoryless Channels
, 2001
"... Tradeoffs between the information rate and fidelity of quantum errorcorrecting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive linear map, where the dimension of the underlying H ..."
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Tradeoffs between the information rate and fidelity of quantum errorcorrecting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive linear map, where the dimension of the underlying Hilbert space is a prime number. On such a quantum channel, the highest fidelity of a quantum errorcorrecting code of length n and rate R is proven to be lower bounded by 1 − exp[−nE(R)] for some function E(R). The E(R) is positive below some threshold, which is therefore a lower bound on the quantum capacity.