In the present paper, the trace distance is exposed within the quantum operations formalism. The definition of the trace distance in terms of a maximum over all quantum operations is given. It is shown that for any pair of different states, there are an uncountably infinite number of maximizing quantum operations. Conversely, for any operation of the described type, there are an uncountably infinite number of those pairs of states that the maximum is reached by the operation. A behavior of the trace distance under considered operations is studied. Relations and distinctions between the trace distance and the sine distance are discussed.
Mixed states are introduced in physics in order to express our ignorance about the actual state of a physical system and are represented in standard quantum mechanics (QM) by density operators. Such operators also appear if one considers a (pure) entangled state of a compound system $Ω$ and performs partial traces on the projection operator representing it. Yet, they do not represent mixed states (or proper mixtures) of the subsystems in this case, but improper mixtures, since the coefficients in the convex sums expressing them never bear the ignorance interpretation. Hence, one cannot attribu…
Losses in quantum communication lines severely affect the rates of reliable information transmission and are usually considered to be state-independent. However, the loss probability does depend on the system state in general, with the polarization dependent losses being a prominent example. Here we analyze biased trace decreasing quantum operations that assign different loss probabilities to states and introduce the concept of a generalized erasure channel. We find lower and upper bounds for the classical and quantum capacities of the generalized erasure channel as well as characterize its de…
We represent generalized density matrices of a $d$-complex dimensional quantum system as a subcone of a real pointed cone of revolution in $\mathbb{R}^{d^2}$, or indeed a Minkowskian cone in $\mathbb{E}^{1,d^2-1}$. Generalized pure states correspond to certain future-directed light-like vectors of $\mathbb{E}^{1,d^2-1}$. This extension of the Generalized Bloch Sphere enables us to cater for non-trace-preserving quantum operations, and in particluar to view the per-outcome effects of generalized measurements. We show that these consist of the product of an orthogonal transform about the axis of…
Within the Hamiltonian framework, the propositions about a classical physical system are described in the Borel σ-algebra of a symplectic manifold (the phase space) where logical connectives are the standard set operations. Considering the geometric formulation of quantum mechanics we give a description of quantum propositions in terms of fuzzy events in a complex projective space equipped with Kähler structure (the quantum phase space) obtaining a quantized version of a fuzzy logic by deformation of the product t-norm.
We show that both information erasure process and trace process can be realized by projective measurement. And a partial trace operation consists to a projective measurement on a subsystem. We show that a nonunitary operation will destroy the wave-behavior of a particle. We give a quantum manifestation of Maxwell's demon and give a quantum manifestation of the second law of therodynamics. We show that, considering the law of memontum-energy conversation, the evolution of a closed system should be unitary and the von Neumann entropy of the closed quantum system should be least.
It is shown that the observability of a large class of operations on mixed states is fundamentally limited. We consider trace preserving, unital operations. This class includes unitary and perfect premeasurement operations. An upper bound on the trace distance between an untransformed state and a state transformed by one of these operations is derived. The bound is dependent only on the purity of the state. In the case of maximal mixedness, the bound implies all operations of this class are unobservable.
Cryptographic security of quantum key distribution is currently based on a trace distance criterion. The widespread misinterpretation of the criterion as failure probability and also its actual scope have been discussed previously. Recently its distinguishability advantage interpretation is re-emphasized as an operational guarantee, and the failure probability misinterpretation is maintained with a further failure probability per bit interpretation. In this paper we explain the basic perpetuating error as a confusion on the correspondence between mathematics and reality. We note that the assig…
A measure of quantum non-Markovianity for an open system dynamics, based on revivals of the distinguishability between system states, has been introduced in the literature using the trace distance as quantifier for distinguishability. Recently it has been suggested to use as measure for the distinguishability of quantum states the trace norm of Helstrom matrices, given by weighted differences of statistical operators. Here we show that this new approach, which generalizes the original one, is consistent with the interpretation of information flow between the system and its environment associat…
In quantum information, most information processing processes involve quantum channels. One manifestation of a quantum channel is quantum operation acting on quantum states. The coherence of quantum operations can be considered as a quantum resource, which can be exploited to perform certain quantum tasks. From the viewpoint of Choi-Jamiołkowski isomorphism, we study the coherence of quantum operations in the framework of resource theory. We define the phase-out superoperation and give the operation which transforms the Choi-Jamiołkowski state of a quantum operation to the Choi-Jamiołkowski st…
Emergence of the Partial Trace from Classical Probability Theory
ITEAM Research Institute, Universitat Politècnica de València（瓦伦西亚理工大学 ITEAM 研究所）；Universidad de Vigo（维戈大学）；iPronics, Programmable Photonics, S.L（iPronics 可编程光子学有限公司）
Partial trace is a very important mathematical operation in quantum mechanics. It is not only helpful in studying the subsystems of a composite quantum system but also used in computing a vast majority of quantum entanglement measures. Calculating partial trace becomes computationally very intensive with increasing number of qubits as the Hilbert space dimension increases exponentially. In this paper we discuss about our new method of partial tracing which is based on set theory and it is more efficient. The proposed method of partial tracing overcomes all the limitations of the other well kno…
The development of techniques for manipulation of quantum information has opened the door to a variety of protocols for accomplishing unprecedented tasks. In particular, a new protocol of quantum teleportation was proposed in 2008 to achieve effective energy transportation simply via local operations and classical communication without breaking any known physical laws. This is achieved by extracting zero-point energy of entangled many-body systems by local operations dependent on information obtained by a distant measurement. The concept is reviewed from an introductory viewpoint herein.
We introduce the notion of trace-norm isometric encoding and explore its implications for passive and active methods to protect quantum information against errors. Beside providing an operational foundations to the "subsystems principle" [E. Knill, Phys. Rev. A 74, 042301 (2006)] for faithfully realizing quantum information in physical systems, our approach allows additional explicit connections between noiseless, protectable, and correctable quantum codes to be identified. Robustness properties of isometric encodings against imperfect initialization and/or deviations from the intended error m…
Recently, trace distance measure of coherence has been proposed for characterizing the coherence of a given quantum state. However, it seems difficult to estimate the optimal incoherent state for high dimensional states. An explicit expression for the trace distance measure of coherence for a class of qudit states is provided. We show that the closest incoherent state to the class of qudit states is just the diagonal matrix obtained from it by deleting all off-diagonal elements. It is also shown that the measure of coherence induced by trace distance can act as a measure of coherence for this …
Minimum error state discrimination between two mixed states ρand σcan be aided by the receipt of "classical side information" specifying which states from some convex decompositions of ρand σapply in each run. We quantify this phenomena by the average trace distance, and give lower and upper bounds on this quantity as functions of ρand σ. The lower bound is simply the trace distance between ρand σ, trivially seen to be tight. The upper bound is \sqrt{1 - tr(ρσ)}, and we conjecture that this is also tight. We reformulate this conjecture in terms of the existence of a pair of "unbiased decomposi…
Assuming that quantum states, including pure states, represent subjective degrees of belief rather than objective properties of systems, the question of what other elements of the quantum formalism must also be taken as subjective is addressed. In particular, we ask this of the dynamical aspects of the formalism, such as Hamiltonians and unitary operators. Whilst some operations, such as the update maps corresponding to a complete projective measurement, must be subjective, the situation is not so clear in other cases. Here, it is argued that all trace preserving completely positive maps, incl…
Trace decreasing quantum operations naturally emerge in experiments involving postselection. However, the experiments usually focus on dynamics of the conditional output states as if the dynamics were trace preserving. Here we show that this approach leads to incorrect conclusions about the dynamics divisibility, namely, one can observe an increase in the trace distance or the system-ancilla entanglement although the trace decreasing dynamics is completely positive divisible. We propose solutions to that problem and introduce proper indicators of the information backflow and the indivisibility…
In this paper, we show that the quantum Zeno effect occurs for any frequent quantum measurements or operations. As a result of the Zeno effect, for non-selective measurements (or trace preserving completely positive maps), the evolution of a measurement invariant state is governed by an effective Hamiltonian defined by the measurements and the free-evolution Hamiltonian. For selective measurements, the state may change randomly with time according to measurement outcomes, while some physical quantities (operators) still evolve as the effective dynamics.
The trace distance between two quantum states, $ρ$ and $σ$, is an operationally meaningful quantity in quantum information theory. However, in general it is difficult to compute, involving the diagonalization of $ρ- σ$. In contrast, the Hilbert-Schmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the Hilbert-Schmidt distance with a bound that is particularly strong when either $ρ$ or $σ$ is low rank. Our bound is stronger than the bound one could obtain via the norm equivalence of the Frobenius and trace…
Motivated by the engineering applications of uncertainty quantification, in this work we draw connections between the notions of random quantum states and operations in quantum information with probability distributions commonly encountered in the field of orientation statistics. This approach identifies natural probability distributions that can be used in the analysis, simulation, and inference of quantum information systems. The theory of exponential families on Stiefel manifolds provides the appropriate generalization to the classical case, and fortunately there are many existing technique…
Nonlocality is one unique property of quantum mechanics differing from classical world. One of its quantifications can be properly described as the maximum global effect caused by locally invariant measurements, termed as measurement-induced nonlocality (MIN) (2011 \emph{Phys. Rev. Lett.} {\bf 106} 120401). Here, we propose to quantify the MIN by the trace norm. We show explicitly that this measure is monotonically decreasing under the action of completely positive trace-preserving map, which is the general local quantum operation, on the unmeasured party for the bipartite state. This property…
Introducing contravariant trace-densities for quantum states, we restore one to one correspondence between quantum operations described by normal CP maps and their trace densities as Hermitian positive operator-valued contravariant kernels. The CB-norm distance between two quantum operations with type one input algebras is explicitly expressed in terms of these densities, and this formula is also extended to a generalized CB-distances between quantum operations with type two inputs. A larger C-distance is given as the natural norm-distance for the channel densities, and another, Helinger type …
New quantum distance is introduced as a half-sum of several singular values of difference between two density operators. This is, up to factor, the metric induced by so-called Ky Fan norm. The partitioned trace distances enjoy similar properties to the standard trace distance, including the unitary invariance, the strong convexity and the close relations to the classical distances. The partitioned distances cannot increase under quantum operations of certain kind including bistochastic maps. All the basic properties are re-formulated as majorization relations. Possible applications to quantum …
We examine the use of weight enumerators for analyzing tensor network constructions, and specifically the quantum lego framework recently introduced. We extend the notion of quantum weight enumerators to so-called tensor enumerators, and prove that the trace operation on tensor networks is compatible with a trace operation on tensor enumerators. This allows us to compute quantum weight enumerators of larger codes such as the ones constructed through tensor network methods more efficiently. We also provide a general framework for quantum MacWilliams identities that includes tensor enumerators a…
Quantum energy teleportation (QET) is, from an operational viewpoint of distant protocol users, the transportation of energy via local operations and classical communication. QET has various links to fundamental research fields, including black hole physics, the quantum theory of Maxwell's demon, and condensed-matter entanglement. There are promising signs that QET will be experimentally verified using the chiral boson fields of quantum Hall edge currents. In this Letter, we prove that, using the vacuum state of a quantum field, the upper bound of the amount of energy teleported by QET is inve…
