We propose that the quantum conditional mutual information (QCMI), computed with a suitably chosen partition of the system, serves as a powerful probe for detecting measurement-induced entanglement phase transitions in monitored quantum circuits. To demonstrate this, we investigate monitored variable-range Clifford circuits and identify the phase boundary between volume-law and area-law entanglement phases by performing finite-size scaling analyses of the QCMI. Assuming that the entanglement entropy exhibits a logarithmic dependence on system size at criticality in short-range interacting cases, we further show that the QCMI allows for the simultaneous determination of both the critical point and the universal coefficient of the logarithmic term in the entanglement entropy via a crossing-point analysis. For the shortest-range interacting case studied, we obtain the thermodynamic-limit value of the coefficient as $\tilde{c}=1.519(3)$, which is significantly smaller than values reported in previous studies.
We propose entanglement negativity as a fine-grained probe of measurement-induced criticality. We motivate this proposal in stabilizer states, where for two disjoint subregions, comparing their "mutual negativity" and their mutual information leads to a precise distinction between bipartite and multipartite entanglement. In a measurement-only stabilizer circuit that maps exactly to two-dimensional critical percolation, we show that the mutual information and the mutual negativity are governed by boundary conformal fields of different scaling dimensions at long distances. We then consider a cla…
I study the mutual information between spatial subsystems in a variety of scale invariant quantum field theories. While it is derived from the bare entanglement entropy, the mutual information offers a more refined probe of the entanglement structure of quantum field theories because it remains finite in the continuum limit. I argue that the mutual information has certain universal singularities that are a manifestation of the idea of "entanglement per scale". Moreover, I propose a method, based on an ansatz for higher dimensional twist operators, to compute the entanglement entropy, Renyi ent…
In previous papers, we expressed the Entanglement of Formation in terms of Conditional Mutual Information (CMI). In this brief paper, we express the Entanglement of Distillation in terms of CMI.
Based on the monogamy of entanglement, we develop the technique of quantum conditioning to build an {\it additive} entanglement measure: the conditional entanglement of mutual information. Its {\it operational} meaning is elaborated to be the minimal net "flow of qubits" in the process of partial state merging. The result and conclusion can also be generalized to multipartite entanglement cases.
We study the classical, classical-quantum, and quantum parts of conditional mutual information in the ``system-environment-ancilla'' setting of open quantum systems. We perform the separation of conditional mutual information by generalizing the classification of correlations of quantum states. The condition for identifying the classical part of conditional mutual information is given by adapting the no-local-broadcasting theorem to this setting, while the condition for classical-quantum part of conditional mutual information is obtained by considering the multipartite quantum discord and the …
In this paper, we study measures of quantum non-Markovianity based on the conditional mutual information. We obtain such measures by considering multiple parts of the total environment such that the conditional mutual information can be defined in this multipartite setup. The benefit of this approach is that the conditional mutual information is closely related to recovery maps and Markov chains; we also point out its relations with the change of distinguishability. We study along the way the properties of leaked information which is the conditional mutual information that can be back flowed, …
We prove that the quantum Gibbs states of spin systems above a certain threshold temperature are approximate quantum Markov networks, meaning that the conditional mutual information decays rapidly with distance. We demonstrate the exponential decay for short-ranged interacting systems and power-law decay for long-ranged interacting systems. Consequently, we establish the efficiency of quantum Gibbs sampling algorithms, a strong version of the area law, the quasi-locality of effective Hamiltonians on subsystems, a clustering theorem for mutual information, and a polynomial-time algorithm for cl…
We study the relation between the quantum conditional mutual information and the quantum $α$-Rényi divergences. Considering the totally antisymmetric state we show that it is not possible to attain a proper generalization of the quantum conditional mutual information by optimizing the distance in terms of quantum $α$-Rényi divergences over the set of all Markov states. The failure of the approach considered arises from the observation that a small quantum conditional mutual information does not imply that the state is close to a quantum Markov state.
A simpler approach to the characterization of vanishing conditional mutual information is presented. Some remarks are given as well. More specifically, relating the conditional mutual information to a commutator is a very promising approach towards the approximate version of SSA. That is, it is conjectured that small conditional mutual information implies small perturbation of quantum Markov chain.
We study entanglement witness and present a construction of entanglement witnesses in terms of the mutually unbiased measurements (MUMs). These witnesses include the entanglement witnesses constructed from mutually unbiased bases (MUBs) as a special case. Comparing with the dimension dependence of MUBs, the witnesses can be always constructed from a complete set of $d+1$ MUMs for any dimension $d$. We show that our witness can detect entanglement better than previous separability criterion given also by MUMs. And the approach can be experimentally implemented.
We study the entanglement detection by using mutually unbiased measurements and provide a quantum separability criterion that can be experimentally implemented for arbitrary $d$-dimensional bipartite systems. We show that this criterion is more effective than the criterion based on mutually unbiased bases. For isotropic states our criterion becomes both necessary and sufficient.
A new family of positive, trace-preserving maps is introduced. It is defined using the mutually unbiased measurements, which generalize the notion of mutual unbiasedness of orthonormal bases. This family allows one to define entanglement witnesses whose indecomposability depends on the characteristics of the associated measurement operators.
Kang et al. [Chin. Phys. B 24 (2015) 090306] proposed a controlled mutual quantum entity authentication protocol. We find that the proposed protocol is not secure, that is, Charlie can eavesdrop the shared keys between Alice and Bob without being detected.
A complementary behavior between local mutual information and average output entanglement is derived for arbitrary bipartite ensembles. This leads to bounds on the yield of entanglement in distillation protocols that involve disinguishing. This bound is saturated in the hashing protocol for distillation, for Bell-diagonal states.
Entanglement and information are powerful lenses to probe phases transitions in many-body systems. Motivated by recent cold atom experiments, which are now able to measure the corresponding information-theoretic quantities, we study the Mott transition in the half-filled two-dimensional Hubbard model using cellular dynamical mean-field theory, and focus on two key measures of quantum correlations: entanglement entropy and mutual information. We show that they detect the first-order nature of the transition, the universality class of the endpoint, and the crossover emanating from the endpoint.
Entanglement properties of excited eigenstates (or of thermal mixed states) are difficult to study with conventional analytical methods. We approach this problem for random spin chains using a recently developed real-space renormalization group technique for excited states ("RSRG-X"). For the random $XX$ and quantum Ising chains, which have logarithmic divergences in the entanglement entropy of their (infinite-randomness) critical ground states, we show that the entanglement entropy of excited eigenstates retains a logarithmic divergence while the mutual information of thermal mixed states doe…
We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of en…
We give two strengthenings of an inequality for the quantum conditional mutual information of a tripartite quantum state recently proved by Fawzi and Renner, connecting it with the ability to reconstruct the state from its bipartite reductions. Namely we show that the conditional mutual information is an upper bound on the regularised relative entropy distance between the quantum state and its reconstructed version. It is also an upper bound for the measured relative entropy distance of the state to its reconstructed version. The main ingredient of the proof is the fact that the conditional mu…
One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon is most strongly manifested when observables are related to mutually unbiased bases. In this paper, we shed some light on the connection between mutually unbiased bases and another essential feature of quantum mechanics, quantum entanglement. It is shown that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independently of the choice of the set. This has implications for entanglement distribution among t…
An uncertainty relation is introduced for a symmetric arrangement of three mutually unbiased bases in continuous variable phase space, and then used to derive a bipartite entanglement criterion based on the variance of global operators composed of these three phase space variables. We test this criterion using spatial variables of photon pairs, and show that the entangled photons are correlated in three pairs of bases.
In this paper, a lower bound of quantum conditional mutual information is obtained by employing the Peierls-Bogoliubov inequality and Golden Thompson inequality. Comparison with the bounds obtained by other researchers indicates that our result is independent of any measurements. It may give some new insights over squashed entanglement and perturbations of Markov chain states.
We introduce a new concept called as the mutual uncertainty between two observables in a given quantum state which enjoys similar features like the mutual information for two random variables. Further, we define the conditional uncertainty as well as conditional variance and show that conditioning on more observable reduces the uncertainty. Given three observables, we prove a 'strong sub-additivity' relation for the conditional uncertainty under certain condition. As an application, we show that using the conditional variance one can detect bipartite higher dimensional entangled states. The ef…
Renormalization group and long-range conditional mutual information in hierarchical models
University of California at San Diego（圣地亚哥加利福尼亚大学）；Kavli Institute for Theoretical Physics, University of California, Santa Barbara（加州大学圣巴巴拉分校卡弗里理论物理研究所）
The mathematical structure of quantum entanglement is studied and classified from the point of view of quantum compound states. We show that t he classical-quantum correspondences such as encodings can be treated as dia gonal (d-) entanglements. The mutual entropy of the d-compound and entangled states lead to two different types of entropies for a given quantum state: t he von Neumann entropy, which is achieved as the supremum of the information over all d-entanglements, and the dimensional entropy, which is achieved at the standard entanglement, the true quantum entanglement, coinciding with…
We study the growth of entanglement between two adjacent regions in a tripartite, one-dimensional many-body system after a quantum quench. Combining a replica trick with a space-time duality transformation, we derive an exact, universal relation between the entanglement negativity and Renyi-1/2 mutual information which holds at times shorter than the sizes of all subsystems. Our proof is directly applicable to any translationally invariant local quantum circuit, i.e., any lattice system in discrete time characterised by local interactions, irrespective of the nature of its dynamics. Our deriva…
Based on the ideas of {\it quantum extension} and {\it quantum conditioning}, we propose a generic approach to construct a new kind of entanglement measures called {\it conditional entanglement}. The new measures, built from the known entanglement measures, are convex, automatically {\it super-additive}, and even smaller than the regularized versions of the generating measures. More importantly, new measures can also be built directly from measures of correlations, enabling us to introduce an {\it additive} measure and generalize it to a multipartite entanglement measure.
We use the structure of conditionally independent states to analyze the stability of topological entanglement entropy. For the ground state of quantum double or Levin-Wen model, we obtain a bound on the first order perturbation of topological entanglement entropy in terms of its energy gap and subsystem size. The bound decreases superpolynomially with the size of the subsystem, provided the energy gap is nonzero. We also study the finite temperature stability of stabilizer models, for which we prove a stronger statement than the strong subadditivity of entropy. Using this statement and assumin…
In this paper we study the entanglement properties of free {\em non-relativistic} Bose gases. At zero temperature, we calculate the bipartite block entanglement entropy of the system, and find it diverges logarithmically with the particle number in the subsystem. For finite temperatures, we study the mutual information between the two blocks. We first analytically study an infinite-range hopping model, then numerically study a set of long-range hopping models in one-deimension that exhibit Bose-Einstein condensation. In both cases we find that a Bose-Einstein condensate, if present, makes a di…
We discuss conditional Renyi and Tsallis entropies for bipartite quantum systems of finite dimension. We investigate the relation between the positivity of conditional entropies and entanglement properties. It is in particular shown that any state having a negative conditional entropy with respect to any value of the entropic parameter is distillable since it violates the reduction criterion. Moreover we show that the entanglement of Werner states in odd dimensions can neither be detected by entropic criteria nor by any other spectral criterion.
An efficient method for assessing the quality of quantum state tomography is developed. Special attention is paid to the tomography of multipartite systems in terms of unbiased measurements. Although the overall reconstruction errors of different sets of mutually unbiased bases are the same, differences appear when particular aspects of the measured system are contemplated. This point is illustrated by estimating the fidelities of genuinely tripartite entangled states.
We propose a scheme to for fast conditional phase shift and creation entanglement of two qubits that interact with a common heat bath. Dynamical decoupling is applied in the scheme so that it works even in the regime of strong interaction between qubits and environment. Our scheme does not request any direct interaction between the two qubits.
We find a sufficient condition to imprint the single-mode bosonic phase-space nonclassicality onto a bipartite state as modal entanglement and vice versa using an arbitrary beam splitter. Surprisingly, the entanglement produced or detected in this way depends only on the nonclassicality of the marginal input or output states, regardless of their purity and separability. In this way, our result provides a sufficient condition for generating entangled states of arbitrary high temperature and arbitrary large number of particles. We also study the evolution of the entanglement within a lossy Mach-…
We show that the separability of states in quantum mechanics has a close counterpart in classical physics, and that conditional mutual information (a.k.a. conditional information transmission) is a very useful quantity in the study of both quantum and classical separabilities. We also show how to define entanglement of formation in terms of conditional mutual information. This paper lays the theoretical foundations for a sequel paper which will present a computer program that can calculate a decomposition of any separable quantum or classical state.
Quantum steering has recently been formalized in the framework of a resource theory of steering, and several quantifiers have already been introduced. Here, we propose an information-theoretic quantifier for steering called intrinsic steerability, which uses conditional mutual information to measure the deviation of a given assemblage from one having a local hidden-state model. We thus relate conditional mutual information to quantum steering and introduce monotones that satisfy certain desirable properties. The idea behind the quantifier is to suppress the correlations that can be explained b…
A state on a tripartite quantum system $A \otimes B \otimes C$ forms a Markov chain if it can be reconstructed from its marginal on $A \otimes B$ by a quantum operation from $B$ to $B \otimes C$. We show that the quantum conditional mutual information $I(A: C | B)$ of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus quantifies how well the Markov chain property is approximated.
We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state of size $N$ in $L$ orthogonal bases. Lower bounds lead to novel entropic uncertainty relations, while upper bounds allow us to formulate universal certainty relations. For $L=2$ the maximal average entropy saturates at $\log N$ as there exists a mutually coherent state, but certainty relations are shown to be nontrivial for $L \ge 3$ measurements. In the case of a prime power dimension, $N=p^k$, and the number of measurements $L=N+1$, the upper bound for the average entropy becomes minimal for…
The conditional quantum mutual information $I(A;B|C)$ of a tripartite state $ρ_{ABC}$ is an information quantity which lies at the center of many problems in quantum information theory. Three of its main properties are that it is non-negative for any tripartite state, that it decreases under local operations applied to systems $A$ and $B$, and that it obeys the duality relation $I(A;B|C)=I(A;B|D)$ for a four-party pure state on systems $ABCD$. The conditional mutual information also underlies the squashed entanglement, an entanglement measure that satisfies all of the axioms desired for an ent…
Using the Ryu-Takayanagi conjectured formula for entanglement entropy in the context of gauge-gravity duality, we investigate properties of mutual information between two disjoint rectangular sub-systems in finite temperature relativistic conformal field theories in d-spacetime dimensions and non-relativistic scale-invariant theories in some generic examples. In all these cases mutual information undergoes a transition beyond which it is identically zero. We study this transition in detail and find universal qualitative features for the above class of theories which has holographic dual descri…
We study a generalized double Jaynes-Cummings (JC) model where two entangled pairs of two-level atoms interact indirectly. We focus on the case where the cavities and the entangled pairs are uncorrelated. We show that there exist initial states of the qubit system so that two entangled pairs are available at all times. In particular, the minimum entanglement in the pairs as a function of the initial state is studied. Finally, we extend our findings to a model consisting of multi-mode atom-cavity interactions. We use a non-Markovian quantum state diffusion (QSD) equation to obtain the steady-st…
Enhancing quantum entanglement is important for many quantum information processing applications. In this paper, we consider a protocol for entanglement enhancing in a two-mode squeezed vacuum state (TMSVS), attained based on photon subtraction, photon catalysis, and photon addition. Central to such an operation is the task of mixing and detecting number states with each mode of TMSVS. We analyze various settings and find an optimal setup for improving the entanglement of the state.
It is well known that the quantum mutual information and its conditional version do not increase under local channels. I this paper we show that the recently established lower semicontinuity of the quantum conditional mutual information implies (in fact, is equivalent to) the lower semicontinuity of the loss of the quantum (conditional) mutual information under local channels considered as a function on the Cartesian product of the set of all states of a composite system and the sets of all local channels (equipped with the strong convergence). Some applications of this property are considered…
Conditional interference patterns can be obtained with twin photons from spontaneous parametric down-conversion and the phase of the pattern can be controlled by the relative transverse position of the signal and idler detectors. Using a configuration that produces entangled photons in both polarization and transverse momentum we report on the control of the conditional patterns by acting on the polarization degree of freedom.
We study the ground state of a gapped quantum many-body system whose entanglement entropy $S_A$ can be expressed as $S_A = a|\partial A| - γ$, where $a, γ$ are some constants and $|\partial A|$ is an area of the subsystem $A$. By using a recently proved operator extension of strong subadditivity of entropy,[I. H. Kim, J. Math. Phys. 53, 122204 (2012)] we show that certain linear combination of the real-space entanglement spectrum has a small correlation with almost any local operator. Our result implies that there exists a structure relating the real-space entanglement spectrum over different …
We present a quantum information theoretic version of the Klein-Nishina formula. This formulation singles out the quantity, the a priori visibility, that quantifies the ability to deduce the polarisation property of single photons. The Kraus-type structure allows a straightforward generalisation to the multiphoton cases, relevant in the decay of positronium which is utilized e.g. for metabolic PET-imaging (Positron-Emission-Tomograph). Predicted by theory but never experimentally proven, the two- or three-photon states should be entangled. We provide an experimentally feasible method to witnes…
The operational structure of quantum couplings and entanglements is studied and classified for semifinite von Neumann algebras. We show that the classical-quantum correspondences such as quantum encodings can be treated as diagonal semi-classical (d-) couplings, and the entanglements characterized by truly quantum (q-) couplings, can be regarded as truly quantum encodings. The relative entropy of the d-compound and entangled states leads to two different types of entropy for a given quantum state: the von Neumann entropy, which is achieved as the maximum of mutual information over all d-entang…
It is presented a generalization of the von Neumann mutual information in the context of Tsallis' nonextensive statistics. As an example, entanglement between two (two-level) quantum subsystems is discussed. Important changes occur in the generalized mutual information, which measures the degree of entanglement, depending on the entropic index q.
Quantum conditional mutual information and channel capacity
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
Entanglement between quantum dots transmitted via Majorana wire: Insights from the fermionic negativity, concurrence and quantum mutual information
Rzeszów University of Technology（热舒夫理工大学）；University of Opole（奥波莱大学）；A. Mickiewicz University（亚当·密茨凯维奇大学）；M. Curie-Skłodowska University（玛丽亚·居里-斯克沃多夫斯卡大学）
We discuss the notions of mutual information and conditional information for noncomposite systems, classical and quantum; both the mutual information and the conditional information are associated with the presence of hidden correlations in the state of a single qudit. We consider analogs of the entanglement phenomena in the systems without subsystems related to strong hidden quantum correlations.
The connection between quantum state recovery and quantum conditional mutual information (QCMI) is studied for the class of purely generated finitely correlated states (pgFCS) of one-dimensional quantum spin chains. For a tripartition of the chain into two subsystems separated by a buffer region, it is shown that a pgFCS is an approximate quantum Markov chain, and stronger, may be approximated by a quantum Markov chain in trace distance, with an error exponentially small in the buffer size. This implies that, (1) a locally corrupted state can be approximately recovered by action of a quantum c…
The study of mutual entropy (information) and capacity in classica l system was extensively done after Shannon by several authors like Kolmogor ov and Gelfand. In quantum systems, there have been several definitions of t he mutual entropy for classical input and quantum output. In 1983, the autho r defined the fully quantum mechanical mutual entropy by means of the relati ve entropy of Umegaki, and it has been used to compute the capacity of quant um channel for quantum communication process; quantum input-quantum output. Recently, a correlated state in quantum syatems, so-called quantum entan…
