Quantum criticality under decoherence or weak measurement

Decoherence inevitably happens when a quantum state is exposed to its environment, which can affect quantum critical points (QCP) in a nontrivial way. As was pointed out in recent literature on $(1+1)d$ conformal field theory (CFT), the effect of weak measurement can be mathematically mapped to the problem of boundary CFT. In this work, we focus on the $(2+1)d$ QCPs, whose boundary and defect effects have attracted enormous theoretical and numerical interests very recently. We focus on decoherence caused by weak measurements with and without post-selecting the measurement outcomes. Our main results are: (1) for an O(N) Wilson-Fisher QCP under weak measurement with post-selection, an observer would in general observe two different types of boundary/defect criticality with very different behaviors from the well-known Wilson-Fisher fixed points; in particular, it is possible to observe the recently proposed exotic"extraordinary-log"correlation. (2) An extra quantum phase transition can be driven by decoherence, if we consider quantities nonlinear with the decohered density matrix, such as the Renyi entropy. We demonstrate the connection between this transition to the information-theoretic transition driven by an error in the toric code model. (3) When there is no post-selection, though correlation functions between local operators remain the same as the undecohered pure state, nonlocal operators such as the"disorder operator"would have qualitatively distinct behaviors; and we also show that the decoherence can lead to confinement.


I. INTRODUCTION
When a quantum state is exposed to an environment, it is being constantly probed and "measured", forming entanglement with the degrees of freedom in the environment.If one is ignorant of the environment and the measurement outcome is lost, it amounts to tracing out the environment's degrees of freedom and the original pure quantum state becomes a mixed state, which results in the loss of coherent quantum information.This process is referred to as quantum decoherence, and it is the bridge between the quantum mechanics that governs the microscopic nature, and our classical macroscopic world [2].More generally, if a quantum state is weakly measured and the measurement outcome is still accessible, one can consider the effect of postselecting the measurement outcome and study process that is more general than decoherence.In recent years there has been a surge of progress in simulating quantum states of matter with nontrivial entanglement on platforms summarized as the noisy intermediate-scale quantum (NISQ) technology [3], including simulating exotic quantum many-body states such as topological order, spin liquids, and symmetry protected topological (SPT) states [4][5][6][7][8], which have long been discussed in condensed-matter physics.In these platforms, decoherence can happen due to various reasons, which motivated recent inspection of the fate of SPT states under decoherence [9][10][11][12] (related studies motivated from other contexts were also conducted [13,14]).
Quantum criticality represents another class of quantum many-body states with peculiar and universal entanglement.Recently a class of (1 + 1)d conformal field theory (CFT), i.e., the Luttinger liquid under weak measurements has been studied, and it was pointed out that the effect of weak measurements can be mathematically mapped to the problem of the boundary of the CFT [1], which is a subject that was studied extensively in the past [15].The same trick was used in the recent study of SPT states under decoherence [12], which exploited the observation that the wave function of the SPT states can be mapped to the partition function of the boundary states of the system after a space-time rotation [16,17].
In this work, we focus on quantum critical points in (2 + 1)d.The connection between the decohered QCPs (or QCPs under weak measurement) in the bulk and the boundary criticality still holds, but the boundary criticality of (2 + 1)d QCPs is a subject that has only been carefully studied very recently, and it has attracted enormous interests from both the theoretical and numerical communities [18][19][20][21][22][23][24][25][26][27][28][29][30].It has been understood since long back that there exists an ordinary boundary condition of a (2 + 1)d Wilson-Fisher QCP (or a 3d classical Wilson-Fisher critical point), where the Landau order parameter φ has a scaling dimension ∆ b φ > 1, which is far greater than the bulk scaling dimension of the order parameter (which is slightly greater than 1/2).Only recently it has become clear that at the boundary of an O(N ) Wilson-Fisher critical point, in addition to the well-known ordinary boundary criticality, there is a socalled "extraordinary-log" boundary criticality, meaning the correlation function of the order parameter at the boundary reads where q depends on N .This peculiar scaling was proposed theoretically [25,26] and recently confirmed numerically in Monte Carlo simulations [28,29].
Our current work will bridge these two directions that are under active studying, and we will demonstrate that a (2+1)d QCP under decoherence or weak measurements naturally exhibits the peculiar boundary criticality studied recently.This work is organized as follows: In section II, we develop the general formalism of analyzing decohered quantum critical states, including the general connection between the decohered or weakly measured bulk state and the boundary/defect criticality.
In section III we discuss the (2 + 1)d O(N ) Wilson-Fisher quantum critical points under decoherence or weak measurements, and in section III A we focus on the quantities linear with the density matrix, which can be observed directly in experiments.We demonstrate that weak measurements generally render the observed quantities rather different from the bulk QCP (with certain postselection that preserves the O(N ) symmetry in the mixed state ensemble), and we may observe the exotic extraordinary-log correlation mentioned above.
In section III B we discuss quantities nonlinear with the density matrix, which reveals a lot more structures of the mixed state density matrix of the critical state under decoherence.In particular, we discuss a quantum information phase transition that can be diagnosed through the 2nd Renyi entropy of the decohered system, along with other correlations defined in this section.
Here we would like to point out an important difference between the physical scenarios to be considered in section III A and III B. In section III A we discuss physics under weak measurements on quantities such as energy density, and we allow a postselection on the measurement outcomes.Section III B considers quantities nonlinear with the density matrix where postselection is not needed in this scenario; hence it can genuinely correspond to physics under decoherence due to coupling to the environment.
In section III D we demonstrate that there is an explicit lattice model with an information transition driven by the strength of decoherence (or weak measurement) analogous to the decoherence-driven transition in section III B. We also show that the transition in this lattice model is dual to an information transition in the toric code model, which is related to the error threshold of the toric code [31].
If we forbid postselection, local correlation functions of the (locally) decohered density matrix would remain largely unchanged from the undecohered pure state.In recent years the nonlocal disorder operator has become a very important diagnosis of quantum states of matter.In section IV we demonstrate that the nonlocal disorder operator can still have qualitatively different behavior from the undecohered pure state density matrix, even if there is no postselection.In particular, we show that in several examples, decoherence can lead to confinement.
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II. GENERAL FORMALISM
In order to discuss quantum states under decoherence, one approach is to first explicitly derive the ground-state wave function |Ψ in either exactly soluble lattice models or effective field theories, then construct the density matrix ρD under decoherence [12].This approach relies on an explicit derivation of the ground-state wave function.The ground-state wave function can be derived for gapped states such as SPT phases [9,16,17], and also gapless phases with a Gaussian (free boson) Lagrangian, such as the (1 + 1)d Luttinger liquid [1], or the Rokhsar-Kivelson (RK) point in (2 + 1)d models [32].But for general interacting theories, such as systems near the Wilson-Fisher quantum critical points, deriving the ground-state wave function is cumbersome.In this section, we follow a more general procedure to study interacting systems under decoherence.
Let us prepare an interacting quantum state which is the ground state of a Hamiltonian, whose pure state density matrix is given as ρ0 .After the quantum state is prepared, we turn off the Hamiltonian and expose the system to local decoherence.For example, for a lattice model of qubits, a decohered density matrix may be represented as ( where ρD describes a mixed state (ensemble) and E is given as the composition of local decoherence channels E x .One interpretation of this decoherence channel is that, there is a certain probability for the environment to measure qubits in the Z basis (weak measurements) at each location x, and the measurement outcomes are "lost", which amounts to the dephasing noise in the study of quantum circuits.A generic decoherence channel maps a density matrix ρ0 into ρD = m K m ρ0 K † m , where {K m } is a set of Kraus operators satisfying the condition If we interpret a decoherence channel to be induced by weak measurements, we can consider a postselection followed by the channel where the resulting density matrix is given as where P is a generalized projection onto a subset of mea-surement outcomes P , i.e., postselection.
In this work, we will mostly study systems at or close to a QCP, hence we will use a coarse-grained continuous space rather than a lattice model.In the coarse-grained formalism, the density matrix of a pure state is given by the following imaginary time path-integral where S = β 0 dτ dx L(φ) is the bulk action of the system and x = (x 1 , ..., x d ) is the spatial coordinate.Following Ref. 1 and 12, a class of decoherence problem in the coarse-grained continuous space can be converted into the following imaginary-time path-integral: The effect of decoherence or postselection is captured by an extra interaction term L int in the Lagrangian, and L int has no temporal integral, i.e., it is a "long range" interaction in the Euclidean temporal direction, between fields at τ = 0 and τ = β.The coupling constant in L int is controlled by the strength of decoherence (or strength of weak measurement) p.A natural choice of L int favors configurations with φ(x, 0) ∼ φ(x, β), meaning it enhances the weight of the diagonal components of the density matrix, which drives the system into a mixed state density matrix.
The most important factor that determines the form of L int is still its symmetry.Let us label the symmetry of the original Lagrangian L as G, and assume that the original pure state without decoherence is a symmetric state under G. Then there are two types of symmetry constraints on L int .If the environment is weakly "measuring" quantities that are invariant under G, then L int must be invariant under a "doubled" symmetry transformation, i.e., it is invariant under symmetry transformation G on φ(x, 0) and φ(x, β) separately.In the language of Ref. 12, this is the case that preserves the doubled G u × G l symmetry, where G u and G l correspond to the upper and lower symmetry in the formalism of doubled Hilbert space using the Choi-Jamiolkowski isomorphism [33,34], which maps any density matrix to a pure ket-state in the doubled Hilbert space.However, if the environment is measuring quantities that carry a nontrivial representation of G, but eventually we sum over the measurement outcomes within the same representa-tion of G with equal weight, meaning the symmetry G is broken in each quantum trajectory but still preserved in an average sense, then L int is only invariant under symmetry G, which is the diagonal subgroup of G u × G l , and it corresponds to a simultaneous transformation of φ(x, 0) and φ(x, β).
In Eq. ( 2), if the original pure state is invariant under a Z 2 symmetry action x X x then the pure state density matrix ρ0 is invariant under a doubled Z u 2 × Z l 2 symmetry, where Z u 2 and Z l 2 correspond to the left and right operations of the Z 2 symmetry.But since the decoherence is caused by weakly measuring the Ising variable which carries a nontrivial representation of Z 2 , the decohered density matrix ρD is no longer invariant under separate Z u 2 or Z l 2 action, it is instead only invariant under the simultaneous action of both Z u 2 and Z l 2 , hence ρD is invariant under a diagonal Z 2 symmetry.Here the Z u 2 × Z l 2 symmetry is explicitly broken down to Z 2 by the decoherence; later we will also discuss an example where the decoherence preserves the doubled symmetry but the doubled symmetry can be spontaneously broken down to the diagonal Z 2 .Now suppose we would like to compute the expectation value of a certain quantity O( φ) that is a composite of φ.The expectation value is linear with the density matrix, and it can be evaluated in the path integral form: Euclidean spacetime diagram and its spacetime rotated version.The evaluation of the second Renyi entropy of ρD becomes a path-integral in the Euclidean spacetime with an interaction between fields at τ = 0 and τ = β/2.
In the space-time rotated picture, such an interaction corresponds to the long-range interaction between fields at y = 0 and y = L/2, where L = β.Note that the first coordinate of x = (x, y) is not shown in the diagram.
where we have identified the field configurations φ(x, 0) and φ(x, β) due to the trace.If there is no postselection at all, one can easily show that tr{ρ D O( φ)} and tr{ρ 0 O( φ)} have essentially the same behavior (except for some local corrections) as long as O( φ) is a product of local operators [35], as pictorially illustrated in Fig. 1.For example, the correlation function tr{ρ D φ(x 1 ) φ(x 2 )} should have the same scaling in space as tr{ρ 0 φ(x 1 ) φ(x 2 )}.In the field theory language, this corresponds to S int being trivial when φ(x, 0) = φ(x, β).
But if there is some weak postselection by P even on quantities that are singlet under G, tr{ρ D P φ(x 1 ) φ(x 2 )} can be very different from tr{ρ 0 φ(x 1 ) φ(x 2 )}.With postselection, S int remains non-trivial even at φ(x, 0) = φ(x, β), which effectively corresponds to the insertion of defects at the slab τ = 0 (or τ = β) in the path-integral in the (d + 1)−dimensional Euclidean space-time.This is where we can make a connection to all the recent studies on boundary criticality for systems with d = 2 [36], and the desired decohered correlation functions become the correlation functions restricted on the plane-like defect.
The expectation value of an operator is linear with the density matrix, which is directly related to experimental observables.But a lot of information of the quantum system is encoded in quantities that are nonlinear with the density matrix.The most famous example of such is the von Neumann entropy of the density matrix.In this work, we will evaluate quantities such as the 2nd Renyi entropy of ρD , which is an analogue of the von Neumann entropy, and it also provides an approximate characterization [37] of the amount of quantum information lost due to entangling with the environment: This calculation is schematically shown in Fig. 2, which amounts to evaluating the partition function and free energy of the system with an interaction between fields at imaginary time τ = 0 and τ = β/2.Or we can also rotate the space-time (assuming there is a Lorentz symmetry), then the problem becomes evaluating the partition function of the system with nonlocal interaction in space, but constant in time.
Other quantities of interest include tr{(ρ We will show that these quantities nonlinear with ρD would reveal some novel quantum phase transitions.In the example we will discuss in the next section, the decoherence respects the doubled Z u 2 ×Z l 2 symmetry; but when we increase the decoherence strength, the 2nd Renyi entropy of the system (which captures the information loss to the environment) may encounter singularity at a critical decoherence strength, which corresponds to the spontaneous symmetry breaking from Z u 2 × Z l 2 to the diagonal Z 2 .

III. DECOHERED WILSON-FISHER CRITICAL POINT
In this section, we study two different scenarios for the system under weak measurements: In Sec.III A, we postselect a set of measurement outcomes and show that the resulting correlation functions linear in the density matrix ρD P exhibit novel behaviors.In Sec.III B, we average over all measurement outcomes, which corresponds to genuine decoherence; although correlation functions linear in ρD would exhibit an ordinary Wilson-Fisher behavior in this case, we show that quantities non-linear in ρD still exhibit novel behaviors including the extraordinarylog criticality and information-theoretic phase transition.
A. Quantities linear with ρD P First, we discuss critical physics under postselection as elaborated in Eq. ( 3).The critical system we consider is the O(N ) Wilson-Fisher fixed point in (2 + 1)d, where x = (x, y) is a two-dimensional spatial coordinate.Under postselection onto a set of measurement outcomes P , we would like to consider quantities linear with ρD P , for example the correlation function tr{ρ D P φ(0) • φ(x)}.The evaluation of quantities such as tr{ρ D P O( φ(x))} can be performed by evaluating path-integral Eq. ( 6).Since we would like to keep at least the diagonal O(N ) symmetry, the postselected measurement outcomes P should be invariant under O(N ) symmetry.Under that condition, the corresponding S int becomes a function of the magnitude of the order parameter |φ| at τ = 0, which is always equal to |φ| at τ = β due to the trace.The simplest term (and most relevant term) of S int is The term S int can be interpreted as weakly measuring the energy density or the order parameter φ of the system, followed by postselection.
Although there is only one simple term in S int , the physical consequence is already rather nontrivial.S int is a 2d interface in a (2 + 1)d cylindrical space-time with extra mass ε for the order parameter, and obviously ε is always a relevant perturbation, as the scaling dimension of |φ| 2 is always smaller than 2 at the O(N ) Wilson-Fisher fixed point.When ε > 0, S int suppresses the fields at τ = 0, and "cuts" the connection between τ = 0 + and τ = β − , i.e. τ = 0 + and τ = β − become two boundaries with ordinary boundary condition.In this case, the correlation function tr{ρ D P φ(0) • φ(x)} scales as [38] tr{ρ where is the boundary scaling dimension of the order parameter φ to the first order expansion of 1/N .Note that the value of ∆ b φ is far larger than the bulk scaling dimension of φ.When ε < 0, the latest progress [26,36] of the boundary criticality indicates that S int will drive the interface τ = 0 into an extraordinary-log criticality, which implies that the correlation function becomes Please note that, for a single exposed boundary against the vacuum, the extraordinary-log criticality exists only for N < N c , with the critical N c estimated around N c ∼ 5 [26]; but for an interface defect inserted in space-time, the theoretical prediction is that N c → ∞ [36].
B. Quantities nonlinear with ρD P Now we evaluate quantities nonlinear with ρD .Although our formalism can be straightforwardly generalized for any higher orders of ρD , here we focus on the quantities that are quadratic in ρD .The quantities of interest include the 2nd Renyi entropy S (2) , the correlation function C (2) (x), the "crossed" correlation function C (2) X (x), and the "crossed-Ising" correlation function r w m 2 g 9 j W q R q J 9 B a P + r L F 3 2 5 g O / s 1 D u u + s x M L J g 6 s y e e r M F J S w r w m 2 g 9 j W q R q J 9 B a P + r L F 3 2 5 g O / s 1 D u u + s x M L J g 6 s y e e r M F J S w V Z 5

C
(2) XI (x) defined as follows: The expression of the correlation functions above need to be divided by the purity of the decohered density matrix tr{(ρ D ) 2 } to be properly normalized.We note that these correlation functions are distinguished by their representation under O(N ) u × O(N ) l symmetry, which becomes clear in the doubled Hilbert space as in Fig. 3.
As an example, we consider the following interaction term S int in Eq. 7: Here W is still a function of the magnitude of the order parameter at τ = 0 and τ = β/2.The two terms in Eq. 13 correspond to two different types of weak measurements (decoherence) channels.The first term still corresponds to weakly measuring the energy density of the system, which preserves the doubled symmetry of the system, resulting in the coupling where the "..." part includes quartic and higherorder terms in |φ(x, 0)| and |φ(x, β/2)|, such as The second w term can be rewritten as where is the traceless rank-2 symmetric tensor of the O(N ) vector φ, and the ellipsis are terms that only depend on |φ| and can be absorbed into W .The w term can be interpreted as a weak measurement on the tensor Q ab , and eventually all the measurement outcomes are summed with equal weight.The w term breaks the SO(N ) u × SO(N ) l down to the diagonal SO(N ), but still preserves the Z u 2 × Z l 2 symmetry, where Z 2 corresponds to changing the sign of φ.
One can perform the space-time rotation in the (y, τ ) plane, so the two interfaces at τ = 0, β/2 become spatial interfaces at y = 0 and L/2, with L = β.The interaction term then becomes The first term W will either explicitly include an extra mass term ε|φ| 2 at the interface y = 0 and y = L/2, or generate the mass term through renormalization group flow.Then depending on the sign of ε, there can be three possible scenarios: (1) If ε > 0, the extra mass term ε|φ| 2 at the interface y = 0 and y = L/2 is a relevant perturbation.The role of this extra mass term is to "cut" the system into two halves: the region from y ∈ (0, L/2) and region y ∈ (L/2, L ∼ 0).The relevant mass term will make the two interfaces at y = 0 and y = L/2 both at ordinary boundary criticality.
In this scenario, w is obviously an irrelevant perturbation since both interfaces y = 0 and y = L/2 have ordinary boundary criticality, and the scaling dimension of φ is greater than 1 at the interfaces.Hence the directions of φ at the two interfaces are pretty much uncorrelated to each other.Then we expect the correlation function, the crossed-Ising correlation, and the crossed-correlation functions to behave as For example, the crossed-correlation C (2) X corresponds to the correlation function between order parameters at the two different interfaces; since w is irrelevant when ε > 0, the directions of order parameters at the two interfaces are uncorrelated, hence C (2) X should vanish in the limit β, L → ∞.Another way to perceive these results is that, since w is irrelevant, the system should have a full SO(N ) u × SO(N ) l symmetry in the infrared, but the correlation function C (2)

X and C
(2) XI break the SO(N ) u × SO(N ) l symmetry, hence they must vanish.
(2) In the case with ε < 0, the extra local mass term ε is still a relevant perturbation, and it flows to the extraordinary-log criticality.Then the w term (we assume w > 0) is a very relevant perturbation, and it will flow to w → ∞.
The fate of the system with strong w can be perceived through a mean field decoupling of L int .We first recognize that the w term is analogous to the coupling between two sets of Néel order parameters in the J 1 − J 2 Heisenberg model on the square lattice, which has applications in the context of frustrated magnets and iron-pnictides superconductors [39][40][41][42][43]. Guided by the previous studies in these contexts, the most natural mean field decoupling of the w term is Here we have introduced an Ising order parameter Φ(x, τ ) ∼ φ(x, τ ) y=0 • φ(x, τ ) y=L/2 .The order parameter Φ is analogous to the nematic order parameter in the J 1 − J 2 Heisenberg model on the square lattice, and the phase with large w is likely a phase with condensation of Φ, which spontaneously breaks Z u 2 × Z l 2 down to the diagonal Z 2 .
The condensation of Φ will "pin" φy=0 and φy=L/2 along the parallel direction.With a nonzero condensate of Φ, the three correlation functions mentioned above behave like The crossed-Ising correlation saturates to a nonzero constant in the limit of large |x| due to the condensate of Φ, which also leads to an extraordinary-log correlation of crossed-correlation function.When w flows to infinity and Φ condenses, there is only one O(N ) symmetry in the infrared; hence all these correlations can be nonzero.
(3) Now if ε = 0, with large but finite N , the scaling dimension of the traceless rank-2 symmetric tensor Q ab is ∆ = 1 + 32/(3π 2 N ) to the leading order of 1/N expansion.This means that w is weakly irrelevant with scaling dimension [w] = −64/(3π 2 N ).Then the beta function of w should be where A = 64/(3π 2 ).The constant C can be extracted through some OPE calculation, or simply a one-loop calculation in the large−N limit.Since C is positive, there is a fixed point at finite w * , beyond which w will flow strongly, and the order parameter Φ defined above will condense.

C. The phase diagram
In phase diagram Fig. 4 we summarize the results related to (ρ D ) 2 discussed in this section.The orderdisorder transition of Φ should extend to the region with ε > 0, where there is a competition between ε which drives the interfaces to the ordinary boundary criticality and w which drives the crossed-Ising transition.In this phase diagram, the critical w c is a function of ε: w c (ε) − w c (0) ∼ ε ∆w/∆ε , and ∆ w,ε are the scaling dimensions of parameter w and ε.
Since Φ(x) ∼ φ(x, 0) • φ(x, β/2) is the crossed-Ising order parameter, the crossed-Ising correlation function should show a transition from short-range to correlation while increasing w.If we consider tr{(ρ D ) 2 } as the partition function, and the 2nd Renyi entropy S (2) = − log tr{(ρ D ) 2 } as the free energy, the nature of the transition at w = w c and ε > 0 should belong to a 2d classical Ising universality class.Without coupling to other modes with nonlocal correlations in the infrared, the order-disorder transition of an Ising order parameter in the 2d space should belong to the 2d Ising universality class, and here we should consider the perturbation of the coupling w on top of the 2d Ising transition.In fact, if we start with a 2d classical Ising transition of order parameter Φ, the coupling −wΦ(x) (φ(x, 0) • φ(x, β/2)) is irrelevant knowing the fact that the scaling dimension ∆ b φ > 1 for the ordinary boundary criticality at ε > 0.
c , both sides are characterized by non-vanishing correlation functions of operators, which corresponds to (left) the formation of mean-field for Zv,uZ v,l operator or (right) condensation of a pair of e-anyons eue l .
Hence at w = w c and ε > 0, the crossed-Ising correlation function should scale as Also, when we increase w across w c , the 2nd Renyi entropy S (2) = − log tr{(ρ D ) 2 } should have the same singularity as the free energy of the classical 2d Ising model.

D. Lattice Model and Doubled Hilbert Space
We have shown that a decoherence channel whose Kraus operators are symmetric under O(N ) can drive an Ising-type phase transition that spontaneously breaks the Z u 2 × Z l 2 ⊂ O(N ) u × O(N ) l symmetry down to the diagonal Z 2 symmetry for quantities nonlinear in the density matrix.However, most physical quantities are linear in the density matrix, and this calls for a proper physical interpretation of these quantities.As we will see, our crossed-Ising transition is closely related to a better known information transition in the context of topological surface codes.
In the following, we use another model to illustrate the essential physics of such a spontaneous symmetry breaking (SSB) from Z u 2 × Z l 2 to the diagonal Z 2 .Instead of starting with a state at quantum criticality, we consider a concrete lattice model with a trivially disordered state |Ω 0 = |+ ⊗N on a L × L square lattice with qubits defined on vertices.We consider the following quantum channel as a symmetric local decoherence model: where "e" labels the edge between the nearest-neighbor pair of sites v and v .The decoherence channel is given as the composition of local channels, E = e E e .To proceed further, we apply the Choi-Jamiolkowski isomorphism to map the density matrix into the pure state and decoherence channel into the operator in the doubled Hilbert space.Under this mapping, the Choi state of the pure state density matrix ρ 0 = |Ω 0 Ω 0 | is denoted as ρ 0 ≡ |Ω 0 |Ω 0 , which is nothing but a vectorized density matrix, and the Choi operator of the decoherence channel is given as where tanh τ = p/(1 − p).As illustrated in Fig. 5, the isomorphism effectively maps the density matrix into the pure state in the bilayer system.Now, when the system is subject to the decoherence, we can show that ρ D = Ê ρ 0 is the ground-state wavefunction of the following local Hamiltonian in the doubled-Hilbert space [9,12]: where v ∈ v means that v is a nearest-neighbor site of v.At p = τ = 0, it reduces into two decoupled disordered states.At p > 0, the decoherence gives rise to the (local) coupling between two layers that may induce a phase transition into an SSB phase that breaks the Z u 2 ×Z l 2 into the diagonal Z 2 symmetry.The (unnormalized) groundstate can be written as where the summation is taken over all string configurations l on the edges of the square lattice, |∂l ≡ v∈∂l Z v |Ω 0 , and N v = L 2 is the number of vertices in the square lattice.We stress that Ê ρ 0 is normalized in the sense that its corresponding density matrix ρ D (under the Choi-Jamiolkowski isomorphism) is normalized with tr{ρ D } = 1 in the single Hilbert space while it is unnormalized as a state in the doubled Hilbert space.Then, it is straightforward to show that the norm of the wavefunction Ê ρ 0 , i.e., the purity of the density matrix tr{(ρ D ) 2 } is equivalent to the partition function of the 2d Ising model at the temperature β = 2τ , as explicitly derived in Appendix.A 6. Accordingly, at β = 0.441, which corresponds to p (2) c = 0.178, the wavefunction in the doubled Hilbert space undergoes the transition of 2d Ising universality.Here the superscript in p implies that the transition happens in the quantity that involves the product of two density matrices.
The SSB transition between the Z u 2 × Z l 2 -symmetric phase and the phase with only the diagonal Z 2 symmetry in the doubled Hilbert space is captured by the susceptibility χ to an external symmetry-breaking field coupled to O = v Z v,u Z v,l (in the doubled Hilbert space): which is given by the summation over connected correlation functions of the order parameter Z v,u Z v,l of this SSB transition in the doubled Hilbert space, related to the crossed-Ising correlation function C XI in Eq. (19).χ is closely related to the crossed-Ising correlation functions in Eq. (19).χ diverges as |p − p (2) c | −7/4 at the SSB transition.The exponent 7/4 is the signature of the 2d Ising universality class of this SSB transition.Note that O in the definition of χ is the order parameter that takes a non-zero value in the symmetry-broken phase and should be evaluated with δ = 0 + after the thermodynamical limit is taken.We elaborate more on this in the next section.
In the doubled Hilbert space, the definition of χ and the associated notion of the external symmetry-breaking field are standard for generic condensed-matter systems with global symmetries.In Sec.III E, we explain their physical meaning at the level of the density matrix ρ D .

E. Correlation Functions and Physical Meanings
In the previous sections, we have defined several correlation functions or physical quantities which are nonlinear in decohered density matrices.Accordingly, these quantities are not directly accessible from experiments, raising questions about what they really mean physically.In order to connect these quantities to experiments, we consider the signatures of the transition in the probability distribution of measurement outcomes.More precisely, we will show that χ in Eq. ( 26) corresponds to the sensitivity of the decohered mixed state against small perturbations.This, in turn, is related to the amount of information that can be obtained from measuring the mixed state.
To proceed, we define the notion of distance between two different density matrices as the following: This quantity is the quantum generalization of the Jeffreys divergence [44] (or symmetrized quantum relative entropy), which quantifies the distance between two density matrices.It satisfies good properties to be a valid metric between mixed states: (i) non-negative, (ii) vanishing if and only ifρ = σ, and (iii) monotonically decreasing under the application of quantum channels.
In order to discuss the "sensitivity" of the decohered density matrix, we define an infinitesimal variation of the original density matrix under symmetry breaking "perturbation", defined as the following quantum channel: which amounts to the application of weak measurement in the Z basis.Hence, δ also serves as a symmetrybreaking field in the doubled Hilbert space.Then, D(δ) ≡ D J (ρ D , ρ D δ ) quantifies the difference between the unperturbed and perturbed decohered density matrices.Accordingly, how fast the distance changes with δ tells how sensitive the decohered system is against weak measurement in the Z basis (or how informative the Z measurement is).This is captured by the second derivative of the distance, which is related to the (classical) Fisher information as F ≡ ∂ 2 δ D(δ) δ=0 [45].However, the expression in Eq. ( 27) is very challenging to evaluate.To proceed, we generalize the Jeffreys distance in a way similar to Ref. 46: This n-th Jeffreys distance is symmetric and nonnegative [47].Furthermore, in the limit n → 1 + , D (n) → D J .At n = 2, we find that the expression behaves like an overlap between Choi states of two density matrices in the doubled Hilbert space description: With this definition, we evaluate the derivatives of the "distance" between ρ D and its perturbed version ρ D δ with respect to δ (See Appendix.B).This quantity exhibits interesting behaviors when δ → 0 and L → ∞.The two different orders of limit lead to different results which have natural interpretations from a standard condensedmatter perspective and from a quantum information perspective respectively.
In the limit where the thermodynamical limit L → ∞ is taken first and δ → 0 + afterward, the conventional choice of limit for a condensed-matter system experiencing SSB, the second-order derivative ∂ 2 δ D (2) produces a quantity proportional to the susceptibility: which diverges at the SSB phase transition.Based on the nature of this SSB transition and the interpretation of δ as the symmetry-breaking field, we conclude that the decohered density matrix undergoes a qualitative change (with respect to this distance) as a function of p. Beyond critical p c , the density matrix changes significantly under weak measurements in the Z basis.
However, when we consider taking the limit δ → 0 before the thermodynamical limit, the second-order derivative produces a quantity that aligns with the informationtheoretic intuition: where M can be interpreted as the expectation value of the squared order parameter, which vanishes in the paramagnetic phase but acquires a finite value in the SSB phase.Since this SSB transition belongs to the 2d Ising universality class, the standard phenomenology of the Ising model can be directly translated into the language of our study.In an Ising ferromagnet, the magnetic order of the ground state of the system is very sensitive to an infinitesimal external Zeeman field.This well-known feature corresponds to the sensitivity of the density matrix to the decoherence in the Z v basis (as in Eq. ( 28)), in the phase where the Z u 2 × Z l 2 symmetry is spontaneously broken down to the diagonal Z 2 .This is quantified by the value of ∂ 2 δ D (2)  δ=0 (where δ is set to 0 before L grows large) scaling as L 4 in the SSB phase, which aligns with the Fisher information of the GHZ state in the context of quantum metrology [48].
It is straightforward to argue that the second derivatives of the distance D (n>2) in both limits would also exhibit similar behavior across a certain critical value (that depends on n), particularly because tr((ρ D ) n ) is mathematically equivalent to the partition function of the coupled n-copies of Ising model which is expected to undergo a transition from a paramagnetic to a ferromagnetic phase potentially at a different critical strength p c .Accordingly, we expect the behavior of ∂ 2 δ D (n) to extrapolate in the limit n → 1.Therefore, there should be two different phases separated by a critical point at strength p = p (1) c .This is indeed the case, as we will see by using a dual description of the model.

F. Duality, Intrinsic transition, and Decodability
Under the Kramers-Wannier (KW) duality, the Z 2 paramagnet under symmetric decoherence maps to the toric code under dephasing noise.The toric code under dephasing noise, in turn, is well known to exhibit an information transition at critical noise strength, beyond which the quantum information (logical qubits) stored in the toric code is not decodable [31].This hints at an inti-mate connection between the criticality discussed above and a well-known information transition.
First, we illustrate how the doubled Hilbert space formalism provides us some insights into the transition in the decohered toric code.Under the KW duality (See Appendix.A 5), the Hamiltonian in Eq. ( 24) maps to the two copies of toric code coupled by local anyon tunneling terms, as schematically shown in Fig. 5.At critical strength of the tunneling, the anyon condenses and the topological order reduces into a single toric code.The critical behavior is associated with a Higgs transition signified by the development of the anyon condensation amplitude ∼ ρ D γu γl ρ D , where γu/l = e∈γ ⊥ Z e,u/l creates a pair of e-anyon at the end of the string γ ⊥ .In turn, this implies that ρD and γ ρD γ have an appreciable overlap and they become less and less distinguishable for p > p (2) c in the doubled Hilbert space.In this coupled toric code in the doubled Hilbert space, the existence of the criticality can be directly seen by calculating the norm of the wave function, which is equivalent to the purity of the decohered toric code.The said quantity is given by the partition function of the 2d Ising model at β = tanh −1 (1−2p) 2 as shown in Appendix.A 3, which has a transition at β = ln 1 + √ 2 /2.As expected from the KW duality, this transition from an ordered to disordered phase upon increasing p coincides with the transition of the original Choi state Eq. ( 25) from a disordered to an ordered phase, giving rise to the same critical point p (2) c = 0.178.As a next step, we calculate the von Neumann entropy of the decohered toric code state, a quantity highly nonlinear in the density matrix that enters the expression in Eq. (27).Following the calculations in Appendix.A 2, we show that the von Neumann entropy of the decohered density matrix is proportional to the free energy of the random bond Ising model along the Nishimori line [49,50] at β = tanh −1 (1 − 2p), which is known to be critical at p = 0.1094 [51].Therefore, the transition behavior indeed extends down to the n → 1 limit for a toric code under dephasing noise, and by the KW duality, for the original paramagnetic spin model under symmetric decoherence.As remarked, the transition behavior in the limit n → 1 corresponds to the singularity of the Fisher information, a well-known information-theoretic quantity.We remark that this series of transitions for different n is intrinsic to the decohered density matrix (and the distance in use), as these transitions are associated with spontaneous symmetry breaking.Interestingly, the n → 1 limit of this intrinsic transition coincides with the decodability transition point demonstrated in Ref.31, which is based on a certain decoding procedure.Such a connection may imply that the intrinsic transition point provides an upper bound for some information retrieval protocols to be successful in this setting.

IV. NON-LOCAL OPERATORS
As we mentioned previously, without any postselection, the correlation function for local operators linear with the decohered density matrix should have the same scaling as the undecohered correlation function.But in this section, we will show that nonlocal quantities can still have qualitatively different behaviors even if we only consider expectation values linear with the density matrix.

A. 1d Quantum Rotor
To illustrate the behavior of nonlocal operators under decoherence, let us start with the 1d systems with a description in terms of the quantum rotor, such as the spin-1/2 chain.In terms of the Abelian bosonization, the Néel and valence bond solid (VBS) order parameters of a spin-1/2 chain are represented as (N x , N y , N z , V ) ∼ (sin θ, cos θ, sin φ, cos φ) .(33) Under decoherence or weak measurement of (for example) local operator e i φ, the density matrix becomes ρ0 is the undecohered density matrix of the spin-1/2 chain, and ρD = E[ρ 0 ] still keeps tr[ρ D ] = 1.We can first evaluate the correlation function of local operator O(r): For local bosonic order parameters at position r, such as the Néel and VBS, their nontrivial commutation with e i φ(x) would be exponentially suppressed by the separa-tion |r − x| at long distance, and thus for |r − x| 1, Ô(r) and e i φ(x) should commute.When Ô ∼ cos(φ) or sin(φ), since e i φ(x) always commute with Ô(0) and Ô(r), E x [ Ô(r) Ô(0)] = Ô(r) Ô(0), and the decoherence does not affect the correlator.When Ô = N x or N y , the decoherence channel does not commute with Ô(r) only in the vicinity of r.However, (i) the field (N x , N y ) is already the primary field with the lowest scaling dimension that forms an irreducible representation of the U(1) rotation symmetry generated by x S z x of the spin chain (θ → θ + α), and (ii) the decoherence channel explicitly preserves this U(1) symmetry, e i φ(r) (N x r , N y r )e −i φ(r) belong to the same representation as (N x r , N y r ) and would not generate any field with lower scaling dimension.Therefore, under decoherence the correlation functions of local operators (N x , N y , N z , V ) acquire only a constant amount of multiplicative factor aside from potential subleading power-law contributions, implying that they should still have the same power-law with scaling dimensions of the undecohered spin-1/2 chain [52].
However, for nonlocal operators, the situation can be very different.A particular family of nonlocal operators is the disorder operators [53][54][55][56][57][58][59][60] [61], which have attracted great interests recently.These operators have been used as an auxiliary diagnosis for the states of matter, especially for critical states of matter.For a 1d quantum rotor, if we view φ as the phase angle of a local boson creation operator, then (∇ x θ)/2π is the boson density nφ .The following operator is called a disorder operator If φ carries a full U(1) symmetry, then α ∈ R; if φ only carries a Z N symmetry, then α = 2πk N with k ∈ {1, ..., N }, as nφ is defined modulo N .
For example, for the spin-1/2 chain, if we view φ as a local operator, then e i θ/2 is a disorder operator of φ, and e i θ/2 plays two roles simultaneously: it first creates a fractionalized spin-1/2 excitation (i.e. a spinon), it also creates a domain wall of the VBS order parameter, i.e. e i θ(r)/2 shifts φ(x) → φ(x) + π for x < r.Indeed, it is well-known that a spin-1/2 is localized at the domain wall between two VBS orders.We can evaluate the correlation function of e i θ/2 for the decohered density matrix: where tr{ρ 0 e i θ(r)/2 e −i θ(0)/2 } is the correlation function of e i θ/2 for undecohered spin-1/2 chain, and the "correlation length" ξ is ξ ∼ −1/ ln(1 − 2p) for small p.
Hence our calculation for the 1d quantum rotor system implies that, although the disorder operator has a power-law correlation with the absence of decoherence, it can be rendered short-ranged under decoherence.As we will show in the next subsection, similar behavior of the disorder operator happens in higher dimensions as well.
The spin-1/2 chain is also an example of spin liquid with fractionalized spinon excitations since the spinon correlation function decays as a power-law in the undecohered spin-1/2 chain.But under decoherence or weak measurement on the VBS order parameter the spin chain loses its fractionalization, as the spinon operator decays exponentially.This can be intuitively understood as the fact that, if the VBS operator is "measured", in each measurement outcome the system is pinned to a certain particular VBS pattern, which leads to confinement in this measurement outcome.The confinement persists even if we average over all measurement outcomes.Now let us consider a (2 + 1)d QCP or CFT with a global U(1) symmetry, or Z N symmetry that can be embedded into a U(1) that emerges in the infrared.This U(1) symmetry is dual to a noncompact U(1) gauge field [62][63][64].We always turn on decoherence on the scalar boson creation operator which carries the U(1) charge, or equivalently the monopole operator of the dual U(1) gauge field.The decohered density matrix takes the same form as Eq.34: Here e i φ(x) is the monopole operator, which creates a scalar boson, or a 2π gauge flux at location x.
We evaluate the expectation value of the following quantity defined for a closed loop C = ∂A: In the undecohered density matrix, and in the dual formalism, this quantity reduces to the evaluation of the Wilson loop: and as was shown previously, it should obey a perimeter law, with a universal logarithmic contribution from the sharp corners of the loop C [57,58,65].The coefficient of the universal logarithmic contribution arising from the corner is proportional to the universal conductivity of the scalar boson current at the (2 + 1)d CFT in the AC limit ω/T → ∞.However, under decoherence, the operator ÕC will shift φ(x) by angle 2π/N for x ∈ A. Hence we expect the decoherence to change the behavior of ÕC significantly: namely ÕC should now decay with an area law.Just like the 1d example discussed in the previous subsection, an area law decay of the Wilson loop is a sign of confinement.It means that the vortex of the U(1) boson, which is also the gauge charge of the dual gauge field â, should be confined under decoherence of the scalar boson creation operator.Here we would like to remark that, one of the tools for diagnosing fractionalization and deconfinement is the dynamic structure factor, where the fractionalization would lead to a continuum [66][67][68].Computing real-time dynamics is beyond the current set-up of our current manuscript as it requires the formalism that involves the Lindbladian.Here we use the behavior of the Wilson loop as the sign of confinement/deconfinement.

V. SUMMARY AND FUTURE DIRECTIONS
In this study, we examined the effects of decoherence and weak measurement on quantum critical points in (2+1)d space-time.We found that this problem is mathematically equivalent to the boundary or defect criticality of (2 + 1)d conformal field theories, which have been extensively researched in recent years.Our results indicate that when a QCP is exposed to decoherence or weak measurement, observers may observe peculiar behaviors, including the extraordinary-log correlation recently discovered in the context of boundary criticality.Additionally, as the strength of decoherence or weak measurement increases, the system can experience an informationtheoretic transition that is captured by quantities nonlinear in the decohered density matrix.This transition is linked to spontaneous symmetry breaking when we consider quantities to the n-th power of the density matrix; in particular using the "doubled formalism", we provided a specific example where the transition belongs to the 2d Ising universality class for n = 2.By KW duality, this transition is connected to the error threshold transition in the decohered toric code, which is another type of information-theoretic transition in quantum systems.
There are many related directions that are very much worth exploring in the future.We list two such directions as follows: (1) Unconventional quantum criticality under decoherence: It is known that in the world of quantum manybody systems, there are two types of quantum critical points: conventional and unconventional.Conventional QCPs correspond to quantum phase transitions between a disordered state that can be adiabatically connected to a direct product state, and a state that spontaneously breaks certain symmetry.This type of QCPs has classical analogs such as the Wilson-Fisher fixed points discussed in this work.Well-understood examples of such conventional QCP can be found in Ref. [69], including the QCP of the transverse field quantum Ising model, which is between the Ising ordered phase that spontaneously breaks the Ising symmetry, and a symmetric disordered phase that is adiabatically connected to a direct product state.
For a two-dimensional quantum Ising model, the QCP is described by the 3D Ising Wilson-Fisher fixed point.Another well-known example of such conventional QCP is the quantum phase transition between the superfluid phase and the Mott insulator phase in the Bose Hubbard model at integer filling [70], which belongs to the 3D XY Wilson-Fisher universality class if the model is built on a two-dimensional lattice.
On the other hand, unconventional QCPs are those that do not have a simple classical analog and may involve transitions between two ordered phases with different symmetries, or between an ordered phase and a topological order.The most well-known example of unconventional QCPs is the deconfined QCP [71,72], which has many desirable phenomena such as deconfinement and a duality web as was summarized in Ref. 73.There is another large class of unconventional QCPs which involve quantum disordered states that cannot be adiabatically connected to a direct product state, such as the topological orders.One early example of such quantum phase transitions can be found in Ref. [74].It is reasonable to expect that the unconventional QCPs under decoherence can also be mapped to certain boundary criticality problems, and it is going to be an unusual boundary criticality with unconventional QCP in the bulk.As we have already seen in the current work, decoherence may be at odds with deconfinement, as deconfinement is often signified by nonlocal operators such as the disordered operators or the Wilson loops, whose behavior can be strongly affected by decoherence.It would be interesting to study the fate of unconventional QCPs under decoherence in general in the future.
(2) The Strange Correlator: The notion of a strange correlator was originally proposed as a tool to diagnose SPT states using their bulk wave functions [17], rather than edge states.The strange correlator is defined as the following quantity where |Ψ is the wave function that awaits diagnosis, and |Ω is the trivial direct product disordered state with the same symmetry G and Hilbert space as |Ψ .Ô is an order parameter that carries a nontrivial representation of G.The arguments given in Ref. 17 suggest that, although the ordinary correlation functions in both |Ψ and |Ω must be short-ranged, this strange correlator Eq. 42 must have either long-ranged or power-law correlation, for 1d and 2d states.In the past decade, the strange correlator has been used as a tool for both conceptual understanding and numerical diagnosis for SPT states and also topological states .
One of the future directions worth pursuing is the strange correlator between a quantum critical state |Ω , and an SPT wave function |Ψ .Let us still focus on twodimensional systems.Using the formalism developed in this work, this problem may be mapped to the 2d interface between a quantum critical point on the temporal domain τ < 0, and an SPT state on the other domain τ > 0 in the Euclidean space-time path-integral.Under space-time rotation, the strange correlator is mapped to the spatial interface between a quantum criticality and an SPT state.This kind of interface has two types of boundary effects: the boundary states arising from the bulk topology, and also the boundary criticality originating from the bulk critical modes.This is a subject under very active research lately, both theoretically and numerically [18-22, 24, 27].In particular, some novel interface criticality especially a (1 + 1)d deconfined quantum critical point was identified in the literature [24].One potentially highly interesting direction in the future is to analyze the strange correlator (and its generalized form defined in Ref. 12) between quantum criticality and SPT state, and explore the possible novel phenomena, especially when either the bulk quantum critical state |Ω , or the SPT state |Ψ , or both are under decoherence.
We consider the toric code ground state decohered under the following channel: we have where h(l) ∈ π 1 (T 2 ) is the element of the homotopy group of the torus.For the simplest case where C X 1,2 = 1, we remark that F (l, s • s ) is non-zero if and only ifh(s, s ) is trivial.Then, the overlap in Eq. (A4) is given as tr(E[ρ s,s ]ρ tc ) = where each block is 2 Nv−1 by 2 Nv−1 dimensional matrix labeled by the equivalence class m and its entries are all equal, i.e., where s∼sm |Ω s .There are total 2 Nv+1 equivalence classes originated from (N v − 1) independent stabilizers B p and two logical operators C X i .

Entanglement Entropy
After reorganizing terms, we get which is nothing but a disorder averaged random bond Ising model's free energy along the Nishimori line, whose transition point is located at p c = 0.1094 [51].Therefore, there is an intrinsic phase transition of the entanglement entropy of the decohered toric code state at p c = 0.1094, which coincides with the decodability transition point obtained in Ref. 31.

Purity
Interestingly, the purity of the decohered density matrix maps to the partition function of the Ising model: (A18) a 3 g y C c 6 E s q 0 x x I 1 6 1 Y 3 x 6 b g l M F B r y O j t 2 0 9 6 a 7 v s t y v l t e v + H K v y J k 9 u i S y j b C d 5 4 d t m C h s r Y 4 T a 5 3 Y r U 6 s d X I 3 4 S r 1 7 9 i J 8 M A l N n K v r H 7 j f H s Q P R 3 s / P 7 T 5 u 6 z 5 j r 7 K P g 2 + C 7 4 P o i C n 4 P d 4 C A 4 C c 4 C H O j g r + D v 4 J / + b j / r 8 3 6 5 k j 5 8 0 P h 8 H X S e / v w / / T 9 2 3 g = = < / l a t e x i t > ÿ {mi}oeP < l a t e x i t s h a 1 _ b a s e 6 4 = " e 5 0 W 9 a 5 b w 6 g C S T K r 5 6 x t R a w G W + J I W Z U S U e p O p 2 X O B e I T i m e 6 4 w 7 B k J x 4 4 5 q X c 9 6 h 4 b k g 4 9 X p n T w a e 6 B 3 K W F v 1 b v f p o S v b J y Y i + 5 c a 0 3 U l Y p 6 u S 0 6 g 6 4 b / G G a C H 1 T A z c e D h 5 f w 0 I d 7 l 3 D P g 6 8 y O C w r s a 5 b w 6 g C S T K r 5 6 x t R a w G W + J I W Z U S U e p O p 2 X O B e I T i m e 6 4 w 7 B k J x 4 4 5 q X c 9 6 h 4 b k g 4 9 X p n T w a e 6 B 3 K W F v 1 b v f p o S v b J y Y i + 5 c a 0 3 U l Y p 6 u S 0 6 g 6 4 b / G G a C H 1 T A z c e D h 5 f w 0 I d 7 l 3 D P g 6 8 y O C w r s r 7 a a e 9 N e 3 L f e r 5 X U r / u r 6 C A M I 1 N 4 j u 3 l + 2 I a J / M 5 q O e W d U 3 6 j U 9 4 5 2 Z t w l f q 3 7 E R 4 4 B I b 2 F d W t 3 G + P Q o e j 3 b + + G l z 9 4 m 5 z j 7 w v v W + 8 7 7 3 A u 9 n b 9 c 7 8 E 6 8 M w 9 7 y v v L + 9 v 7 Z 7 g 7 T I Z s W K 6 k 9 + 8 Z n 6 + 9 3 j N c / A e 5 g H Z n < / l a t e x i t > Îfl D ÍÍ < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 w I u o g x w 5 j D 9 l 8 3 C O q g h H S D j m 5 K n L U 9 K 4 K b 4 t H k w I T g U b g F W Z S 0 m d D X I c 4 7 3 R 7 n p I j p 4 n Y x u S w 6 8 e + X j Y x c 3 i x M a N o J n 9 L 0 9 g E l i j r d C x T 9 j 6 7 E U D R b 3 d Q U g q l I o 5 k a j 4 I f t 8 a j n Z 2 t p q V b 1 5 n r 7 a a e 9 N e 3 L f e r 5 X U r / u r 6 C A M I 1 N 4 j u 3 l + 2 I a J / M 5 q O e W d U 3 6 j U 9 4 5 2 Z t w l f q 3 7 E R 4 4 B I b 2 F d W t 3 G + P Q o e j 3 b + + G l z 9 4 m 5 z j 7 w v v W + 8 7 7 3 A u 9 n b 9 c 7 8 E 6 8 M w 9 7 y v v L + 9 v 7 Z 7 g 7 T I Z s W K 6 k 9 + 8 Z n 6 + 9 3 j N c / A e 5 g H Z n < / l a t e x i t > Îfl D ÍÍ < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 w n y I P c J u t y f a 2 + x I p l K L T I Y I M E y e 3 / + Q 0 u H h I S O e 0 0 q O x / / e u / / B h x 9 9 / M m D T w f D z z 7 / 4 s I u o g x w 5 j D 9 l 8 3 C O q g h H S D j m 5 K n L U 9 K 4 K b 4 t H k w I T g U b g F W Z S 0 m d D X I c 4 7 3 R 7 n p I j p 4 n Y x u S w 6 8 e + X j Y x c 3 i x M a N o J n 9 L 0 9 g E l i j r d C x T 9 j 6 7 E U D R b 3 d Q U g q l I o 5 k a j 4 I f t 8 a j n Z 2 t p q V b 1 5 n

FIG. 3 .
FIG. 3. Correlation functions in the doubled Hilbert space.The red and blue lines represent the upper and lower Hilbert spaces respectively in the doubled Hilbert space (See Sec.III D).

FIG. 4 .
FIG. 4. Phase Diagram.The global phase diagram of decohered Wilson-Fisher critical point in terms of quantities nonlinear in ρD .
t e x i t s h a 1 _ b a s e 6 4 = " f b M d J 1 P J K U t P y F n J s A a x 4 1 + Z h A w = " > A A A L 8 n i c j Z b P b 9 s 2 F M f V 7 l d n 7 0 e 6 H X c R F h T t g M y w s q U b d m o a r E 1 Q B M n i J g 1 i e Q F F U T J h U m I o y o 0 r 8 I / Y d b d h 1 w H 7 a 3 b r l 7 P e o R S p J G T W N U b W K / G S K S r z 1 9 r v m E l W c q q I J V 6 G p G O a w l p L S Z K u F T I h g o W 3 3 A s I / p T E l j E v W f c b x 6 V K v p 9 U N B O l I h l e L k F S M l / l f p 1 a f k w l w Y o t o I G w p L C K P p 4 i i b C C B I S h M v I a 5 5 y j L K 5 C E u k q 5 8 h H M 3 3 / q B D P o P z M e D l e m C a g 8 e 8 B G B H X x S Q + 6 G i L C Y Q S e j Y A o X K t 4 K 6 Y w k O 5 s 0 u O N D n 7 e u d P B 0 5 o j c o 4V 3 V m + 1 n + 7 Z s l 9 Q R 2 b 3 S 1 F 1 b I h f x q u Q I u n b 4 D 2 E m + E P M P 3 T X Z v 8 K 7 r t w 5 w r u O P A 0 g c M y l 0 t F X s A N a 0 q U W z / P V m Q F z W 6 S F a W A Q 0 U v 9 3 o 1 K o V g p L 5 s I b n w 9 + E i J O u M t n x o v V T T A o 5 8 U g V C 6 f Z Y o n r V a u f Y F J w S O O i r Q F eb d T 3 p r m 9 T 7 p f L 6 1 Z 8 u V 3 f K G G A 9 m p p 5 v l 6 E y a 6 u n B a T q x 1 Y t c 6 s d b J 3 o T L 1 L 9 h J 8 I D l 9 j A v r K 6 j Z P N Q f B 4 s P X T t + t P n p r r 7 D 3 v C + 9 L 7 5 E X e N 9 5 T 7 x d 7 9 A 7 9 r D 3 p / e X 9 4 / 3 b 1 / 1 f + 3 / 1 v 9 9 K b 1 7 x / h 8 7 n W e / h / / A d O O i 1 w = < / l a t e x i t >

< l a t e x i t s h a 1 _
b a s e 6 4 = " f A P M K Z 7 C z F m r h Q m D 3 X a 7 2 / 9 Y b f Y = " > A A A M F n i c j Z b P b 9 s 2 F M e V d D 8 6 e z / S 7 b i L s K B A B 2 S G l S 3 d s F P T Y G 2 C I k g W N 2 m Q y A 0 o i p I J k x J D U a l d g X / E b v t r d h t 2 H b D T / p T d 9 m j T q U U m W w Q Y J t / n + 0 j p 8 f G R i W C 0 U v 3 + 3 y u r 9 9 5 7 / 4 M P 7 3 / U 6 X 7 8 y a e f r T 3 4

FIG. 5 .
FIG. 5. Choi Isomorphism and Duality.Schematic diagrams illustrate how decohered mixed states map into the coupled bilayer system under the Choi-Jamiolkowski isomorphism.Through the Kramers-Wannier duality, Z2 paramagnet under symmetric decoherence (left) maps into the Z2 toric code under dephasing noise (right).Accordingly, their phase diagrams in the doubled Hilbert space is also dual to each other.At p > p