1 Introduction↩︎

In the problem of binary state discrimination, an experimenter has to read out the value of one bit encoded into the state of a quantum system, say, state \(\rho\) for bit value \(0\) and state \(\sigma\) for bit value \(1\). This is achieved by a measurement with outcomes \(0\) and \(1\), which, in the most general case, the experimenter is free to choose in order to minimize the probability of an erroneous decoding. More precisely, if the measurement operators are \(T_0=T\) and \(T_1=I-T\), the probability of misidentifying the message \(0\) (type I error) is given by \(\varepsilon_0(\rho|T):=\mathop{\mathrm{Tr}}\rho(I-T)\), and the probability of misidentifying the message \(1\) (type II error) is given by \(\varepsilon_1(\sigma|T):=\mathop{\mathrm{Tr}}\sigma T\). These cannot both be made \(0\) unless the two states are orthogonal to each other, and in general, one error probability can be decreased by changing the measurement only at the expense of increasing the other error probability, showing a trade-off between the values of the two error probabilities.

The decoding errors can be decreased by sending the same message multiple times, resulting in the code states \(\rho_n=\rho^{\otimes n}\) (for \(0\)) and \(\sigma_n=\sigma^{\otimes n}\) (for \(1\)) , provided that the encoding device has no memory and its operation does not change by time. The two types of error probabilities then can be made to go to \(0\) with an exponential speed in the number of repetitions \(n\) by the right choice of measurements, and their trade-off can be quantified on the level of the exponents as \[\begin{align} \mathrm{d}_r(\vec{\rho}\|\vec{\sigma})&:= \sup\left\{\lim_{n\to+\infty}-\frac{1}{n}\log\varepsilon_0(\rho_n|T_n)\,\Big|\, \lim_{n\to+\infty}-\frac{1}{n}\log\varepsilon_1(\sigma_n|T_n)>r \right\}\tag{1}\\ &=\sup_{\alpha\in(0,1)}\frac{\alpha-1}{\alpha}\left[r-D_{\alpha,1}(\rho\|\sigma)\right]=:H_r(\rho\|\sigma), \tag{2} \end{align}\] as was shown in [1]–[3]. Here, \(\vec{\rho}:=(\rho_n)_{n\in\mathbb{N}}\), \(\vec{\sigma}:=(\sigma_n)_{n\in\mathbb{N}}\), the supremum is taken over all test sequences \((T_n)_{n\in\mathbb{N}}\) satisfying the given constraint, and \(D_{\alpha,1}(\rho\|\sigma)\) is the Petz-type Rényi \(\alpha\)-divergence of \(\rho\) and \(\sigma\) [4]; see Sections 4.1 and 5.1 for more formal definitions.

In particular, both error probabilities can be made to disappear with an exponential speed as long as the exponent \(r\) of the type II error is such that \(H_r(\rho\|\sigma)>0\), which is known to be equivalent to \(r<D(\rho\|\sigma)\), where \(D(\rho\|\sigma)\) is the Umegaki relative entropy [5] of \(\rho\) and \(\sigma\). If \(r>D(\rho\|\sigma)\) then the type I errors inevitably go to \(1\) exponentially fast, known as the strong converse property, and in this case the trade-off is quantified as \[\begin{align} \mathrm{sc}_r(\vec{\rho}\|\vec{\sigma})&:= \sup\left\{\lim_{n\to+\infty}-\frac{1}{n}\log(1-\varepsilon_0(\rho_n|T_n))\,\Big|\, \lim_{n\to+\infty}-\frac{1}{n}\log\varepsilon_1(\sigma_n|T_n)>r \right\}\tag{3}\\ &=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[r-D_{\alpha,\alpha}(\rho\|\sigma)\right]=:H_r^*(\rho\|\sigma), \tag{4} \end{align}\] as was shown in [6]. Here, \(D_{\alpha,\alpha}(\rho\|\sigma)\) is the sandwiched Rényi \(\alpha\)-divergence [7], [8] of \(\rho\) and \(\sigma\).

Remarkably, the expressions in 2 and 4 only involve a single copy of \(\rho\) and \(\sigma\), and hence may be explicitly computable, at least numerically. The situation changes when the encoding device is not assumed to be memoryless anymore. In fact, the direct exponents \(\mathrm{d}_r(\vec{\rho}\|\vec{\sigma})\) in 1 and the strong converse exponents \(\mathrm{sc}_r(\vec{\rho}\|\vec{\sigma})\) in 3 may be defined more generally to quantify the trade-off between the two error probabilities in the asymptotic discrimination of an arbitrary pair of sequences of density operators \((\rho_n)_{n\in\mathbb{N}}\), \((\sigma_n)_{n\in\mathbb{N}}\). It was shown in [9] and [10] that the equalities in 1 –4 still hold under fairly general conditions, with the single-copy Rényi divergences in 2 and 4 replaced with their regularized versions \[\begin{align} D^{\mathrm{reg}}_{\alpha,\gamma}(\vec{\rho}\|\vec{\sigma}) &:= \lim_{n\to+\infty}\frac{1}{n}D_{\alpha,\gamma}(\rho_n\|\sigma_n), \gamma\in\{1,\alpha\}. \end{align}\] More precisely, what was shown in [9], [10] is that \[\begin{align} \mathrm{d}_r(\vec{\rho}\|\vec{\sigma}) &= \sup_{\alpha\in(0,1)}\frac{\alpha-1}{\alpha}\left[r-D^{\mathrm{reg}}_{\alpha,1}(\vec{\rho}\|\vec{\sigma})\right],\tag{5}\\ \mathrm{sc}_r(\vec{\rho}\|\vec{\sigma}) &= \sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[r-D^{\mathrm{reg}}_{\alpha,\alpha} (\vec{\rho}\|\vec{\sigma})\right],\tag{6} \end{align}\] hold whenever \(D^{\mathrm{reg}}_{\alpha,1}(\vec{\rho}\|\vec{\sigma})\) exists for every \(\alpha\in(0,1)\) and it is a differentiable function of \(\alpha\) (for the validity of 5 ), and \(D^{\mathrm{reg}}_{\alpha,\alpha}(\vec{\rho}\|\vec{\sigma})\) exists for every \(\alpha\in(1,+\infty)\) and it is a differentiable function of \(\alpha\) (for the validity of 6 ).

While these are natural and conceptually relevant generalizations of the i.i.d. results in 1 –4 , their practical relevance is limited unless one can express the regularized Rényi divergences in an explicitly computable form. Examples of classes of states where this is possible include translation-invariant and gauge-invariant quasi-free states of a fermionic lattice systems with one single fermionic mode at each physical site, as was demonstrated in [10], [11]. Such a state is specified by a translation-invariant operator (Toeplitz operator) \(Q\) on \(\ell^2(\mathbb{Z})\), such that \(Q\) and \(I-Q\) are both positive semi-definite, or equivalently, via Fourier transformation, by a measurable function \(\hat{q}\) on the one-dimensional torus \(\mathbb{T}\), taking values between \(0\) and \(1\). (For simplicity, here we only consider a one-dimensional lattice, i.e., a chain of fermions; the case of higher-dimensional lattices is very similar.) These are called the symbol operator and the symbol function, respectively, of the state \(\omega_Q\) on the CAR (Canonical Anti-commutation Relation) algebra built on the single-particle Hilbert space \(\ell^2(\mathbb{Z})\). The outcome probabilities of any measurement on a length \(n\) portion of the chain are determined by the quasifree state with density operator \(\widehat\omega_{Q_n}\) on the fermionic Fock space built on \(\ell^2([n])\), which is specified by the symbol operator \(Q_n=P_nQP_n\), where \(P_n\) is the projection from \(\ell^2(\mathbb{Z})\) onto \(\ell^2([n])\).

Given two such states \(\omega_Q\) and \(\omega_R\) encoding \(0\) and \(1\), respectively, the aim is to read out the value of the bit by making a measurement on a finite (say, length \(n\)) portion of the chain, the outcome probabilities of which are determined by the density operators \(\rho_n:=\widehat\omega_{Q_n}\) and \(\sigma_n:=\widehat\omega_{R_n}\). As it was shown in [10], [11], the regularized Rényi divergences are then given by \[\begin{align} D_{\alpha,1}^{\mathrm{reg}}(\omega_Q\|\omega_R) = D_{\alpha,\alpha}^{\mathrm{reg}}(\omega_Q\|\omega_R) = \frac{1}{2\pi}\int_0^{2\pi}\frac{1}{\alpha-1}\log\left[\hat{q}(x)^{\alpha}\hat{r}(x)^{1-\alpha}+ (1-\hat{q}(x))^{\alpha}(1-\hat{r}(x))^{1-\alpha}\right]\,\mathrm{d}x \end{align}\] under the assumption that both \(\hat{q}\) and \(\hat{r}\) are strictly bounded away from \(0\) and \(1\) as \(c\le\hat{q}(x),\hat{r}(x)\le(1-c)\), \(x\in\mathbb{T}\), for some \(c\in(0,1/2)\). (Here we identify \(\vec{\rho}=(\omega_{Q_n})_{n\in\mathbb{N}}\) with \(\omega_Q\), and \(\vec{\sigma}=(\omega_{R_n})_{n\in\mathbb{N}}\) with \(\omega_R\).) Moreover, the differentiability conditions are also satisfied, and hence 5 –6 hold. These examples are asymptotically classical in the sense that the symbols \(Q\) and \(R\) of the infinite systems commute (because they are both mapped into multiplication operators by the Fourier transform), and the regularizations of the different types of Rényi divergences coincide and are determined by the classical (commuting) objects \(\hat{q}\) and \(\hat{r}\). One might then suspect this asymptotic commutativity to be the reason why the regularized Rényi divergences can be given in closed forms, which, however, is not true, as we demonstrate here.

In this paper, we consider a generalization of the above problem, where instead of a single mode per site, we allow a fixed finite number of modes at each site, resulting in the single-particle Hilbert space \(\ell^2(\mathbb{Z})\otimes\mathbb{C}^d\) for some \(d\in\mathbb{N}\), and quasi-free states specified by block Toeplitz operator symbols \(Q,R\) on \(\ell^2(\mathbb{Z})\otimes\mathbb{C}^d\), or equivalently, matrix-valued symbol functions \(\hat{\mathbf{q}}(x),\hat{\mathbf{r}}(x)\in\mathbb{C}^{d\times d}\), \(x\in\mathbb{T}\). Moreover, we consider the regularization of a large variety of quantum Rényi divergences on top of the Petz-type and the sandwiched ones considered previously, and show that if the symbols are bounded in the positive semi-definite order as \(cI_d\le \hat{\mathbf{q}}(x),\hat{\mathbf{r}}(x)\le(1-c)I_d\), \(x\in\mathbb{T}\), for some \(c\in(0,1/2)\), then the regularized Rényi divergences exist and can be given in the closed form \[\begin{align} D_{\alpha,q}^{\mathrm{reg}}(\omega_Q\|\omega_R) = \frac{1}{2\pi}\int_0^{2\pi} D_{\alpha,q}(\widehat\omega_{\hat{\mathbf{q}}(x)}\|\widehat\omega_{\hat{\mathbf{r}}(x)})\,\mathrm{d}x. \end{align}\] Here, \(\widehat\omega_{\hat{\mathbf{q}}(x)}\) and \(\widehat\omega_{\hat{\mathbf{r}}(x)}\) are density operators of quasi-free states of a fermion system with a \(d\)-dimensional single-particle Hilbert space at each point \(x\) of the torus, and \(D_{\alpha,q}\) may be any Rényi \((\alpha,z)\)-divergence [12] (including the Petz-type and the sandwiched Rényi divergences), the log-Euclidean Rényi divergence [13], or the geometric Rényi divergence [14]–[16]. We also evaluate the regularized measured Rényi divergences as \[\begin{align} D^{\mathrm{reg}}_{\alpha,\mathrm{meas}}(\omega_Q\|\omega_R)= \begin{cases} D^{\mathrm{reg}}_{\alpha,\alpha}(\omega_Q\|\omega_R) = \frac{1}{2\pi} \int_0^{2\pi}D_{\alpha,\alpha}(\widehat\omega_{\hat{\mathbf{q}}(x)}\|\widehat\omega_{\hat{\mathbf{r}}(x)})\,\mathrm{d}x ,&\alpha\in[1/2,+\infty),\\ D^{\mathrm{reg}}_{\alpha,1-\alpha}(\omega_Q\|\omega_R) = \frac{1}{2\pi} \int_0^{2\pi}D_{\alpha,1-\alpha}(\widehat\omega_{\hat{\mathbf{q}}(x)}\|\widehat\omega_{\hat{\mathbf{r}}(x)})\,\mathrm{d}x ,&\alpha\in(0,1/2], \end{cases} \end{align}\] and the regularized version of the recently introduced integral, or hockey-stick Rényi divergences [17], [18] as \[\begin{align} D^{\mathrm{reg}}_{\alpha,\mathrm{hs}}(\omega_Q\|\omega_R)= \begin{cases} D^{\mathrm{reg}}_{\alpha,1}(\omega_Q\|\omega_R) = \frac{1}{2\pi} \int_0^{2\pi}D_{\alpha,1}(\widehat\omega_{\hat{\mathbf{q}}(x)}\|\widehat\omega_{\hat{\mathbf{r}}(x)})\,\mathrm{d}x ,&\alpha\in(0,1),\\ D^{\mathrm{reg}}_{\alpha,\alpha}(\omega_Q\|\omega_R) = \frac{1}{2\pi} \int_0^{2\pi}D_{\alpha,\alpha}(\widehat\omega_{\hat{\mathbf{q}}(x)}\|\widehat\omega_{\hat{\mathbf{r}}(x)})\,\mathrm{d}x ,&\alpha\in(1,+\infty). \end{cases} \end{align}\] In particular, we show that the regularized Petz-type and sandwiched Rényi divergences are differentiable in \(\alpha\) on \((0,1)\) and on \((1,+\infty)\), respectively, and hence the the direct and the strong converse exponents can be expressed as in 5 and 6 .

In a different direction of generalization, one may wonder whether the exponential scale is the only reasonable choice on which the asymptotics of the error probabilities can be studied, or if it is possible to obtain faster convergence to zero. Such super-exponential error decay was demonstrated in [19], again by translation-invariant and gauge-invariant fermionic quasifree states on a one-dimensional chain with a single mode at each site. More precisely, it was shown in [19] that if there exists a non-degenerate sub-interval \([\mu,\nu]\) of the torus on which the symbol function \(\hat{q}\) is constant \(0\), while \(\hat{r}\) is constant \(1\), then \[\begin{align} \label{eq:superexp} \varepsilon_0(\widehat\omega_{Q_n}|T_n)\le e^{-cn\log n}\,, \varepsilon_1(\widehat\omega_{R_n}|T_n)\le e^{-cn\log n}\,, \end{align}\tag{7}\] for some positive constant \(c\) and some test sequence \((T_n)_{n\in\mathbb{N}}\). The condition imposed on the symbol functions is very rigid, and does not really allow any modification that would be useful in exploring the phenomenon of super-exponential state discrimination any further.

In this paper we generalize the above result to the case where there are \(d\) modes at each site of the chain, and show that 7 still holds with some positive constant \(c\) and test sequence \((T_n)_{n\in\mathbb{N}}\), which we explicitly construct, provided that there exists a non-degenerate sub-interval \([\mu,\nu]\) of the torus on which the symbol functions \(\hat{q}\) and \(\hat{r}\) are Lipschitz continuous, and one of the following holds:

1. For every \(x\in[\mu, \nu]\), \(\hat{\mathbf{q}}(x)\) and \(\hat{\mathbf{r}}(x)\) are orthogonal, i.e., \(\hat{\mathbf{q}}(x)\hat{\mathbf{r}}(x)=0\), and \(\hat{\mathbf{r}}(x)\) is a non-zero projection;

2. for every \(x\in[\mu, \nu]\), \(I_d-\hat{\mathbf{q}}(x)\) and \(I_d-\hat{\mathbf{r}}(x)\) are orthogonal, i.e., \((I_d-\hat{\mathbf{q}}(x))(I_d-\hat{\mathbf{r}}(x))=0\), and \(I_d-\hat{\mathbf{q}}(x)\) is a non-zero projection.

While the bound on the speed of convergence to \(0\) that we can prove here is the same as the one in [19], the above construction obviously offers a lot more flexibility to modify the parameters and to explore potentially different error asymptotics, which, however, we leave for future work.

The structure of the paper is as follows. In Section 2 we collect the necessary preliminaries, especially on block Toeplitz operators and the mathematical description of fermionic systems. In Section 3, we extend various Szegő-type limit theorems from [11] to the case of block Toeplitz operators. This will provide the main technical ingredient to evaluate the various regularized Rényi divergences in Section 4. Finally, in Section 5.1 we prove 5 –6 for the class of states described above, and in Section 5.2 we prove the above result on super-exponential error asymptotics.

2 Preliminaries↩︎

2.1 General↩︎

By \(\log\) we will denote the natural logarithm, with its extension to \([0,+\infty]\) as \(\log 0:=-\infty\), \(\log+\infty:=+\infty\). For a natural number \(n\in\mathbb{N}=\{1,2,\ldots\}\), we will use the notations \([n]:=\{1,2,\ldots,n\}\), \([n]^*:=\{0,1,\ldots,n-1\}\).

By a Hilbert space we always mean a complex separable Hilbert space. We will denote the inner product on a Hilbert space by \(\Braket{\cdot|\cdot}\) and follow the convention that it is linear in its second and conjugate linear in its first variable. We will also use the Dirac notation: for any vectors \(x,y\) in a Hilbert space \(\mathcal{H}\), the operator \(|y\rangle\langle x|\) is defined by \(|y\rangle\langle x|z:=\Braket{x|z}y\), \(z\in\mathcal{H}\).

For a linear operator \(A\) on a Hilbert space \(\mathcal{H}\), we will use the notations \(\norm{A}:=\norm{A}_{\infty}:=\sup\{\norm{A\psi}:\,\psi\in\mathcal{H},\,\norm{\psi}\le 1\}\) for the operator norm, and \(\mathcal{B}(\mathcal{H}):=\{A:\,\mathcal{H}\to\mathcal{H}\text{ linear },\norm{A}_{\infty}<+\infty\}\) will denote the set of all bounded linear operators on \(\mathcal{H}\). We will use the notation \(\mathcal{B}(\mathcal{H})_{\mathrm{sa}}\) for the set of self-adjoint operators on \(\mathcal{H}\). For an interval \(J\subseteq\mathbb{R}\), \(\mathcal{B}(\mathcal{H})_{J}:=\{A\in\mathcal{B}(\mathcal{H})_{\mathrm{sa}}:\,\mathop{\mathrm{spec}}(A)\subseteq J\}\), i.e., it is the set of self-adjoint operators on \(\mathcal{H}\) with their spectra in \(J\). We will use the shorthand notations \(\mathcal{B}(\mathcal{H})_{\ge 0}:=\mathcal{B}(\mathcal{H})_{[0,+\infty)}\) for the set of positive semi-definite (PSD) operators on \(\mathcal{H}\), and \(\mathcal{B}(\mathcal{H})_{>0}:=\mathcal{B}(\mathcal{H})_{(0,+\infty)}\) for the set of positive definite operators, and we will denote by \(\mathcal{B}(\mathcal{H})_{\gneq 0}\) the set of non-zero PSD operators on \(\mathcal{H}\). An inequality \(A\le B\) between operators \(A,B\in\mathcal{B}(\mathcal{H})\) is always interpreted in the Löwner (or PSD) order, meaning \(B-A\in\mathcal{B}(\mathcal{H})_{\ge 0}\). Elements of the set \[\begin{align} \mathcal{B}(\mathcal{H})_{[0,1]}:=\set{T\in\mathcal{B}(\mathcal{H})_{\mathrm{sa}}|0\le T\le I} \end{align}\] are called tests on \(\mathcal{H}\).

The set of (orthogonal) projections on \(\mathcal{H}\) will be denoted by \[\begin{align} \mathbb{P}(\mathcal{H}):=\mathcal{B}(\mathcal{H})_{\{0,1\}}=\set{P\in\mathcal{B}(\mathcal{H})_{\mathrm{sa}}|P^2=P}. \end{align}\] For a positive semi-definite operator \(A\in\mathcal{B}(\mathcal{H})_{\ge 0}\), we will use the notation \[\begin{align} \label{eq:supppr} A^0:=\lim_{t\searrow 0}A^t \end{align}\tag{8}\] for the projection onto \(\operatorname{supp}A:=(\ker A)^{\perp}\).

The set of states (density operators) on a finite-dimensional Hilbert space \(\mathcal{H}\) is \(\mathcal{S}(\mathcal{H}):=\set{\rho\in\mathcal{B}(\mathcal{H})_{\ge 0}|\mathop{\mathrm{Tr}}\rho=1}\). For any finite set \(\mathcal{X}\), the set of positive operator-valued measures (POVMs) on \(\mathcal{H}\) with outcomes in \(\mathcal{X}\) is defined as \[\begin{align} \mathrm{POVM}(\mathcal{H},\mathcal{X}):=\Set{(M_x)_{x\in\mathcal{X}}\in\mathcal{B}(\mathcal{H})_{\ge 0}^{\mathcal{X}}|\sum\nolimits_{x\in\mathcal{X}}M_x=I}. \end{align}\] The map \(T\mapsto (T,I-T)\) gives an identification between \(\mathcal{B}(\mathcal{H})_{[0,1]}\) and \(\mathrm{POVM}(\mathcal{H},\{0,1\})\). For any POVM \(M\in\mathrm{POVM}(\mathcal{H},\mathcal{X})\), the corresponding measurement channel \(\mathcal{M}\) is defined as \[\begin{align} \mathcal{M}(A):=\sum_{x\in\mathcal{X}}(\mathop{\mathrm{Tr}}M_xA)|x\rangle\langle x|\in\mathcal{B}(\ell^2(\mathcal{X})),A\in\mathcal{B}(\mathcal{H}). \end{align}\]

For a differentiable function \(f\) defined on an interval \(J\subseteq\mathbb{R}\), let \(f^{[1]}:\,J\times J\to\mathbb{R}\) be its first divided difference function, defined as \[\begin{align} f^{[1]}(a,b):=\begin{cases} \frac{f(a)-f(b)}{a-b},&a\ne b,\\ f'(a),&a=b, \end{cases}a,b\in J. \end{align}\] If \(f\) is a continuously differentiable function on an open interval \(J\subseteq\mathbb{R}\) then for any finite-dimensional Hilbert space \(\mathcal{H}\), \(A\mapsto f(A)\) is Fréchet differentiable on \(\mathcal{B}(\mathcal{H})_{J}\), and its Fréchet derivative \((Df)[A]\) at a point \(A\in \mathcal{B}(\mathcal{H})_{J}\) is given by \[\begin{align} \label{eq:opfunction32derivative} (Df)[A](Y)=\sum_{a,b\in\mathop{\mathrm{spec}}(A)}f^{[1]}(a,b)P_a^{A}YP_b^A,Y\in\mathcal{B}(\mathcal{H})_{\mathrm{sa}}, \end{align}\tag{9}\] where \(P^A_a\) denotes the spectral projection of \(A\) corresponding to an eigenvalue \(a\in\mathop{\mathrm{spec}}(A)\). See, e.g., [20] or [21]. It is easy to see from this that if \(f\) is as above, and \((a,b)\ni t\mapsto A(t)\in\mathcal{B}(\mathcal{H})_{J}\) is continuously differentiable, then so is \(\mathop{\mathrm{Tr}}f(A(t))\) as well, and \[\begin{align} \label{eq:Tr32derivative} \frac{d}{dt}\mathop{\mathrm{Tr}}f(A(t))=\mathop{\mathrm{Tr}}\left[f'(A(t))\frac{d}{dt}A(t)\right],t\in(a,b). \end{align}\tag{10}\]

2.2 Block Toeplitz operators↩︎

For a measure space \((\mathcal{X},\mathcal{F},\mu)\) and a finite-dimensional Hilbert space \(\mathcal{H}\), let \[\begin{align} L^2(\mathcal{X},\mathcal{H}):=\left\{f\in\mathcal{H}^{\mathcal{X}}\text{ measurable, }\norm{f}_2^2:=\int_{\mathcal{X}}\norm{f(t)}^2\,\mathrm{d}\mu(t)<+\infty \right\}, \end{align}\] and \[\begin{align} L^{\infty}(\mathcal{X},\mathcal{B}(\mathcal{H}))&:=\big\{A\in\mathcal{B}(\mathcal{H})^{\mathcal{X}}\text{ measurable, }\nonumber\\ &\norm{A}_{\infty}:= \inf\{C>0:\,\mu(\{t\in\mathcal{X}:\,\|A(t)\|_{\infty}>C\})=0\}<+\infty\big\}\label{eq:bounded32opfunctions}\\ &\subseteq\mathcal{B}\left(L^2(\mathcal{X},\mathcal{H})\right),\nonumber \end{align}\tag{11}\] where \(A\in L^{\infty}(\mathcal{X},\mathcal{B}(\mathcal{H}))\) acts on \(f\in L^{2}(\mathcal{X},\mathcal{H})\) as \((Af)(x):=A(x)f(x)\), \(x\in\mathcal{X}\). It is easy to see that the operator norm of such an operator coincides with its norm defined in 11 , justifying the same notation for the two. In particular, when \(\mathcal{H}=\mathbb{C}^d:=\mathbb{C}^{[d]^*}\) for some \(d\in\mathbb{N}\), and \(\mathcal{X}=\mathbb{Z}\) or \(\mathcal{X}=[n]^*\) for some \(n\in\mathbb{N}\), \(\mathcal{F}\) is its full power set, and \(\mu\) is the counting measure, we will use the notations \[\begin{align} \ell^2_d(\mathcal{X})&:=L^2(\mathcal{X},\mathbb{C}^d)=\set{f:\mathcal{X}\to\mathbb{C}^d|\|f\|_2<\infty},\qquad \|f\|_2^2=\sum_{x\in\mathcal{X}}\|f_x\|^2_{\mathbb{C}^d},\\ \ell^{\infty}_{d\times d}(\mathcal{X})&:=L^{\infty}(\mathcal{X},\mathcal{B}(\mathbb{C}^d)), \end{align}\] where \(\|\cdot\|_{\mathbb{C}^d}\) denotes the usual norm of \(\mathbb{C}^d\). We will often use the following natural identifications in the above case: \[\ell^2_d(\mathcal{X})\equiv\bigoplus_{k=0}^{d-1}\ell^2(\mathcal{X})\equiv\ell^2(\mathcal{X})\otimes\mathbb{C}^d,\] which in turn gives that any bounded operator \(A\in\mathcal{B}(\ell^2_d(\mathcal{X}))\) can be decomposed as \[\begin{align} A=\left[ A_{k,l} \right]_{k,l=0}^{d-1}\equiv\sum_{k,l=0}^{d-1} A_{k,l}\otimes |k\rangle\langle l|, \end{align}\] where \(A_{k,l}\in\mathcal{B}(\ell^2(\mathcal{X}))\), \(k,l\in[d]^*\), and we use the standard shorthand notation \[\begin{align} \ket{k}:=\ket{1_{\{k\}}},k\in[d]^*. \end{align}\]

The translation operator \(T\) on \(\ell^2(\mathbb{Z})\) is given by \(T1_{\{k\}}=1_{\{k+1\}}\), \(k\in\mathbb{Z}\), and its extension to \(\ell^2_d(\mathbb{Z})\) is \[\begin{align} \mathcal{T}=\bigoplus_{k=0}^{d-1}T\equiv T\otimes I. \end{align}\] An operator \(A\in\mathcal{B}(\ell_d^2(\mathbb{Z}))\) is said to be translation-invariant, or a block Toeplitz operator, if \[\begin{align} \label{eqn:shift-inv} \mathcal{T}A\mathcal{T}^{-1}=A,\text{or equivalently,} TA_{k,l}T^{-1}=A_{k,l},k,l\in[d]^*, \end{align}\tag{12}\] i.e., if every one of its blocks \(A_{k,l}\), \(k,l\in[d]^*\), is translation-invariant (also called a Toeplitz operator).

Let \(\mathbb{T}=[0,2\pi)\) denote the one-dimensional torus equipped with its canonical rotation (equivalently, modulo \(2\pi\) translation) and the Lebesgue measure. The Fourier transform is given by \[\begin{align} F:\,\ell^2(\mathbb{Z})\to L^2(\mathbb{T}), F1_{\{k\}}:=\chi_k:=\frac{1}{\sqrt{2\pi}}e^{ik(\cdot)},k\in\mathbb{Z}. \end{align}\] Its canonical extension from \(\ell^2_d(\mathbb{Z})\) to \[\begin{align} L^2_d(\mathbb{T}):=L^2(\mathbb{T},\mathbb{C}^d)\equiv\bigoplus_{k=0}^{d-1}L^2(\mathbb{T})\equiv L^2(\mathbb{T})\otimes\mathbb{C}^d \end{align}\] is given by \[\begin{align} \mathop{\mathrm{\mathcal{F}}}=\bigoplus_{k=0}^{d-1}F\equiv F\otimes I. \end{align}\]

The matrix elements of a translation-invariant operator \(A\in\mathcal{B}(\ell^2(\mathbb{Z}))\) in the canonical orthonormal basis \(\{1_{\{k\}}\}_{k\in\mathbb{Z}}\) are given by \[\begin{align} \Braket{1_{\{k\}}|A1_{\{l\}}}=\Braket{1_{\{k\}}|T^lAT^{-l}1_{\{l\}}} =\Braket{1_{\{k-l\}}|A1_{\{0\}}}=a(k-l), \end{align}\] where \(a(k):=\Braket{1_{\{k\}}|A1_{\{0\}}}\), \(k\in\mathbb{Z}\). Since \(a\in\ell^2(\mathbb{Z})\), the function \[\begin{align} \hat{a}:=\sqrt{2\pi}\sum_{k\in\mathbb{Z}}a(k)\chi_k=\sum_{k\in\mathbb{Z}}a(k)e^{ik(\cdot)} \end{align}\] is well defined as an element of \(L^2(\mathbb{T})\). Let \(M_{\hat{a}}:\,f\mapsto \hat{a} f\) denote the corresponding multiplication operator on \(L^2(\mathbb{T})\). Then \[\begin{align} \Braket{1_{\{k\}}|(F^{-1} M_{\hat{a}}F)1_{\{l\}}}&= \Braket{\chi_k|\hat{a}\chi_l}=\int_0^{2\pi}\frac{1}{2\pi}\hat{a}(x)e^{i(l-k)x}\,dx\\ &=\frac{1}{\sqrt{2\pi}}\Braket{\chi_{k-l}|\hat{a}}=a(k-l)=\Braket{1_{\{k\}}|A1_{\{l\}}}. \end{align}\] Since this holds for every \(k,l\in\mathbb{Z}\), we get that \[\begin{align} \label{eq:tiop32Fourier} A=F^{-1} M_{\hat{a}}F. \end{align}\tag{13}\] Thus, every translation-invariant operator on \(\ell^2(\mathbb{Z})\) is mapped into a multiplication operator by the Fourier transform. This implies that any translation-invariant operator is normal, and any two translation-invariant operators commute with each other. In particular, if \(A^{(j)}\in\mathcal{B}(\ell^2_d(\mathbb{Z}))\), \(j=1,2\), are translation-invariant, then any of their blocks commute, i.e., \(A^{(j)}_{k,l}A^{(j')}_{k',l'}=A^{(j')}_{k',l'}A^{(j)}_{k,l}\), \(j,j'\in\{1,2\}\), \(k,k',l, l'\in[d]^*\).

Now, if \(A \in \mathcal{B}(\ell^2_d(\mathbb{Z}))\) is translation-invariant then by (12 ) and 13 , \[\begin{align} A &= \sum_{k,l=0}^{d-1} A_{k,l}\otimes\ket{k}\bra{l} = \sum_{k,l=0}^{d-1} (F^{-1}M_{\hat{a}_{kl}}F)\otimes\ket{k}\bra{l} = \mathop{\mathrm{\mathcal{F}}}^{-1} \underbrace{\left(\sum_{k,l=0}^{d-1} M_{\hat{a}_{kl}}\otimes\ket{k}\bra{l}\right)}_{ =:M_{\hat{\mathbf{a}}}}\mathop{\mathrm{\mathcal{F}}} =\mathop{\mathrm{\mathcal{F}}}^{-1}M_{\hat{\mathbf{a}}}\mathop{\mathrm{\mathcal{F}}}^{-1}, \end{align}\] where \(\hat{a}_{kl}\in L^\infty(\mathbb{T})\) for all \(k,l\in[d]^*\), and we introduce the notation \[\hat{\mathbf{a}} := \sum_{k,l=0}^{d-1}\hat{a}_{kl} \otimes \ket{k}\bra{l} \in L^{\infty}(\mathbb{T},\mathcal{B}(\mathbb{C}^d))=:L^{\infty}_{d\times d}(\mathbb{T}).\]

2.3 Fermionic systems↩︎

For vectors \(\varphi_1,\ldots,\varphi_k\) in a complex Hilbert space \(\mathcal{H}\), let \[\begin{align} \label{eq:as32product} \varphi_1\wedge\ldots\wedge\varphi_k:=\frac{1}{\sqrt{k!}}\sum_{\sigma\in \mathfrak{S}_k}\varepsilon(\sigma)\varphi_{\sigma(1)}\mathop{\mathrm{\otimes\ldots\otimes}}\varphi_{\sigma(k)} \end{align}\tag{14}\] denote their anti-symmetrized tensor product, where \(\mathfrak{S}_k\) stands for the set of permutations of \(k\) elements and \(\varepsilon(\sigma)\) for the sign of the permutation \(\sigma\). For any \(k\in\mathbb{N}\), the \(k\)-th anti-symmetric tensor power \(\wedge^k\mathcal{H}=\mathcal{H}^{\wedge k}\) of \(\mathcal{H}\) is the closure of the subspace of \(\mathcal{H}^{\otimes k}\) spanned by all vectors of the form 14 , and we define \(\mathcal{H}^{\wedge 0}:=\mathbb{C}\).

The Hilbert space of a fermionic system with single-particle Hilbert space \(\mathcal{H}\) is the anti-symmetric Fock space (or fermionic Fock space) \[\begin{align} \Gamma\!\left(\mathcal{H}\right):=\mathop{\mathrm{\scalerel*{\oplus}{\textstyle\sum}}}_{k=0}^{\operatorname{dim}\mathcal{H}}\mathcal{H}^{\wedge k}. \end{align}\] For any operator \(A\in\mathcal{B}(\mathcal{H})\) and any \(k\in\mathbb{N}\), \(A^{\otimes k}\) leaves the subspace \(\mathcal{H}^{\wedge k}\) invariant, and we define \[\begin{align} A^{\wedge k}:=A^{\otimes k}\vert_{\mathcal{H}^{\wedge^k}} \end{align}\] as an operator on \(\mathcal{H}^{\wedge k}\). If \(A\in\mathcal{B}(\mathcal{H})\) is a contraction or it is compact then \[\begin{align} \Gamma\!\left(A\right):=\mathop{\mathrm{\scalerel*{\oplus}{\textstyle\sum}}}_{k=0}^{\operatorname{dim}\mathcal{H}} A^{\wedge k} \end{align}\] defines a bounded operator on \(\Gamma\!\left(\mathcal{H}\right)\), where \(A^{\wedge 0}:=1\in\mathcal{B}(\mathbb{C})\). It is easy to see that if \(\mathcal{H}\) is finite dimensional then \[\begin{align} \label{eq:Fockop32trace} \mathop{\mathrm{Tr}}\Gamma\!\left(A\right) = \det (I+A). \end{align}\tag{15}\] We will also often use the easily verifiable fact that for any positive definite \(A,B\in\mathcal{B}(\mathcal{H})_{>0}\) and \(x,y\in\mathbb{R}\), \[\begin{align} \Gamma\!\left(A\right)^x\Gamma\!\left(B\right)^y=\Gamma\!\left(A^xB^y\right). \end{align}\]

For each \(\varphi\in\mathcal{H}\), the corresponding creation operator \(c(\varphi)\) is the unique bounded linear extension of the map \[\begin{align} \varphi_1\wedge\ldots\wedge\varphi_k\mapsto\varphi\wedge\varphi_1\wedge\ldots\wedge\varphi_k, \varphi_1,\ldots,\varphi_k\in\mathcal{H}, \end{align}\] and the corresponding annihilation operator is its adjoint, \(a(\varphi):=c(\varphi)^*\). These operators satisfy the canonical anti-commutation relations (CARs), \[\begin{align} \label{eqn:cars} \left\{a(\varphi),a(\psi)\right\}= 0,\left\{a(\varphi),a^*(\psi)\right\}= \Braket{\varphi|\psi}I,\varphi,\psi\in\mathcal{H}. \end{align}\tag{16}\] The \(C^*\)-subalgebra of \(\mathcal{B}(\Gamma\!\left(\mathcal{H}\right))\) generated by \(\{a(\varphi):\,\varphi\in\mathcal{H}\}\) is called the algebra of the canonical anti-commutation relations (or CAR-algebra) corresponding to the single-particle Hilbert space \(\mathcal{H}\), and is denoted by \(\mathrm{CAR}\!\left(\mathcal{H}\right)\).

A state on \(\mathrm{CAR}\!\left(\mathcal{H}\right)\) is a positive linear functional that takes the value \(1\) on \(I\). For any positive semi-definite operator \(Q\in\mathcal{B}(\mathcal{H})\) with \(Q\le I\) there exists a unique state \(\omega_Q\) on \(\mathrm{CAR}\!\left(\mathcal{H}\right)\) (called the gauge-invariant quasi-free state with symbol \(Q\)) with the property \[\begin{align} \label{eqn:quasifree95functional} \omega_Q\left(a(\varphi_1)^*\ldots a(\varphi_n)^*a(\psi_m)\ldots a(\psi_1)\right) = \delta_{mn}\det\left\{\Braket{\psi_i|Q\varphi_j}\right\}_{i,j=1}^n. \end{align}\tag{17}\] It is easy to verify that when \(d:=\operatorname{dim}\mathcal{H}\) is finite, the density operator \(\widehat\omega_Q\) of \(\omega_Q\) can be explicitly given as \[\begin{align} \label{quasifree32density} \widehat\omega_Q=\prod_{j=1}^d\left(q_ja(e_j)^*a(e_j)+(1-q_j)a(e_j)a(e_j)^*\right) \end{align}\tag{18}\] where \(Q=\sum_{j=1}^d q_j|e_j\rangle\langle e_j|\) is any eigen-decomposition of \(Q\). Note that for all \(1\le i_1<\ldots<i_k\le d\), \(e_{i_1}\wedge\ldots\wedge e_{i_k}\) is an eigenvector of \(\widehat\omega_Q\) with eigenvalue \(\left(\prod_{j\in\{i_1,\ldots,i_k\}}q_j\right)\cdot\left(\prod_{j\in[d]\setminus\{i_1,\ldots,i_k\}}(1-q_j)\right)\). This implies immediately that if \(1\) is not an eigenvalue of \(Q\) then \(\widehat\omega_Q\) can be written as \[\begin{align} \label{eq:quasifree32density} \widehat\omega_Q=\det(I-Q)\Gamma\!\left(W_{Q}\right), W_{Q}:=\frac{Q}{I-Q}. \end{align}\tag{19}\]

Since in this paper we only consider quasi-free states that are gauge-invariant, in the following we will drop “gauge-invariant” from the terminology, i.e., by a quasi-free state we will always mean a gauge-invariant quasi-free state.

Quasi-free states emerge as equilibrium states of non-interacting fermionic systems. For instance, if the single-particle Hamiltonian \(H\) of a system of non-interacting fermions is such that \(e^{-\beta H}\) is trace-class then the Gibbs state of the system at inverse temperature \(\beta\) is the quasi-free state with symbol \(Q=\frac{e^{-\beta H}}{I+e^{-\beta H}}\) (see, e.g., [22]).

Consider now a fermionic chain with \(d\) modes at each site, the single-particle Hilbert space of which is \(\mathcal{H}=\ell^2_d(\mathbb{Z})\). The translation operator \(\mathcal{T}\) on \(\ell^2_d(\mathbb{Z})\) defines the translation automorphism \(\tau\) on \(\mathrm{CAR}\!\left(\ell^2_d(\mathbb{Z})\right)\) via \(\tau(a(\varphi)):=a(\mathcal{T}\varphi)\), \(\varphi\in\ell^2_d(\mathbb{Z})\). A quasi-free state \(\omega_Q\) on \(\mathrm{CAR}\!\left(\ell^2_d(\mathbb{Z})\right)\) is called translation-invariant if \(\omega_Q\circ\tau=\omega_Q\), which is easily seen to be equivalent to \(\mathcal{T}Q\mathcal{T}^{-1}=Q\), i.e., the translation-invariance of the symbol \(Q\in\mathcal{B}(\ell^2_d(\mathbb{Z}))\). For instance, in the above example a translation-invariant single-particle Hamiltonian \(H\) yields a translation-invariant quasi-free state as the equilibrium state of the system.

A measurement on a subsystem corresponding to modes at the sites \([n]^*:=\{0,\ldots,n-1\}\) has measurement operators in the \(C^*\)-subalgebra \(\mathcal{A}_n\subseteq\mathrm{CAR}\!\left(\ell^2_d(\mathbb{Z})\right)\) generated by \(\{a(\varphi):\,\varphi\in\mathcal{H}_n\}\), \[\begin{align} \mathcal{H}_n:=\ell^2_d([n]^*)\equiv\mathop{\mathrm{span}}\{1_{\{k\}}\otimes1_{\{j\}}:\,k\in[n]^*,\,j\in[d]^*\}\subseteq \ell^2(\mathbb{Z})\otimes\mathbb{C}^d\equiv \ell^2_d(\mathbb{Z}). \end{align}\] This subalgebra is naturally isomorphic to \(\mathrm{CAR}\!\left(\ell^2_d([n]^*)\right)\). It is easy to see that if the state of the infinite chain is given by a quasi-free state with symbol \(Q\) then the statistics of any such local measurement is given by the quasi-free state \(\omega_{Q_n}\) with symbol \[\begin{align} \label{eq:cutoff32def} Q_n:=(P_n\otimes I_d)^*Q(P_n\otimes I_d), \text{where} P_n:=\sum_{k,l=0}^{n-1}|1_{\{k\}}\rangle\langle 1_{\{k\}}|. \end{align}\tag{20}\]

3 Szegő-type theorems for block Toeplitz operators↩︎

In this section, we will consider generalizations of the Szegő-type result given in Lemma 1. For a proof of the latter, see [11].

4 Rényi divergences of quasi-free states↩︎

4.1 Finite dimension↩︎

For two positive definite operators \(A,B\in\mathcal{B}(\mathcal{H})_{>0}\) and \(\alpha\in(0,+\infty)\), let \[\begin{align} \mathop{\mathrm{\mathcal{Q}}}_{\alpha,z}^{\mathrm{op}}&:=\left(A^{\frac{\alpha}{2z}}B^{\frac{1-\alpha}{z}}A^{\frac{\alpha}{2z}}\right)^z, z\in(0,+\infty),\\ \mathop{\mathrm{\mathcal{Q}}}_{\alpha,+\infty}^{\mathrm{op}}(A\|B)&:=\lim_{z\to+\infty}\mathop{\mathrm{\mathcal{Q}}}^{\mathrm{op}}_{\alpha,z}(A\|B) =e^{\alpha\log A+(1-\alpha)\log B},\\ \mathop{\mathrm{\mathcal{Q}}}^{\mathrm{op}}_{\alpha,\max}(A\|B)&:=B^{1/2}\left(B^{-1/2}AB^{-1/2}\right)^{\alpha}B^{1/2} =A^{1/2}\left(A^{-1/2}BA^{-1/2}\right)^{1-\alpha}A^{1/2},\label{eq:maxRenyi32def} \end{align}\tag{35}\] and for every \(\gamma\in(0,+\infty]\cup\{\max\}\), let \[\begin{align} \mathop{\mathrm{\mathcal{Q}}}_{\alpha,\gamma}(A\|B)&:=\mathop{\mathrm{Tr}}\mathop{\mathrm{\mathcal{Q}}}^{\mathrm{op}}_{\alpha,\gamma}(A\|B),\\ \psi_{\alpha,\gamma}(A\|B)&:=\log \mathop{\mathrm{\mathcal{Q}}}_{\alpha,\gamma}(A\|B),\\ D_{\alpha,\gamma}(A\|B)&:=\frac{1}{\alpha-1}\psi_{\alpha,\gamma}(A\|B), \end{align}\] where the last quantity is only defined for \(\alpha\ne 1\). Here, \(D_{\alpha,z}\) is called the Rényi \((\alpha,z)\)-divergence [12], [23], \(D_{\alpha,+\infty}\) is the log-Euclidean Rényi divergence [13], and \(D_{\alpha,\max}\) is the maximal Rényi divergence [14], [15]. The equality in 35 is well known and easy to verify; see, e.g., [14]. The special cases \(D_{\alpha,1}\) and \(D_{\alpha,\alpha}\) are called the Petz-type [4] and the sandwiched [7], [8] Rényi divergences, respectively.

It is known [24], [25] that if \(A\) is a density operator then for any function \((1-\delta,1+\delta)\ni\alpha\mapsto z(\alpha)\in(0,+\infty]\) with \(\liminf_{\alpha\to 1}z(\alpha)>0\), we have \[\begin{align} \label{eq:Umegaki32def} \lim_{\alpha\to 1}D_{\alpha,z(\alpha)}(A\|B) = D(A\|B):=\mathop{\mathrm{Tr}}A(\log A-\log B), \end{align}\tag{36}\] where the latter quantity is the Umegaki relative entropy of \(A\) and \(B\) [5]. We will use the notation \(D_{1,z}(A\|B):=D(A||B)\), \(z\in(0,+\infty]\). On the other hand, for \(\gamma=\max\) we have \[\begin{align} \lim_{\alpha\to 1}D_{\alpha,\max}(A\|B) = D_{1,\max}(A\|B) &:= \mathop{\mathrm{Tr}}A\log (A^{1/2}B^{-1}A^{1/2})\tag{37}\\ &= \mathop{\mathrm{Tr}}B^{1/2}A B^{-1/2}\log (B^{-1/2}A B^{-1/2}), \tag{38} \end{align}\] as one can easily verify. Here, \(D_{1,\max}(A\|B)\) is the Belavkin-Staszewski relative entropy [26].

For any \(\gamma\in(0,+\infty]\cup\{\max\}\), the above definitions can be extended to pairs of non-zero PSD operators \(\rho,\sigma\in\mathcal{B}(\mathcal{H})_{\gneq 0}\) by \(D_{\alpha,\gamma}(\rho\|\sigma):=\lim_{\varepsilon\searrow 0}D_{\alpha,\gamma}(\rho+\varepsilon I\|\sigma+\varepsilon I)\). This definition is consistent in the sense that the above definition becomes an identity for pairs of positive definite operators. It is straightforward to verify that for any \(\gamma\in(0,+\infty]\cup\{\max\}\), \(D_{\alpha,\gamma}\) is a quantum Rényi \(\alpha\)-divergence in the following sense: for any two commuting operators \(\rho,\sigma\), both diagonal in some orthonormal basis \((\ket{\omega})_{\omega\in\Omega}\) as \(\rho=\sum_{\omega\in\Omega}\tilde{\rho}(\omega)|\omega\rangle\langle \omega|\), \(\sigma=\sum_{\omega\in\Omega}\tilde{\sigma}(\omega)|\omega\rangle\langle \omega|\), \[\begin{align} Q_{\alpha,\gamma}(\rho\|\sigma)&=Q_{\alpha}\left(\tilde{\rho}\|\tilde{\sigma}\right):= \lim_{\varepsilon\searrow 0}\sum_{\omega\in\Omega}(\tilde{\rho}(\omega)+\varepsilon)^{\alpha}(\tilde{\sigma}(\omega)+\varepsilon)^{1-\alpha},\gamma\in(0,+\infty]\cup\{\max\}, \end{align}\] where \(D_{\alpha}(\tilde{\rho}\|\tilde{\sigma}):=(\alpha-1)^{-1}\log Q_{\alpha}(\tilde{\rho}\|\tilde{\sigma})\) is the classical Rényi \(\alpha\)-divergence [27] of the non-negative functions \(\tilde{\rho}\) and \(\tilde{\omega}\).

In Lemma 4 below, we give explicit formulas for the above Rényi divergences of two quasi-free states in terms of their symbols. For \(D_{\alpha,+\infty}\), we will use the following simple identities.

For the Petz-type Rényi quantities one may obtain a different expression as follows.

4.2 Regularized Rényi divergences of translation-invariant states↩︎

Let us consider a doubly infinite fermion chain with \(d\) internal degrees of freedom, described by the single-particle Hilbert space \(\ell^2_d(\mathbb{Z})\). For the rest of the section, let \(\omega_Q\) and \(\omega_R\) be translation-invariant quasi-free states of this system with corresponding symbol functions \(\hat{\mathbf{q}},\hat{\mathbf{r}}\in L^{\infty}_{d\times d}(\mathbb{T})\), i.e., \(Q = \mathop{\mathrm{\mathcal{F}}}^{-1} M_{\hat{\mathbf{q}}} \mathop{\mathrm{\mathcal{F}}}\) and \(R = \mathop{\mathrm{\mathcal{F}}}^{-1} M_{\hat{\mathbf{r}}} \mathop{\mathrm{\mathcal{F}}}\). We will assume throughout this section that there exists a constant \(c \in (0, 1/2)\) such that \[\begin{align} \label{eq:strict32bound1} c I_d \le \hat{\mathbf{q}}(x), \hat{\mathbf{r}}(x) \le (1-c)I_d \end{align}\tag{54}\] almost everywhere, or equivalently, that \[\begin{align} \label{eq:strict32bound2} cI\le Q,R\le (1-c)I, \end{align}\tag{55}\] and we define \[\begin{align} \mathbf{w}_{Q}(x) := \hat{\mathbf{q}}(x)(I_d - \hat{\mathbf{q}}(x))^{-1},\mathbf{w}_{R}(x) := \hat{\mathbf{r}}(x)(I_d - \hat{\mathbf{r}}(x))^{-1}, x\in\mathbb{T}. \end{align}\] We may and will assume without loss of generality that 54 holds at every \(x\in\mathbb{T}\). Assumptions 54 –55 guarantee that \[\begin{align} \label{eq:strict32bound3} cI_{nd}\le Q_n,R_n\le(1-c)I_{nd},n\in\mathbb{N}, \frac{c}{1-c}I_d\le \mathbf{w}_{Q}(x) ,\mathbf{w}_{R}(x) \le\frac{1-c}{c} I_d,x\in\mathbb{T}, \end{align}\tag{56}\] where \(Q_n\) is as in 20 , and \(R_n\) is defined analogously.

We are interested in the regularized versions of the Rényi quantities, defined as \[\begin{align} \Delta^{\mathrm{reg}}(\omega_Q\|\omega_R):=\lim_{n\to+\infty}\frac{1}{n}\Delta(\widehat\omega_{Q_n}\|\widehat\omega_{R_n}) \end{align}\] whenever the limit exists, where \(\Delta=\psi_{\alpha,\gamma}\) or \(\Delta=D_{\alpha,\gamma}\) for some \(\alpha\in(0,+\infty)\) and \(\gamma\in(0,+\infty]\cup\{\max\}\).

Theorem 2 can be equivalently stated as follows:

4.3 Regularized measured Rényi divergences↩︎

For any notion of quantum Rényi \(\alpha\)-divergence \(D_{\alpha,q}\), one may define its regularized version in a more general setting along arbitrary sequences of pairs of PSD operators. That is, assume that for every \(n\in\mathbb{N}\), \(\rho_n,\sigma_n\) are non-zero PSD operators on some finite-dimensional Hilbert space \(\mathcal{H}_n\). We will use the notations \(\vec{\rho}:=(\rho_n)_{n\in\mathbb{N}}\), \(\vec{\sigma}:=(\sigma_n)_{n\in\mathbb{N}}\). Three notions of regularized \(q\)-Rényi \(\alpha\)-divergence are then defined as \[\begin{align} \underline{D}^{\mathrm{reg}}_{\alpha,q}(\vec{\rho}\|\vec{\sigma}) &:= \liminf_{n\to+\infty}\frac{1}{n}D_{\alpha,q}(\rho_n\|\sigma_n),\\ \overline{D}^{\mathrm{reg}}_{\alpha,q}(\vec{\rho}\|\vec{\sigma}) &:= \limsup_{n\to+\infty}\frac{1}{n}D_{\alpha,q}(\rho_n\|\sigma_n),\\ D^{\mathrm{reg}}_{\alpha,q}(\vec{\rho}\|\vec{\sigma}) &:= \lim_{n\to+\infty}\frac{1}{n}D_{\alpha,q}(\rho_n\|\sigma_n), \end{align}\] where the last quantity is only defined when the limit exists, or equivalently, \(\underline{D}^{\mathrm{reg}}_{\alpha,q}(\vec{\rho}\|\vec{\sigma}) = \overline{D}^{\mathrm{reg}}_{\alpha,q}(\vec{\rho}\|\vec{\sigma})\).

We will consider such regularized quantities for two further notions of quantum Rényi divergences beyond the ones considered in Section [sec:regularized32Renyi]: the measured Rényi divergences in this section, and the integral Rényi divergences in Section 4.4.

For any pair of non-zero PSD operators \(\rho,\sigma\in\mathcal{B}(\mathcal{H})_{\gneq 0}\), their measured Rényi \(\alpha\)-divergence for some \(\alpha\in[0,+\infty)\) is defined as \[\begin{align} D_{\alpha,\mathrm{meas}}(\rho\|\sigma):=\max\set{D_{\alpha}\left(\mathcal{M}(\rho)\|\mathcal{M}(\sigma)\right)|M\in\mathrm{POVM}(\mathcal{H},[d]),\,d\in\mathbb{N}}. \end{align}\] Note that here \(\mathcal{M}(\rho)\) and \(\mathcal{M}(\sigma)\) commute, and hence for any fixed \(\alpha\), all previously considered notions of Rényi \(\alpha\)-divergences coincide on them, and are equal to the classical Rényi \(\alpha\)-divergence [27] of \((\mathop{\mathrm{Tr}}M_i\rho)_{i\in[d]}\) and \((\mathop{\mathrm{Tr}}M_i\sigma)_{i\in[d]}\). Hence, we simply write \(D_{\alpha}\left(\mathcal{M}(\rho)\|\mathcal{M}(\sigma)\right)\) for this unique Rényi \(\alpha\)-divergence. It is well known and easy to verify that the measured Rényi divergences are quantum Rényi divergences in the sense explained at the beginning of Section 4.1.

According to [29] and the proof of [6], for any \(\rho,\omega\in\mathcal{B}(\mathcal{H})_{\gneq 0}\), \[\begin{align} \label{eq:pinching32bound} D_{\alpha,\alpha}(\rho\|\omega)-\kappa(\alpha)\log|\mathop{\mathrm{spec}}(\omega)| &\le D_{\alpha,\mathrm{meas}}(\rho\|\omega) \le D_{\alpha,\alpha}(\rho\|\omega),\alpha\in[1/2,+\infty), \end{align}\tag{69}\] where \(|\mathop{\mathrm{spec}}(\omega)|\) is the number of different eigenvalues of \(\omega\). From this it follows immediately that for a sequence of i.i.d. states \(\rho_n=\rho^{\otimes n}\), \(\sigma_n=\sigma^{\otimes n}\), \(n\in\mathbb{N}\), the regularized measured Rényi \(\alpha\)-divergence along this sequence is equal to the sandwiched Rényi \(\alpha\)-divergence of \(\rho\) and \(\sigma\) for any \(\alpha\in[1/2,+\infty)\); see [6], [29]. This is due to the fact that \(|\mathop{\mathrm{spec}}(\sigma^{\otimes})|\) grows only polynomially in \(n\). This latter property, however, need not hold anymore for \(|\mathop{\mathrm{spec}}(\widehat\omega_{Q_n})|\) for a quasi-free state \(\omega_Q\) of a fermion chain, which may grow exponentially in \(n\).

An alternative bound to circumvent this problem was given in [10], based on a technique of grouping the eigenvalues, introduced in [30]. Lemma 10 below is a variant and slight improvement of [10]. To state it, we follow [30] and introduce for any non-zero PSD operator \(\omega\in\mathcal{B}(\mathcal{H})_{\gneq 0}\) the quantity \[\begin{align} \Theta(\omega):=\min\left\{|\mathop{\mathrm{spec}}(\omega)|,2+\left\lceil \log\lambda_{\max}(\omega)-\log\lambda_{\min}(\omega)\right\rceil\right\}, \end{align}\] where \(\lambda_{\max}(\omega)=\norm{\omega}\) is the largest eigenvalue of \(\omega\), and \[\begin{align} \lambda_{\min}(\omega):=\max\set{\lambda>0|\lambda\omega^0\le\omega} \end{align}\] is the smallest non-zero eigenvalue of \(\omega\). For any \(\alpha\in[0,+\infty]\), let \[\begin{align} \kappa(\alpha):=\begin{cases} 1,&\alpha\in[1/2,2]\cup\{+\infty\},\\ \frac{\alpha}{\alpha-1},&\alpha>2. \end{cases} \end{align}\]

4.4 Regularized hockey stick Rényi divergences↩︎

In this section we consider the recently introduced integral Rényi divergences, or hockey-stick Rényi divergences [17], [18]. The definitions given in [17], [18] can easily be seen to be equivalent to \[\begin{align} D_{\alpha,\mathrm{hs}}(\rho\|\sigma):=\frac{1}{\alpha-1}\log Q_{\alpha,\mathrm{hs}}(\rho\|\sigma), \alpha\in(0,1)\cup(1,+\infty), \end{align}\] where for two states \(\rho,\sigma\in\mathcal{S}(\mathcal{H})\), \[\begin{align} \label{eq:hsq32def} Q_{\alpha,\mathrm{hs}}(\rho\|\sigma) &:= \alpha(\alpha-1)\left[\int_0^1\mathop{\mathrm{Tr}}(\rho-t\sigma)_-\,t^{\alpha-2}\,dt +\int_1^{+\infty}\mathop{\mathrm{Tr}}(\rho-t\sigma)_+\,t^{\alpha-2}\,dt\right]. \end{align}\tag{74}\] Here, \((\rho-t\sigma)_-\) is the negative part and \((\rho-t\sigma)_+\) is the positive part of the self-adjoint operator \(\rho-t\sigma\). These are quantum Rényi \(\alpha\)-divergences in the sense that for commuting operators \(\rho,\sigma\), \(D_{\alpha,\mathrm{hs}}(\rho\|\sigma)\) is the classical Rényi \(\alpha\)-divergence of the diagonal elements of the matrices of \(\rho\) and \(\sigma\) in any orthonormal basis diagonalizing both; see Section 4.1.

The following bounds have been shown in [18], [33], [34]. More precisely, the first inequality in 75 was shown in [33], the second inequality in 75 is Eq. (3.55) in [18], the second inequality in 76 is from [34], and the first inequality in 76 is due to the fact that the hockey stick Rényi divergences are manifestly monotone under positive trace-preserving maps, and the measured Rényi divergences are the smallest Rényi divergences that are monotone under positive trace-preserving maps, which can be easily verified from their definition.

5 Asymptotic discrimination of translation-invariant quasi-free states↩︎

5.1 Error exponents↩︎

Let us now turn to the problem of asymptotic discrimination of two quasifree states \(\omega_Q\) and \(\omega_R\) on \(\mathrm{CAR}\!\left(\ell^2_d(\mathbb{Z})\right)\) as explained in the Introduction. Recall that for a test \(T_n\in\mathcal{B}\left(\Gamma\!\left(\ell^2_d([n]^*)\right)\right)_{[0,1]}\), the corresponding type I and type II error probabilities are defined as \[\begin{align} \varepsilon_0(\omega_{Q_n}|T):=\mathop{\mathrm{Tr}}\widehat\omega_{Q_n}(I-T_n),\text{(type I)}, \varepsilon_1(\omega_{R_n}|T):=\mathop{\mathrm{Tr}}\widehat\omega_{R_n} T_n\text{(type II)}, \end{align}\] where \(Q_n:=(P_n\otimes I_d)^*Q(P_n\otimes I_d)\), \(R_n:=(P_n\otimes I_d)^*R(P_n\otimes I_d)\), as in 20 . We define the direct exponents corresponding to a fixed type II error exponent \(r\) as \[\begin{align} \overline{\mathrm{d}}_r(\omega_Q\|\omega_R) &:= \sup\left\{\limsup_{n\to+\infty}-\frac{1}{n}\log\varepsilon_0(\omega_{Q_n}|T_n)\,\Big|\, \liminf_{n\to+\infty}-\frac{1}{n}\log\varepsilon_1(\omega_{R_n}|T_n)>r \right\},\tag{77}\\ \underline{\mathrm{d}}_r(\omega_Q\|\omega_R) &:= \sup\left\{\liminf_{n\to+\infty}-\frac{1}{n}\log\varepsilon_0(\omega_{Q_n}|T_n)\,\Big|\, \liminf_{n\to+\infty}-\frac{1}{n}\log\varepsilon_1(\omega_{R_n}|T_n)>r \right\},\tag{78}\\ \mathrm{d}_r(\omega_Q\|\omega_R) &:= \sup\left\{\lim_{n\to+\infty}-\frac{1}{n}\log\varepsilon_0(\omega_{Q_n}|T_n)\,\Big|\, \liminf_{n\to+\infty}-\frac{1}{n}\log\varepsilon_1(\omega_{R_n}|T_n)>r \right\},\tag{79} \end{align}\] where in the last definition we only optimize over test sequence for which the indicated limit exist. Obviously, \[\begin{align} \mathrm{d}_r(\omega_Q\|\omega_R)\le\underline{\mathrm{d}}_r(\omega_Q\|\omega_R)\le\overline{\mathrm{d}}_r(\omega_Q\|\omega_R). \end{align}\] Similarly, the strong converse exponents corresponding to a fixed type II error exponent \(r\) are defined as \[\begin{align} \overline{\mathrm{sc}}_r(\omega_Q\|\omega_R) &:= \inf\left\{\limsup_{n\to+\infty}-\frac{1}{n}\log(1-\varepsilon_0(\omega_{Q_n}|T_n))\,\Big|\, \liminf_{n\to+\infty}-\frac{1}{n}\log\varepsilon_1(\omega_{R_n}|T_n)>r \right\},\tag{80}\\ \underline{\mathrm{sc}}_r(\omega_Q\|\omega_R) &:= \inf\left\{\liminf_{n\to+\infty}-\frac{1}{n}\log(1-\varepsilon_0(\omega_{Q_n}|T_n))\,\Big|\, \liminf_{n\to+\infty}-\frac{1}{n}\log\varepsilon_1(\omega_{R_n}|T_n)>r \right\},\tag{81}\\ \mathrm{sc}_r(\omega_Q\|\omega_R) &:= \inf\left\{\lim_{n\to+\infty}-\frac{1}{n}\log(1-\varepsilon_0(\omega_{Q_n}|T_n))\,\Big|\, \liminf_{n\to+\infty}-\frac{1}{n}\log\varepsilon_1(\omega_{R_n}|T_n)>r \right\},\tag{82} \end{align}\] and \[\begin{align} \underline{\mathrm{sc}}_r(\omega_Q\|\omega_R)\le\overline{\mathrm{sc}}_r(\omega_Q\|\omega_R)\le \mathrm{sc}_r(\omega_Q\|\omega_R) \end{align}\] holds trivially.

5.2 Super-exponential state discrimination↩︎

We say that two sequences of states \(\rho_n,\sigma_n\in\mathcal{S}(\mathcal{H}_n)\), \(n\in\mathbb{N}\), can be super-exponentially discriminated, if there exists a test sequence \(T_n\in\mathcal{B}(\mathcal{H}_n)_{[0,1]}\), \(n\in\mathbb{N}\), such that \[\begin{align} \label{eq:superexp32def} \lim_{n\to+\infty}\frac{1}{\log\operatorname{dim}\mathcal{H}_n}\log\varepsilon_0(\rho_n|T_n)=+\infty= \lim_{n\to+\infty}\frac{1}{\log\operatorname{dim}\mathcal{H}_n}\log\varepsilon_1(\sigma_n|T_n), \end{align}\tag{85}\] which is equivalent to the symmetric error \(\varepsilon_{\mathrm{mix},1}^{}(\rho_n\|\sigma_n):=\min\left\{\varepsilon_0(\rho|T)+\varepsilon_1(\sigma|T)\,|\,T\in\mathcal{B}(\mathcal{H})_{[0,1]}\right\}=(1-\norm{\rho_n-\sigma_n}_1/2)/2\) decaying with a super-exponential speed, i.e., \[\begin{align} \lim_{n\to+\infty}\frac{1}{\log\operatorname{dim}\mathcal{H}_n}\log\varepsilon_{\mathrm{mix},1}^{}(\rho_n\|\sigma_n)=+\infty. \end{align}\]

Below we show that for any quasi-free states \(\omega_Q\) and \(\omega_R\) on \(\mathrm{CAR}\!\left(\ell^2_d(\mathbb{Z})\right)\), their local restrictions \((\widehat\omega_{Q_n})_{n\in\mathbb{N}}\) and \((\widehat\omega_{R_n})_{n\in\mathbb{N}}\) can be super-exponentially discriminated. Equivalently, the quasi-free states can be super-exponentially discriminated by a sequence of tests \(T_n\) performed on length \(n\) portions of the whole chain for every \(n\in\mathbb{N}\). More formally, there exists a sequence of tests \(T_n\in\mathcal{B}\left(\ell^2([n]^*)\otimes\mathbb{C}^d\right)_{[0,1]}\), \(n\in\mathbb{N}\), such that \[\begin{align} \label{eq:qf32superexp32def} \lim_{n\to+\infty}\frac{1}{n}\log\mathop{\mathrm{Tr}}\widehat\omega_{Q_n}(I-T_n)=+\infty= \lim_{n\to+\infty}\frac{1}{n}\log\mathop{\mathrm{Tr}}\widehat\omega_{R_n}T_n. \end{align}\tag{86}\] This is equivalent to 85 with \(\rho_n:=\widehat\omega_{Q_n}\), \(\sigma_n:=\widehat\omega_{R_n}\) given on \(\mathcal{H}_n=\Gamma\!\left(\ell^2([n]^*)\otimes \mathbb{C}^d\right)\), since in this case, \(\log\operatorname{dim}\mathcal{H}_n=nd\log 2\).

Since quasi-free states are defined by single-particle symbol operators, it is not surprising that a sequence of tests with the above properties can be obtained from a sequence of operators on the single-particle Hilbert spaces \(\ell^2([n]^*)\otimes\mathbb{C}^d\). Indeed, by Lemma 3.2 and Corollary 3.3 of [19], we have the following:

Hence, our goal is to construct a sequence of projections \((E_n)_{n\in\mathbb{N}}\) such that the upper bounds in 87 decay super-exponentially fast for \(G_n=Q_n\) in the first, and \(G_n=R_n\) in the second inequality. To construct such a sequence of projections, we will use an idea from [19] based on discrete Fourier transform, and apply a suitable generalization of it to the \(d\) mode/site setting considered here.

Recall that the discrete Fourier transform \(F_n\) on \(\ell^2([n]^*)\) is defined by its action on the canonical basis of \(\ell^2([n]^*)\) as \[\begin{align} F_n 1_{\{k\}} := \frac{1}{\sqrt{n}} \sum_{j=0}^{n-1} e^{i \frac{2\pi}{n} j k} 1_{\{j\}}, k \in [n]^*. \end{align}\] For the matrix-valued symbol case, we define \(\mathcal{F}_{n} := F_n \otimes I_d\) acting on \(\operatorname{ran}\mathcal{P}_n= \ell^2([n]^*)\otimes\mathbb{C}^d\).

For convenience, we will identify the one-dimensional torus \(\mathbb{T}\) with \([-\pi,\pi)\) below instead of \([0,2\pi)\) as before, and consider the symbol functions of quasi-free states by first periodically extending them and then restricting to \([-\pi,\pi)\). The following is an extension of [19] from the case of a scalar to the case of a matrix-valued symbol function. To state it, we will need the definition of the \(n\)-th Fejér kernel \(\Phi_n\), given by \[\begin{align} \Phi_n(y):=\frac{1}{ n} \frac{\sin^2(ny/2)}{\sin^2(y/2)},y\in[-\pi,\pi). \end{align}\]

For Lemma 15 below, we will need some well-known properties of the Fejér kernel, which we summarize below for readers’ convenience.

For a self-adjoint operator \(A\), we will use the shorthand notation \[\begin{align} \{A>0\}:=\sum_{a>0}P^A_a \end{align}\] for the projection onto to the support of the positive part of \(A\).

Acknowledgments↩︎

This work was partially funded by the National Research, Development and Innovation Office of Hungary (NKFIH) via the research grants K 146380 and EXCELLENCE 151342, and by the Ministry of Culture and Innovation and the National Research, Development and Innovation Office within the Quantum Information National Laboratory of Hungary (Grant No. 2022-2.1.1-NL-2022-00004). MM was partially supported by the Ministry of Education, Singapore, through grant T2EP20124-0005. GMZ was partially supported by the QuantERA II project HQCC-101017733 (Grant No. 2019-2.1.7-ERA-NET-2022-00052). The authors are grateful to Zoltán Zimborás for discussions.

References↩︎