{'fraction_non_alphanumeric': 0.04191646682653877, 'fraction_numerical': 0.016936450839328536, 'mean_word_length': 4.015534953645703, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 9, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Invariance of separability probability over reduced states in 4x4 bipartite systems

The geometric separability probability of composite quantum systems is extensively studied in the last decades. One of most simple but strikingly difficult problem is to compute the separability probability of qubit-qubit and rebit-rebit quantum states with respect to the Hilbert-Schmidt measure. A lot of numerical simulations confirm the P(rebit-rebit)=29/64 and P(qubit-qubit)=8/33 conjectured probabilities. Milz and Strunz studied the separability probability with respect to given subsystems. They conjectured that the separability probability of qubit-qubit (and qubit-qutrit) depends on sum of single qubit subsystems (D), moreover it depends just on the Bloch radii (r) of D and it is constant in r. Using the Peres-Horodecki criterion for separability we give mathematical proof for the P(rebit-rebit)=29/64 probability and we present an integral formula for the complex case which hopefully will help to prove the P(qubit-qubit)=8/33 probability too. We prove Milz and Strunz's conjecture for rebit-rebit and qubit-qubit states. The case, when the state space is endowed with the volume form generated by the operator monotone function f(x)=sqrt(x) is studied too in detail. We show, that even in this setting the Milz and Strunz's conjecture holds and we give an integral formula for separability probability according to this measure.


Introduction
Since entanglement is one of the most striking features of composite quantum systems, it is natural to ask what the probability is that a given quantum state is entangled (or separable). "Is the world more classical or more quantum? Does it contain more quantum correlated (entangled) states than classically correlated ones?" These questions were addressed to physicists in 1998 by Zyczkowski, Horodecki, Sanpera and Lewenstein [6]. The first question is a rather philosophical one, the second is easier to formulate mathematically, although more specification is needed. It has turned out during the recent years that even in the simplest quantum case, when one considers only qubit-qubit states over real or complex Hilbert-space, to determine the separability probability of a given state is a highly nontrivial problem. Many researchers agree and emphasize the philosophical and experimental interest of the separability probability. First, one should specify a natural measure on the state space and then should compute somehow the volume of the separable states and the volume of the state space. In this paper we endow the state space with Hilbert-Schmidt measure which is induced by the Hilbert-Schmidt metric. We note here that other measures are also relevant, as it was pointed out by Slater [11], mainly those which are generated by monotone metrics [10]. The volume of the state space with respect to the Hilbert-Schmidt measure was computed by Zyczkowski [16] and Andai [1]. There are several good separability criteria, we use the Peres-Horodecki criterion [5] which is a simple necessary and sufficient condition for separability of qubit-qubit states. To compute the volume of separable states is a much more complicated task.
So far only extensive numerical studies and some related conjectures have existed for the separability probability. Numerical simulations give rise to an intriguing formula for separability probability, presented in 2013 by Slater [12], which was tested in real, complex and even in quaternionic Hilbert-spaces [14,13,4]. Based on this formula and on numerical simulations the separability probability for real qubit-qubit state is 29 64 and for complex state is 8 33 . Now we give mathematical proof for the 29 64 probability and we present an integral formula for the complex case which hopefully will help to prove the 8 33 probability, too. One of the most useful conjecture about separability probability was presented by Milz and Strunz in 2015 [7]. They conjectured that the separability probability of qubit-qubit (and qubit-qutrit) states of the form of D 1 C C * D 2 depends on D = D 1 + D 2 (on single qubit subsystems), moreover it depends only on the Bloch radii (r) of D and it is constant in r. In this paper we prove this conjecture for real and complex qubit-qubit states.
We study the case in detail when the state space is endowed with the volume form generated by the operator monotone function f (x) = √ x. We show that the volume of rebit-rebit and qubitqubit states are infinite, although there is a simple and reasonable method to define the separability probabilities. We present integral formulas for separability probabilities in this setting, too. We argue that from the separability probability point of view, the main difference between the Hilbert-Schmidt measure and the volume form generated by the operator monotone function x → √ x is a special distribution on the unit ball in operator norm of 2 × 2 matrices, more precisely in the Hilbert-Schmidt case one faces with a uniform distribution on the whole unit ball and for monotone volume forms one gets uniform distribution on the surface of the unit ball.
The paper is organized as follows. In Section 2, we fix the notations for further computations and we mention some elementary lemmas which will be used in the sequel. In Section 3, we present our main results, namely an explicit integral formula for the volume of separable qubit-qubit states over real and complex Hilbert-space, a proof for Milz and Strunz's conjecture [7] and an analytical proof for the rebit-rebit separability probability. In Section 4 we endow the state space with the volume measure which induced by the operator monotone function f (x) = √ x, and we show, that even in this setting the Milz and Strunz's conjecture holds and we give an integral formula for separability probability according to this measure. As a kind of checking, in Section 5, we compute the volume of the real and complex qubit-qubit state space with methods introduced in Section 3 and we compare our results to the previously published ones. In the second part of Section 5 we prove that the volume of the qubit-qubit state space is infinite if the volume measure comes from the function f (x) = √ x, but still there is a natural way to define the separability probability.

Basic lemmas and notations
The quantum mechanical state space consists of real and complex self-adjoint positive matrices with trace 1. We consider only the set of faithful states with real and complex entries. In our notation the state space is The space of n×n self-adjoint matrices is denoted by M sa n,K (K = R, C). Let us introduce the notation E K for the operator interval where "<" denotes the partial ordering of self-adjoint matrices defined by the cone of positive matrices.
The following lemma is an essential ingredient of the proof of our main theorem. It gives a characterization of positive definite matrices in terms of their Schur complement.
Proof. The statement is well-known in linear algebra. For the proof see for example p. 34 in [15].
Lemma 2. For every matrix V ∈ K n×n there exists a factorization, called a singular value decomposition of the form where U 1 , U 2 ∈ K n×n are unitary matrices and Σ ∈ K n×n is a diagonal matrix with real non-negative entries.
Proof. The proof can be found for example in Bathia's book (See p. 6 in [2]).
Lemma 3. Let X ∈ K n×n be an arbitrary matrix. The matrix X * X is positive semidefinite and the following equivalence holds X * X < I ⇔ X < 1, where · denotes the usual operator norm i.e. the largest singular value or Schatten-∞ norm.
Proof. The inequality v, X * Xv = Xv 2 ≥ 0 holds for all v ∈ K n which proves the first part of the statement. By the definition of operator norm, we can write To a matrix D ∈ K n , one can associate the left and right multiplication operators L D , R D : It is obvious that L D and R D are invertible if and only if D ∈ Gl(n, K). By a straightforward computation, one can show that det(L D ) = det(R D ) = det(D) n .
In integral transformations, the n × n complex matrix D is regarded as a 2n × 2n real matrix that acts on R 2n ∼ = C n therefore the Jacobian of L D and R D is det(D) 2n in the complex case.
The vector space of n × n matrices is the direct sum of the space of n × n self-adjoint matrices and n × n anti self-adjoint matrices For any self-adjoint matrix D ∈ M sa n,K , the map L D • R D : K n × K n preserves the direct sum decomposition i.e.
This observation lead us to the following lemma.
Proof. In the real case, M sa 2,R = R 0 −1 1 0 . One can verify that In the complex case, we have an isomorphism M sa 2,C = i M sa 2,C and thus which completes the proof.
Note that the Jacobian of the transformation is det(D) 2d−d 2 /2 because D is regarded in integral transformations as a 4 × 4 real matrix that acts on Lemma 5. Let A be a 2 × 2 invertible matrix with singular values σ 1 > σ 2 > 0. The operator norm and the singular value ratio of A, which is defined as σ(A) := σ 2 /σ 1 , can be expressed as follows where · HS denotes the Hilbert-Schmidt norm.
Proof. By definition, singular values of A are the eigenvalues of √ A * A that are which completes the proof.
The standard unit ball in the normed vector space of 2 × 2 matrices is denoted by B 1 K 2×2 and the notation ∂B 1 K 2×2 stands for the surface of the unit ball. We set the notation 1 A for the indicator function of the set A ⊆ K 2×2 .
Clearly, these functions are reciprocal symmetric i.e. (1) function would have a similar probabilistic meaning, but we will see that η d (1) = ∞ therefore we should find an other way to calculateη d (ε).
can be expressed as follows Proof. The proof, which is elementary but somewhat lengthy, can be found in Appendix A. The functionχ 1 (ε) can be written in a closed form using polylogarithmic functions but this is unnecessary for our purposes. It is somewhat interesting, that the identity function approximates wellχ 1 (ε) (See Fig. 2).
Recall that Pauli matrices σ 1 = 0 1 1 0 , σ 2 = 0 − i i 0 and σ 3 = 1 0 0 −1 with I = 1 0 0 1 form an orthogonal basis of the space of 2 × 2 self-adjoint matrices. We parametrize the spaces M sa 2,R and M sa 2,C in the following way This parametrization is very convenient because eigenvalues of R(θ, x, y) and R(θ, φ, x, y) can be directly read out from the parametrization. We introduce the notation O(φ) for the standard 2 × 2 rotation matrix that rotates points counterclockwise through an angle φ about the origin. Let us denote by Λ(x, y) the 2 × 2 diagonal matrix that contains x, y in its diagonal.
Polar decomposition will be utilized to parametrize the space of 2 × 2 complex matrices. The space of 2 × 2 real and complex density matrices will be parametrized by the canonical Bloch sphere parametrization as follows.
In Table 2, we summarize the normalization constants corresponding to χ d and η d d = 1, 2. As an example, we calculate here χ 2 (1). By definition, we can write Now we apply the parametrization and volume form presented in Table 2 and we obtain To make the explanation precise, we defineη d (ε) as which limit exists because the measures converge in weak- * topology to a measure concentrated on {(x, y) ∈ [0, 1] | x = 1 ∨ y = 1} as δ → 1−0. By the unitary symmetry, we can conclude that the measure converges in weak- * topology to the uniform distribution on ∂B 1 K 2×2 as δ → 1 − 0. The next lemma states that with this definition we get backχ 1 . We conjecture that this identity also holds true forχ 2 andη 2 .
Proof. The proof of this theorem is provided in Appendix B.

Definition 2. The polylogarithmic function is defined by the infinite sum
for arbitrary complex s and for all complex arguments z with |z| < 1.

The main result
We parametrize the space of 4 × 4 density matrices (D 4,K ) in the following way where D 1 , D 2 > 0, D 1 + D 2 ∈ D 2,K and C ∈ K 2×2 . Note that, with this parametrization For a given state D ∈ D 2,K we define that is the set of those states, which partial trace with respect to the system 2, respectively yield the matrix D ∈ D 2,K . Let us introduce the involution that is just the composition of partial transpose and element-wise conjugation which is a positive map. Consequently, the aforementioned Peres-Horodecki positive partial transpose criterion can be reformulated as Now we are in the position to state one of our main results.
and the volume of the space D s 4,K can be expressed as where d = dim R (K) = 1, 2.
Proof. For fixed D 1 , D 2 ∈ M sa 2,K , we set By Fubini's theorem, we have If D 1 , D 2 > 0 and Tr(D 1 + D 2 ) = 1 fixed, then a matrix C ∈ K 2×2 belongs to the set C(D 1 , D 2 ) if and only if D 1 C C * D 2 > 0 and D 1 C * C D 2 > 0 holds. This condition can be reformulated by Lemma 1. as where · denotes the usual operator norm.
To compute the inner integral, we substitute The Jacobian of this transform is det Observe that the last term depends only on the singular value ratio of V , because taking the singular value decomposition of V : V = U 1 ΣU 2 , we have and the transformation X → U 2 XU 1 is isometric with respect to the Hilbert-Schmidt norm. It means that holds. By Lemma 5, the singular value ratio of V is hence for the volume of separable states (21) we obtain We introduce the parametrization where D takes values in D 2,K and A runs on self-adjoint 2 × 2 matrices that satisfy the condition −D < A < D.
By the invariance of trace under cyclic permutations, the previous integral can be written as According to the remark after Lemma 4, the Jacobian of this transformation is which completes the proof.
The next Corollary proves Milz and Strunz's conjecture on the behavior of the conditioned volume over reduced states (See equation (23) in [7]).

Corollary 1. In the complex case, the conditioned volume can be expressed as
where K 1 , K 2 are constants and r is the radius of D in the Bloch sphere.
Proof. We set d = 2 in (19) and obtain the first equality. According to the parametrization of D 2,C (14), det(D) = 1 4 (1 − r 2 ) which proves the second equality. Corollary 2. If D ∈ D 2,K is a fixed density matrix, then the probability to find a separable state in D 4,K (D) can be written as where It is apparent that this probability is not depend on D that proves the conjecture of Milz and Strunz [7].
Proof. In a similar way, we can calculate the volume of the whole space and thus we have which completes the proof.
Using the fact that µ d+2 and σ Proof. According to Equation (23) and the above mentioned unitary invariance, we can write where the denominator is equal to 16 35 . In the numerator, we substitute u = 1−x 1+x , v = 1−y 1+y . The Jacobian of this transformation is 4 (1+u) 2 (1+v) 2 . Not that, the map z → 1−z 1+z is a monotone decreasing involution that maps (−1, 1) onto (0, ∞). After substitution, the numerator gains the following form We substitute again. Let u = ts and v = s t . The Jacobian of this substitution is 2s t and the domain of integration is 0 < s < ∞, 0 < t < 1. So, we have Now we integrate by parts in the inner integral and we get 1 0χ With this, the numerator can be written as where we have interchanged the order of integration in the last term. One can check that where we applied Lemma 6. Using this, one can conclude that 4 Generalization to D 4,K , g √ x The operator monotone function f : R + → R is said to be symmetric and normalized if f (x) = xf (x −1 ) holds for every positive argument x and f (1) = 1. The set of symmetric and normalized operator monotone functions plays an important role in quantum information geometry [9,3]. Petz's classification theorem states that there exists a bijective correspondence between the set of symmetric and normalized operator monotone functions and the family of monotone metrics [10]. The metric associated to the operator monotone function f is given by for all n ∈ N + , D ∈ D n,K and X, Y ∈ T D D n,K . The space of n × n density matrices endowed with the accompanying monotone metric of the operator monotone function f is denoted by (D n,K , g f ).
In this point, we generalize our results to the space D 4,K , g √ x . According to theorem 6 in [1], the volume form of (D n,K , g f ) can be expressed as where d = dim R K, µ i -s are the eigenvalues of D and c f (x, y) = A slight modification of the previous proofs gives the following Theorem.
Theorem 3. Let D ∈ D 2,K be a fixed density matrix. The volume of the submanifold D s 4,K (D), g √ x can be formally written as and the volume of the space D s 4,K , g √ x can be formally expressed as where d = dim R (K) = 1, 2.
Proof. By the factorization det(D) = det(D 1 D 2 ) det( where C(D 1 , D 2 ) is the same as in Theorem 1. Using the substitution , the inner integral can be written in the following form . By a similar argument, the last term depends only on the singular value ratio of V hence it can be written as η d • σ(V ). As a result, for the volume we get Using the parametrization (22) and the substitution Y = D −1/2 AD −1/2 , we obtain Corollary 3. For a fixed density matrix D ∈ D 2,K , g √ x the probability to find a separable state in where This probability is also independent from D which means that the conjecture of Milz and Strunz holds true for the statistical manifold D 4,K , g √ x .
Proof. Similarly, one can show that then we take the ratio x (D 4,K (D)) and we get the desired result. Now, we are in the position to calculate the separability probability for rebit-rebit systems in this setting.

Theorem 4. The separability probability in the statistical manifold
where K is the complete elliptic integral of the first kind and E is the elliptic integral of the second kind, that is Proof. Due to the fact thatη 1 =χ 1 (See Lemma 7 and Appendix B.) and by the unitary invariance, we can write where the denominator is equal to 2π 3 . To evaluate the numerator, we use the same strategy that we have applied in the Hilbert-Schmidt case. After the first substitution, the numerator gains the following form After the second substitution, we have We interchange the order of integration and obtain 1 0 that can be evaluate only numerically. For the separability probability, we have which completes the proof.

Examples
To verify the results, first we calculate the volume of 4 × 4 density matrices with respect to the standard Lebesgue measure. As we mentioned in the proof of Corollary 2, the volume of D 4,K can be expressed as which can be written in the following product form In the real case we have .
In the complex case we have If we put all together, we get which is equal to the volume obtained byŻyczkowski and Sommers [16] and Andai (See Theorem 1 and 2 in [1]) up to a factor that comes from the difference between the Lebesgue measure and the Hilbert-Schmidt measure. Contrary to the 2 × 2 case (See Corollary 1 in [1]), the volume of the statistical manifold D 4,K , g √ x is infinite in both of the real and complex cases because η d (1) = ∞ (See Table 2.) and the volume admits the following factorization

Conclusion
The structure of the unit ball in operator norm of 2 × 2 matrices plays a critical role in separability probability of qubit-qubit and rebit-rebit quantum systems. It is quite surprising that the space of 2 × 2 real or complex matrices seems simple, but to compute the volume of the set for a given parameter ε ∈ [0, 1], which is the value of the function χ d (ε), is a very challenging problem. The gist of our considerations is that the behavior of the function χ d (ε) determines the separability probabilities with respect to the Hilbert-Schmidt measure. When the volume form generated by the operator monotone function x → √ x, a reasonable normalization can be given to define the separability probability and in this case the probability is determined by the structure of the surface of the unit ball.
Although the metric tensor has a complicated form, the volume form is quite simple det(g(r, t, ρ, φ)) = r 3 .
We can write Note that Y ± (t, ρ, φ) ∈ SL 2 (R) and by Lemma 5, we have which means